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A refinement of Heath-Brown's theorem on quadratic forms S. G. Vlăduţab, A. V. Dymovcde, S. B. Kuksinfgc, A. Maiocchih a Aix-Marseille Université, CNRS, I2M UMR 7373, Marseille, France b Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow, Russia c Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia d National Research University Higher School of Economics, Moscow, Russia e Skolkovo Institute of Science and Technology, Moscow, Russia f Université Paris Cité, Sorbonne Université, CNRS, IMJ-PRG, Paris, France g Peoples' Friendship University of Russia, Moscow, Russia h Università degli Studi di Milano-Bicocca, Milano, Italy
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     § 1. Introduction1.1. The setting and resultConsider a nondegenerate quadratic form with integer coefficients on $\mathbb{R}^d$, $d\geqslant 4$, which implies that $A$ can be chosen as a nondegenerate symmetric matrix with integer elements whose diagonal elements are even. If $F$ is sign-definite, then for $t\in \mathbb{R}$ the quadric
$$
\begin{equation}
\Sigma_t=\{\mathbf z\colon F^t(\mathbf z)=0\}, \qquad F^t=F-t,
\end{equation}
\tag{1.2}
$$
is either an ellipsoid or an empty set, while in the non-sign-definite case $\Sigma_t$ is an unbounded hypersurface in $\mathbb{R}^d$, which is smooth if $t\ne0$, while $\Sigma_0$ is a cone and has a singular point at zero. Let $\mathbb{Z}^d_L$ be the lattice of small period $L^{-1}$,
$$
\begin{equation*}
\mathbb Z^d_L:=L^{-1} \mathbb Z^d, \qquad L\geqslant 1,
\end{equation*}
\notag
$$
and let $w$ be a regular real function on $\mathbb{R}^d$, which means that $w$ and its Fourier transform $\widehat w(\xi)$ are continuous functions decaying sufficiently rapidly at infinity:
$$
\begin{equation}
|w(\mathbf z)| \leqslant C |\mathbf z|^{-d-\gamma} \quad\text{and}\quad |\widehat w( \mathbf \xi)| \leqslant C |\xi|^{-d-\gamma}
\end{equation}
\tag{1.3}
$$
for some $\gamma$, $C>0$. Our aim is to study the behaviour of the series
$$
\begin{equation*}
N_L(w; A, m):=\sum_{\mathbf z \in \Sigma_m \cap \mathbb Z^d_L} w(\mathbf z),
\end{equation*}
\notag
$$
where $m\in \mathbb{R}$ is such that ${L^2m}$ is an integer.1 Let Then it is obvious that We also write
$$
\begin{equation*}
N_L(w;A):=N_L(w;A,0)\quad\text{and}\quad N(w_L;A):=N(w_L;A,0).
\end{equation*}
\notag
$$
To investigate $N_L(w;A,m)$ we follow closely the circle method in the form given to it by Heath-Brown in [1]. Our notation differs a bit from that in [1]. Namely, under the scaling $z=z'/L$, $z'\in \mathbb{Z}^d$, we count (with weights) solutions of the equation $F(z') = mL^2$, $z'\in \mathbb{Z}^d$, while Heath-Brown writes the equation as $F(z') = m$, $z'\in \mathbb{Z}^d$, so that his $m$ corresponds to our $L^2m$. We start with a key theorem, which expresses the analogue of the Dirac delta function on the integers, that is, the function $\delta\colon\mathbb{Z}\to\mathbb{R}$ such that
$$
\begin{equation*}
\delta(n):=\begin{cases} 1 & \text{for } n=0, \\ 0 & \text{for }n\neq 0, \end{cases}
\end{equation*}
\notag
$$
in terms of a sort of Fourier representation. This result goes back at least to Duke, Friedlander and Iwaniec [2] (also see [3]), and we state it in the form given to it in [1], Theorem 1; basically, it replaces the trivial identity employed in the usual circle method. In the theorem below, given $q\in \mathbb{N}$, we denote by $e_q$ the exponential function $e_q(x):=e^{2\pi i x/q}$ and by $\sum_{a(\operatorname{mod}q)}^*$ the sum over the residues $a$ such that $(a,q)=1$, that is, over all integers $a\in [1, q-1]$ that are relatively prime with $q$. Theorem 1.1. For any $Q\geqslant1$ there exists $c_Q>0$ and a smooth function $h(x,y)$: $\mathbb{R}_{>0}\times \mathbb{R}\to \mathbb{R}$ such that Since for any function $\widetilde w$ on $\mathbb{R}^d$ the quantity $N(\widetilde w;A,t)$ can be written as $\sum_{\mathbf{z}\in\mathbb{Z}^d} \widetilde w(\mathbf{z}) \delta (F^t(\mathbf{z}))$, Theorem 1.1 allows us to represent the series $N(\widetilde w;A,t)$ as an iterated sum. Transforming this sum further using Poisson’s summation formula as in [1], Theorem 2, we arrive at the following result.2 Theorem 1.2 (Theorem 2 in [1]). For any regular function $\widetilde w$, any $t$ and any ${Q\!\geqslant\! 1}$, We use Theorem 1.2 to examine the sum $N(w_L;A,L^2 m)= N_L(w;A,m)$ for large $L$, by choosing $\widetilde w=w_L$, $t=L^2m$ and $Q=L\geqslant1$ and estimating explicitly the leading terms with respect to $L$ of the sums $S_q(\mathbf{c})$ and $I^0_q(\mathbf{c})$, as well as the remainders. The answer will be given in terms of the integral
$$
\begin{equation}
\sigma_{\infty}(w)=\sigma_{\infty}(w; A,t)=\int_{\Sigma_t} w(\mathbf z)\mu^{\Sigma_t}\,(d\mathbf z)
\end{equation}
\tag{1.9}
$$
(which is singular if $t=0$). Here
$$
\begin{equation*}
\mu^{\Sigma_t}(d\mathbf z)=|\nabla F(\mathbf z) |^{-1}\,dz|_{\Sigma_t}= |A\mathbf z|^{-1}\,dz|_{\Sigma_t},
\end{equation*}
\notag
$$
where $dz|_{\Sigma_t}$ represents the volume element on $\Sigma_t$ induced by the standard Euclidean structure on $\mathbb{R}^{d}$, and $A$ is the symmetric matrix in (1.1). For regular functions $w$ this integral converges (see § 7). To write down the asymptotic for $N_L(w;A,m)$ we will need the following quantities, where $p$ ranges over all prime numbers and $c\in\mathbb{Z}^d$:
$$
\begin{equation}
\sigma_p^\mathbf c=\sigma_p^\mathbf c(A,{L^2m}):=\sum_{l=0}^\infty p^{-dl}S_{p^l}(\mathbf c; A,{L^2m}), \qquad \sigma_p:=\sigma_p^{\mathbf 0},
\end{equation}
\tag{1.10}
$$
where $S_1\equiv 1$,
$$
\begin{equation*}
\sigma^*_\mathbf c(A):=\prod_p (1-p^{-1})\sigma_p^\mathbf c(A,0), \qquad \sigma^*(A):=\sigma^*_{\mathbf 0}(A)=\prod_p(1-p^{-1}) \sigma_p(A,0),
\end{equation*}
\notag
$$
and
$$
\begin{equation}
\sigma(A,{L^2m})=\prod_p\sigma_p^{\mathbf 0}(A,{L^2m})= \prod_p\sigma_p(A,{L^2m}).
\end{equation}
\tag{1.11}
$$
The products in the above formulae are taken over all prime numbers. In the asymptotics, where these quantities are used, they are bounded uniformly in $L$ (see Theorems 1.3 and 1.4, as well as Proposition 1.6). Throughout, for a function $f\in C^k(\mathbb{R}^d)$ we set
$$
\begin{equation*}
\| f\|_{n_1, n_2}=\sup_{\mathbf z\in \mathbb R^d} \max_{|\alpha|_1\leqslant n_1} |\partial^\alpha f(\mathbf z)| \langle \mathbf z\rangle^{n_2},
\end{equation*}
\notag
$$
where $n_1 \in \mathbb{N}\cup\{0\}$, $n_1\leqslant k$ and $n_2\in \mathbb{R}$. Here
$$
\begin{equation*}
\langle \mathbf x\rangle:=\max\{1,|\mathbf x|\} \quad \text{for } \mathbf x\in \mathbb R^l, \ l\in \mathbb N,
\end{equation*}
\notag
$$
and $|\alpha|_1 \equiv \sum\alpha_j$ for any integer vector $\alpha \in (\mathbb{N}\cup\{0\})^d$. We let $\mathcal{C}^{n_1,n_2}(\mathbb{R}^d)$ denote the linear space of $C^{n_1}$-smooth functions $f\colon\mathbb{R}^d\to \mathbb{R}$, satisfying $\|f\|_{n_1,n_2}<\infty$. Note that if $w\in \mathcal{C}^{d+1,d+1}(\mathbb{R}^d)$, then the function $w$ is regular, so Theorem 1.2 applies. Indeed, the first relation in (1.3) is obvious. To prove the second note that for any integer vector $\alpha \in (\mathbb{N}\cup\{0\})^d$,
$$
\begin{equation*}
\xi^\alpha \widehat w(\xi)=\biggl( \frac{i}{2\pi}\biggr)^{|\alpha|_1} \widehat{\partial_\mathbf x^\alpha w}(\xi).
\end{equation*}
\notag
$$
But if $|\alpha|_1 \leqslant d+1$, then $|\partial_\mathbf{x}^\alpha w| \leqslant C \langle \mathbf{x}\rangle^{-d-1}$, so $\partial_\mathbf{x}^\alpha w$ is an $L_1$-function. Thus its Fourier transform $\widehat{\partial_\mathbf{x}^\alpha w}$ is a bounded continuous function for each $|\alpha|_1\leqslant d+1$ and the second relation in (1.3) also holds. Now we formulate our main results. First we treat the case $d\geqslant 5$. Theorem 1.3. Assume that $d\geqslant5$. Then for any $\varepsilon$, $0<\varepsilon\leqslant1$, there exist positive constants $K_1(d,\varepsilon)$, $K_2(d,\varepsilon)$ and $K_3(d,\varepsilon)$, where $K_2(d,\varepsilon)\leqslant K_3(d,\varepsilon)$, such that if $w \in \mathcal{C}^{K_1,K_2}(\mathbb{R}^{d})\cap \mathcal{C}^{0,K_3}(\mathbb{R}^{d})$ and a real number $m$ satisfies ${L^2m} \in \mathbb{Z}$, then where the constant $C$ depends on $d$, $\varepsilon$, $m$ and $A$. The constant $\sigma(A,L^2m)$ is bounded uniformly in $L$ and $m$. In particular, if $\varepsilon=1/2$, then one can take $K_1= 2d(d^2+ d- 1)$, $K_2=4(d+1)^2+3d+1$ and $K_3=K_1+3d+4$. Next we consider the case $d=4$, limiting ourselves to the situation when ${m=0}$. Theorem 1.4. Assume that $d=4$ and $m=0$. Then for any $0<\varepsilon < 1/5$ there exist positive constants $K_1(\varepsilon)$ and $K_2(\varepsilon)$ such that for $w \in \mathcal{C}^{K_1,K_2}(\mathbb{R}^d)$ If $\eta(0) \sigma^*(A) =0$, then the asymptotic (1.13) degenerates. In a similar way (1.12) also degenerates to an upper bound for $N_L$, unless we know that $\sigma(A,L^2m)$ admits a suitable positive lower bound, for all $L$. Luckily enough, the required lower bounds often exist; see Proposition 1.6 below. Theorems 1.3 and 1.4 refine Theorems 5–7 from [1] in three respects:
These improvements are crucial for us since in our work [6], dedicated to the problem of wave turbulence, the two theorems above are used in the situation when $w(0) \ne0$ and the support of $w$ is not compact. A similar specification of the Heath-Brown method was obtained in [7], § 5, to study an averaging problem related to the questions considered in [6]. Apart from wave turbulence and averaging, the replacement of sums over integer points of a quadric by integrals, with careful estimates for the remainders, is needed in the Kolmogorov-Arnold-Moser theory for partial differential equations (see (C.2) in [8], for example). The publications [6]–[8] are recent. We are certain that these days, when people working in PDEs and dynamical systems treat complicated nonlinear phenomena with resonances more and more often, there will be an increasing demand for the asymptotics (1.12) and (1.13) and their variations. Our paper uses only basic results from number theory and is well accessible to readers from Analysis. We note that the papers [9] and [10] treat the sums $N_L(w; A, m)$ for even and odd dimensions $d$, respectively, without the restriction that $w(0)\ne 0$, and in a more general context than our Theorems 1.3 and 1.4 do. However, because of this generality, the corresponding constants in the asymptotic (in $L$) formulae in [9] and [10] are very implicit (for example, the question of whether or not they vanish is highly nontrivial). The connection of the constants with singular integrals like (1.9) and the dependence of the remainders in asymptotic formulae on the weight function $w$, which is crucial for application to analysis, is not clear. Another feature of [9] and [10] is the use of rather advanced adelic technique, which makes it difficult for readers without serious number-theoretic background to use the result and method of that work. Remark 1.5. 1) Theorem 1.3 is a refinement of Theorem 5 in [1], while Theorem 1.4 refines Theorems 6 and 7 in [1]. Also, [1] provides some asymptotic (in $L$) information about the behaviour of the sums $N_L(w;A,m)$ for $d=4$, $m\neq 0$ and $d=3$, $m=0$. Since our proof of Theorems 1.3 and 1.4 is based on ideas from [1], strengthened by Theorem 7.3, which is valid for $d\geqslant3$, our approach most likely allows one to generalise the above-mentioned results of [1] for $d=3,4$ to the case when $w\in \mathcal{C}^{K_1,K_2}(\mathbb{R}^d)$ for suitable $K_1$ and $K_2$. 2) In our work the dependence of constants in estimates on $m$ is uniform on compact intervals, while the dependence on the operator $A$ is only via the norms of $A$ and $A^{-1}$. 3) The values of the constants $K_j(d, \varepsilon)$ in (1.12) that are provided by Theorem 1.3 are far from optimal since it was not our goal to optimise them. 4) As the proofs of the theorems are based on the representation (1.6), the function $w$ must be regular (see (1.3)). But this holds true if $w\in\mathcal{C}^{d+1,d+1}$, and so the proof is valid if the constants $K_1$ and $K_2$ are sufficiently large (for example, if $K_1$ and $K_2$ are as large as in the last line of the statement of Theorem 1.3). 1.1.1. A brief discussion of the proofsWe present in full only the proof of Theorem 1.3, which resembles that of Theorem 5 in [1], with an additional control of how the constants depend on $w$. A significant difference from the argument of Heath-Brown shows up in §§ 3 and 4, where we do not assume that the function $w$ vanishes near the origin, the last assumption being crucial for the analysis of integrals in §§ 6 and 7 of [1]. To cope with this difficulty, which becomes apparent, for example, in Proposition 3.8 below, we have to examine the smoothness of the function at zero and its decay at infinity. The corresponding analysis is performed in § 7. There, using the techniques developed in [11] to study integrals in (1.9), we prove that the function (1.15) is $(\lceil d/2\rceil-2)$-smooth but, in general, for even $d$ its derivative of order $(d/2-1)$ can have a logarithmic singularity at zero. We also estimate there the rate of decay of the function (1.15) at infinity.The proof of Theorem 1.4 resembles that of Theorems 6 and 7 in [1], with a new addition given by Proposition 3.8, which is based on a result from § 7. We thus limit ourselves to a sketch of the proof of this theorem, which is presented in § 1.3 in parallel to that of Theorem 1.3, and point out the main differences between the two proofs. In establishing Theorem 1.4 we use certain results from [1] (namely, Lemmas 30 and 31) without proof. 1.1.2. Lower bounds for the constants in the asymptoticsLet us now discuss lower bounds for the constants $\sigma(A, L^2m)$ and $\sigma^*(A)$ from Theorems 1.3 and 1.4 (see Theorems 4, 6 and 7 in [1]). Proposition 1.6. (i) If $d\geqslant 5$ then there exist positive constants $c(A)<C(A)$ such that $0<c(A)\leqslant \sigma(A, L^2m)\leqslant C(A)<\infty $ for any nonsingular matrix $A$, uniformly in $L$ and $m$. (ii) If $d=4$ and $m=0$, then $\sigma^*(A)>0 $ for any nonsingular matrix $A$ such that the corresponding equation $2F(\mathbf z)=A \mathbf z\cdot \mathbf z=0$ has nontrivial solutions in every $p$-adic field (in particular this holds if the equation has a nontrivial solution in $\mathbb{Z}^4$). We do not prove this result, but just note that its demonstration uses a refinement of the calculation in the second part of the proof of Lemma 2.3. Namely, while the lemma gives an upper bound for the required quantity, a more thorough analysis also allows us to establish the claimed lower bounds. In § 8.2 we give essentially a complete calculation, by proving Proposition 1.6 in the case of the simplest quadratic form $F=\Sigma_{i=1}^{d/2} x_iy_i$, where $d=2s\geqslant 4$, and $m=0$. A proof of the proposition for an arbitrary $A$ can follow the same lines, by replacing explicit formulae by some general results (Hensel’s lemma, for example). 1.1.3. Nonhomogeneous quadratic polynomialsNow consider a nonhomogeneous quadratic polynomial $\mathcal{F}$ with the second-order part equal to $F$ in (1.1):
$$
\begin{equation*}
\mathcal F(\mathbf z)=\frac12 A\mathbf z\cdot \mathbf z +\mathbf z_* \cdot\mathbf z +\tau, \qquad \mathbf z_*\in \mathbb R^d, \quad \tau \in \mathbb R,
\end{equation*}
\notag
$$
and consider the corresponding set $\Sigma^\mathcal{F} =\{\mathbf{z}\colon \mathcal{F}(\mathbf{z})=0\}$,
$$
\begin{equation*}
N_L(w; \mathcal F)=\sum_{\mathbf z \in \Sigma^\mathcal F\cap \mathbb Z^d_L} w(\mathbf z).
\end{equation*}
\notag
$$
Set
$$
\begin{equation*}
\mathfrak z=A^{-1}\mathbf z_*, \qquad \mathbf z'=\mathbf z+\mathfrak z \quad\text{and}\quad m=\frac12 \mathfrak z\cdot A\mathfrak z- \tau
\end{equation*}
\notag
$$
and assume4 that $\mathfrak{z}\in \mathbb{Z}^d_L$ and $ L^2 \tau \in\mathbb{Z}$. Then $L^2m\in\mathbb{Z}$ and $\mathbf{z}'\in\mathbb{Z}^d_L$ if and only if $\mathbf{z} \in\mathbb{Z}^d_L$, and $\mathcal{F}(\mathbf{z}) = F(\mathbf{z}')-m$. So setting $w^\mathfrak{z}(\mathbf{z}') = w(\mathbf{z}' -\mathfrak{z})$ we have $N_L(w; \mathcal{F}) = N_L(w^\mathfrak{z};A,m)$. Since
$$
\begin{equation*}
\sigma_\infty(w^\mathfrak z; A,m)=\int_{\Sigma_m} w^\mathfrak z(\mathbf z') \, \frac{d\mathbf z'|_{\Sigma_m}}{|\nabla F(\mathbf z')|} =\int_{\Sigma^\mathcal F} w(\mathbf z)\, \frac{d\mathbf z|_{\Sigma^\mathcal F}}{|\nabla \mathcal F(\mathbf z)|}=: \sigma_\infty(w; \mathcal F),
\end{equation*}
\notag
$$
we arrive at the following corollary to Theorem 1.3. Corollary 1.7. If $d\geqslant5$, the quadratic form $F$ is as in Theorem 1.3, $\mathcal{F}$ is a nonhomogeneous quadratic form as above and $L$ is such that $\mathfrak{z}:= A^{-1} \mathbf{z}_* \in \mathbb{Z}^d_L$ and $\tau L^2\in\mathbb{Z}$, then for any $0<\varepsilon\leqslant1$ and $w \in \mathcal{C}^{K_1,K_2}(\mathbb{R}^{d})\cap \mathcal{C}^{0,K_3}(\mathbb{R}^{d})$ we have Here the constants $K_1$, $K_2$ and $K_3$ depend on $d$ and $\varepsilon$, while $C$ depends on $d$, $\varepsilon$, $A$, $\tau$ and $|\mathbf{z}_*|$. Notation and agreements 1.8. We write $A \lesssim_{a,b} B$ if $A\leqslant C B$, where the constant $C$ depends on $a$ and $b$. In a similar way $O_{a,b}(\|w\|_{m_1, m_2}) $ denotes a quantity bounded by $C(a,b) \|w\|_{m_1, m_2}$ in absolute value. We do not indicate the dependence on the matrix norms $\|A\|$ and $\|A^{-1}\|$ or on the dimension $d$ since most of our estimates depend on these quantities. We always assume that the function $w$ belongs to the space $\mathcal{C}^{m,n}(\mathbb{R}^d)$ for sufficiently large $m$ and $n$. If in the statement of an assertion we employ the norm $\|w\|_{a,b}$ then we assume that $w\in\mathcal{C}^{a,b}(\mathbb{R}^d)$. We set $e_q(x) = e^{2\pi ix/q}$ and abbreviate $e_1(x)=:e(x)$. We let $\lceil\,{\cdot}\,\rceil$ denote the ceiling function $\lceil x \rceil =\min_{n\in\mathbb{Z}}\{ n\geqslant x\}.$ We denote the set of positive integers by $\mathbb{N}$. 1.2. The scheme of the proof of Theorem 1.3Let $d\geqslant5$. As already discussed, if $w$ satisfies the assumptions of the theorem for sufficiently large constants $K_i$ then $w$ is regular in the sense of § 1.1, so Theorem 1.2 applies. Then, according to (1.6) and (1.4),
$$
\begin{equation}
N_L(w; A, m)=c_LL^{-2}\sum_{\mathbf c\in \mathbb Z^{d}}\sum_{q=1}^\infty q^{-d}S_q(\mathbf c) I_q(\mathbf c),
\end{equation}
\tag{1.16}
$$
where the sum $S_q(\mathbf{c})=S_q(\mathbf{c};A,L^2m)$ is given by (1.7) for $t={L^2m}$ and the integral $I_q(\mathbf{c})$ is given by (1.8) for $\widetilde w=w_L$, $Q=L$ and $t={L^2m}$,
$$
\begin{equation}
I_q(\mathbf c; A,m,L) :=\int_{\mathbb R^{d}} w\biggl(\frac{\mathbf z}{L}\biggr)h\biggl(\frac qL, \frac{F^{L^2m}(\mathbf z)}{L^2}\biggr) e_q(-\mathbf z\cdot \mathbf c)\,d\mathbf z.
\end{equation}
\tag{1.17}
$$
Setting
$$
\begin{equation*}
n(\mathbf c;A,m,L)=\sum_{q=1}^\infty q^{-d} S_q(\mathbf c) I_q(\mathbf c),
\end{equation*}
\notag
$$
we have
$$
\begin{equation*}
N_L(w;A,m)=c_L L^{-2}\sum_{\mathbf c\in\mathbb Z^{d}} n(\mathbf c).
\end{equation*}
\notag
$$
Then for any $\gamma_1 \in (0,1/2)$ we write $N_L$ as
$$
\begin{equation}
N_L(w;A, m)=c_L L^{-2}\bigl(J_0 + J_{<}^{\gamma_1} + J_{>}^{\gamma_1}\bigr),
\end{equation}
\tag{1.18}
$$
where
$$
\begin{equation}
J_0:=n(0), \qquad J_<^{\gamma_1}:=\sum_{\mathbf c\ne 0,\,|\mathbf c|\leqslant L^{\gamma_1}} n(\mathbf c) \quad\text{and}\quad J_>^{\gamma_1}:=\sum_{|\mathbf c|> L^{\gamma_1}} n(\mathbf c).
\end{equation}
\tag{1.19}
$$
Proposition 5.1 (which is a modification of Lemmas 19 and 25 in [1]) implies that
$$
\begin{equation*}
|J_>^{\gamma_1}|\lesssim_{\gamma_1,m} \|w\|_{N_0,2N_0+d+1},
\end{equation*}
\notag
$$
where $N_0:=\lceil {d+(d+1)/{\gamma_1}} \rceil$ (see Corollary 5.2). In Proposition 6.1, following Lemmas 22 and 28 in [1], we show that
$$
\begin{equation}
|J_<^{\gamma_1}|\lesssim_{\gamma_1,m} L^{d/2+2+\gamma_1(d+1)} \bigl(\|w\|_{\overline N,d+5}+\|w\|_{0,\overline N+3d+4}\bigr) ,
\end{equation}
\tag{1.20}
$$
where $\overline N= \lceil d^2/{\gamma_1}\rceil-2d$. To analyse $J_0$ we write it as $J_0=J_0^++J_0^-$, where
$$
\begin{equation}
J^+_0:=\sum_{q>\rho L} q^{-d}S_q(0) I_q(0)\quad\text{and} \quad J_0^-:=\sum_{q\leqslant \rho L} q^{-d} S_q(0) I_q(0);
\end{equation}
\tag{1.21}
$$
here $\rho = L^{-\gamma_2}$ for some $\gamma_2$, $0<\gamma_2<1$, to be determined. Lemma 4.2, which is a combination of Lemmas 16 and 25 in [1], modified using the results in § 7, implies that
$$
\begin{equation*}
|J_0^+|\lesssim L^{d/2+2+\gamma_2(d/2-1)}|w|_{L_1} \lesssim L^{d/2+2+\gamma_2(d/2-1)}\|w\|_{0,d+1} .
\end{equation*}
\notag
$$
Finally Lemma 4.3, which is a combination of Lemma 13 and simplified Lemma 31 in [1] with results from § 7, establishes that $J_0^- $ equals
$$
\begin{equation*}
L^{d} \sigma_\infty(w) \sigma(A,{L^2m}) + O_{\gamma_2,m}\bigl( (\|w\|_{d/2-2,d-1} +\|w\|_{0,d+1}) L^{d/2+2+\gamma_2(d/2-2)}\bigr)
\end{equation*}
\notag
$$
(see (1.9) and (1.11)). Identity (1.18), together with the above estimates implies the required result if we choose $\gamma_2=\varepsilon/{(d/2-1)}$ and $\gamma_1= \varepsilon/(d+1)$. The uniform boundedness of the product $\sigma(A,L^2m)$ in $L$ and $m$ follows from Lemma 2.3. 1.3. The scheme of the proof of Theorem 1.4In this section we assume that $d=4$ and $m=0$. The proof proceeds exactly as in § 1.2 up to formula (1.20), which is not sharp enough for $d=4$ and should be replaced by
$$
\begin{equation}
\biggl|J_<^{\gamma_1}- L^d\sum_{\mathbf c\neq 0}\eta(\mathbf c){\sigma^*_\mathbf c(A) \sigma_{\infty}^\mathbf c}(w;A,L)\biggr|\lesssim_{\gamma_1} L^{7/2+(d+4)\gamma_1}\|w\|_{\widetilde K_1,\widetilde K_2}
\end{equation}
\tag{1.22}
$$
for appropriate constants $\widetilde K_1$ and $\widetilde K_2$, where the terms $\sigma^*_\mathbf{c}(A)$ are introduced in (1.10), the terms ${\sigma_{\infty}^\mathbf{c}}(w;A)$ are given by
$$
\begin{equation}
{\sigma_{\infty}^\mathbf c}(w;A,L):=L^{-d} \sum_{q=1}^\infty q^{-1}I_q(\mathbf c;A,0,L) ,
\end{equation}
\tag{1.23}
$$
and the constants $\eta(\mathbf{c})=\pm 1$ are defined in Lemma 8.1. In particular, $\eta(0)=1$ if the determinant $\det A$ is the square of an integer and $\eta(0)=0$ otherwise. The proof of the bound (1.22) makes use of Lemma 8.1 (Lemma 30 in [1]), involving only minor modifications of the argument in [1], and is left to the reader. The bound on $J_0$ must be refined too, and this is done in § 8.1. We consider only the case when the determinant $\det A$ is the square of an integer, so, in particular, $\eta(0)=1$. The opposite case can be obtained by a minor modification of this proof, in which we follow [1] (see § 8.1 for a discussion). In Proposition 8.3, which is a combination of Lemmas 13, 16 and 31 in [1] modified using Proposition 3.8, we prove that in the case of a perfect-square determinant $\det A$
$$
\begin{equation*}
\begin{aligned} \, J_0&=\sigma_\infty(w)\sigma^*(A)L^d\log L + K(0)L^d+ O_{\varepsilon}\bigl(L^{d-\varepsilon} \bigl(\|w\|_{d/2-2,d-1}+\|w\|_{0,d+1}\bigr)\bigr) , \end{aligned}
\end{equation*}
\notag
$$
where the constant $K(0)=K(0;w,A)$ is defined in § 8.1.1. Again, identity (1.18), together with the above estimates, implies the required result if we choose $\gamma_1=(1/2-\varepsilon)/(d+4)$ and put
$$
\begin{equation}
\sigma_1(w;A,L):=K(0)+ \sum_{\mathbf c\neq 0}\eta(\mathbf c) \sigma^*_\mathbf c(A) \sigma_{\infty}^\mathbf c(w;A,L).
\end{equation}
\tag{1.24}
$$
Finiteness of the products $\sigma^*_\mathbf{c}(A)$ follows from Lemma 8.2, while the estimate for the constant $\sigma_1(w;A,L)$ claimed in the theorem is established in § 8.1.3.
§ 2. Series $S_q$Now we begin the proof of Theorem 1.3 by following the scheme presented in § 1.2. Part of the assertions forming the proof do not use that $d\geqslant5$. So in all assertions below involving the dimension $d$ we indicate the actual requirements for $d$. We recall that the constants in estimates can depend on $d$ and $A$, but this dependence is not indicated (see section Notation and agreements 1.8). In the present section we analyse the sums entering, in particular, the definitions of the singular series $\sigma(A,{L^2m})$ and $\sigma_p(A,{L^2m})$.Lemma 2.1 (Lemma 25 in [1]). For any $d\geqslant 1$ uniformly in $\mathbf{c}\in\mathbb{Z}^{d}$. Proof. According to (1.7), an application of the Cauchy-Schwarz inequality shows that
$$
\begin{equation}
\begin{aligned} \, \notag |S_q(\mathbf c)|^2 &\leqslant \phi(q) \mathop{{\sum}^*}_{a(\operatorname{mod}q)} \biggl|\sum_{\mathbf b(\operatorname{mod} q)} e_q(aF^{L^2m}(\mathbf b) + \mathbf c\cdot \mathbf b) \biggr|^2 \\ &=\phi(q) \mathop{{\sum}^*}_{a(\operatorname{mod} q)} \sum_{\mathbf u,\mathbf v(\operatorname{mod}q)} e_q\bigl(a(F^{L^2m}(\mathbf u)-F^{L^2m}(\mathbf v)) + \mathbf c\cdot (\mathbf u-\mathbf v)\bigr), \end{aligned}
\end{equation}
\tag{2.1}
$$
where $\phi(q)$ is the Euler totient function. Since $F^t(\mathbf z)=\frac12 A\mathbf z\cdot\mathbf z -t$, we have
$$
\begin{equation*}
F^{L^2m}(\mathbf u)-F^{L^2m}(\mathbf v) =(A\mathbf v)\cdot \mathbf w+ F(\mathbf w) =\mathbf v\cdot A\mathbf w + F(\mathbf w).
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
e_q\bigl(a(F^{L^2m}(\mathbf u)-F^{L^2m}(\mathbf v)) + \mathbf c\cdot (\mathbf u-\mathbf v)\bigr) =e_q\bigl(aF(\mathbf w) + \mathbf c\cdot \mathbf w\bigr)e_q(a\mathbf v\cdot A\mathbf w).
\end{equation*}
\notag
$$
Now we see that the sum over $\mathbf{v}$ in (2.1) produces a zero contribution, unless each component of the vector $A\mathbf{w}$ is divisible by $q$. This property holds for at most a finite number $N$ of vectors $\mathbf{w}$, where the constant $N$ depends only on $\det A$. Thus, The assertion of Lemma 2.1 shows that the sums $\sigma^{\mathbf{c}}_p$, defined in (1.10) are finite. Recall that $\sigma(A,L^2m)=\prod_p \sigma_p(A,L^2m)$ (see (1.11)). Lemma 2.3. For any $d\geqslant5$ and $1\leqslant X \leqslant\infty$ Proof. We begin by showing the multiplicative property of trigonometric sums whenever $(q,q')=1$ (see Lemma 23 in [1]). By definition
$$
\begin{equation*}
S_{qq'}(0)= \mathop{{\sum}^*}_{a(\operatorname{mod} qq')} \sum_{\mathbf v(\operatorname{mod} qq')} e_{qq'}(aF^{L^2m}(\mathbf v)).
\end{equation*}
\notag
$$
When $(q,q') = 1$, we can replace summation over $a\pmod {qq'}$ by double summation over $a_q$ modulo $q$ and $a_{q'}$ modulo $q'$, by writing $a=q a_{q'}+q'a_q$. Thus,
$$
\begin{equation*}
S_{qq'}(0)= \mathop{{\sum}^*}_{a_q(\operatorname{mod} q)}\, \mathop{{\sum}^*}_{a_{q'}(\operatorname{mod} q')} \sum_{\mathbf v(\operatorname{mod} qq')} e_{q}(a_qF^{L^2m}(\mathbf v) ) e_{q'}(a_{q'}F^{L^2m}(\mathbf v) ) .
\end{equation*}
\notag
$$
Then we replace the sum over $\mathbf{v}\pmod{qq'}$ by the double sum over $\mathbf{v}_q$ modulo $q$ and $\mathbf{v}_{q'}$ modulo $q'$ by writing $\mathbf{v}= q\overline q \mathbf{v}_{q'} + q'\overline q' \mathbf{v}_q$, where $\overline q$ and $\overline q'$ are defined by the relations $q\overline q=1\pmod{q'}$ and $q'\overline q' = 1\pmod q$. We observe that
$$
\begin{equation*}
F^{L^2m}(\mathbf v)=q^2\overline q^2F(\mathbf v_{q'}) + q'^2{\overline q}'^2F(\mathbf v_{q}) + q\overline qq'{\overline q}' A\mathbf v_{q'}\cdot \mathbf v_q -{L^2m} ,
\end{equation*}
\notag
$$
so that
$$
\begin{equation*}
e_q(a_qF^{L^2m}(\mathbf v))=e_q(a_qq'^2{\overline q}'^2 F(\mathbf v_q) -a_q{L^2m})=e_q (a_q F^{L^2m} (\mathbf v_q))
\end{equation*}
\notag
$$
by the definition of $\overline q'$ and since $e_q(q N)=1$ for any integer $N$. In a similar way,
$$
\begin{equation*}
e_{q'}(a_{q'}F^{L^2m}(\mathbf v))= e_{q'}(a_{q'} F^{L^2m}(\mathbf v_{q'})) .
\end{equation*}
\notag
$$
This gives (2.2).
Next we note that by Lemma 2.1,
$$
\begin{equation}
\sum_{q\geqslant X}q^{-d} |S_q(0)|\lesssim \sum_{q\geqslant X}q^{-d/2+1} \lesssim X^{-d/2+ 2}.
\end{equation}
\tag{2.3}
$$
By (2.2) and the definition of $\sigma$,
$$
\begin{equation*}
\sigma=\lim_{n\to\infty} \sigma^n, \quad\text{where } \sigma^n=\prod_{p\leqslant n}\sum_{l=0}^n p^{-dl}S_{p^l}(0)= \sum_{q\in P_{n}} q^{-d}S_q(0),
\end{equation*}
\notag
$$
where the $p$ are prime numbers and $P_{n}$ denotes the set of natural numbers $q$ with prime factorization of the form $q=p_1^{k_1}\cdots p_m^{k_m}$, where $2\leqslant p_1<p_2<\dots <p_m\leqslant n$, $k_j\leqslant n$ and $m\geqslant0$ ($m=0$ corresponds to $q=1$). Since any $q\leqslant n$ belongs to $P_n$, according to (2.3) we have
$$
\begin{equation*}
\biggl|\sum_{q\in P_{N}} q^{-d}S_q(0) - \sum_{q\leqslant X} q^{-d}S_q(0)\biggr| \lesssim X^{-d/2 +2} \quad \forall\, N\geqslant X
\end{equation*}
\notag
$$
for any finite $X>0$. Passing to a limit as $N\to\infty$ in this estimate we recover the assertion if $X<\infty$. Then the result for $X=\infty$ follows in an obvious way.
Lemma 2.3 is proved. § 3. Singular integrals $I^0_q$3.1. The properties of $h(x,y)$Following [1], § 3, we construct the function $h(x,y)\in C^\infty(\mathbb{R}_>,\mathbb{R})$ in Theorem 1.1, by starting from the weight function $w_0\in C_0^\infty(\mathbb{R})$ defined by
$$
\begin{equation}
w_0(x)=\begin{cases} \exp\biggl(\dfrac1{x^2-1}\biggr) & \text{for }|x|<1, \\ 0 & \text{for }|x|\geqslant 1. \end{cases}
\end{equation}
\tag{3.1}
$$
We set $c_0:= \displaystyle\int_{-\infty}^{\infty}w_0(x)\,dx$ and introduce the shifted weight function which, of course, belongs to $C^\infty_0(\mathbb{R})$. Clearly, $0\leqslant \omega\leqslant 4e^{-1}/c_0$, $\omega$ has its support on $(1/2,1)$, and $\displaystyle\int_{-\infty}^{\infty}\omega(x)\,dx =1$. The required function $h\colon \mathbb{R}_{>0}\times\mathbb{R}\to\mathbb{R}$ is defined in terms of $\omega$ by where
$$
\begin{equation}
h_1(x):=\sum_{j=1}^\infty\frac{1}{xj}\omega(xj) \quad\text{and}\quad h_2(x,y):=\sum_{j=1}^\infty\frac{1}{xj}\omega \biggl(\frac{|y|}{x j}\biggr) .
\end{equation}
\tag{3.2}
$$
For any fixed pair $(x,y)$, each of the two sums with respect to $j$ contains a finite number of nonzero terms. So $h$ is a smooth function. It was shown in [1], § 3, how to derive Theorem 1.1 from the definition (3.2).5 Here we limit ourselves to providing some relevant properties of $h$, proved in § 4 of [1]. In particular these properties imply that for small $x$, $h(x,y)$ behaves as the Dirac delta function in $y$. Lemma 3.1 (Lemma 4 in [1]). The following hold: Lemma 3.2 (Lemma 5 in [1]). Let $m,n,N\geqslant 0$. Then for any $x$ and $y$ Lemma 3.2 for $m=n=N=0$ immediately implies the following. Lemma 3.4 (Lemma 6 in [1]). Fix $X\in \mathbb{R}_{>0}$ and $0<x<C\min\{1,X\}$ for some ${C>0}$. Then for any $N\geqslant0$, Lemma 3.5 (Lemma 8 in [1]). Fix $X\in \mathbb{R}_{>0}$ and $n\in \mathbb{N}$. Let $x<C\min\{1, X\}$ for ${C>0}$. Then The previous results are used to prove Lemma 9, the key lemma in [1], which can be extended to the following result. Lemma 3.6. Let $f\in \mathcal{C}^{M-1,0}(\mathbb{R})\cap L^1(\mathbb{R})$, $M\geqslant 1$, be a function such that its $(M-1)$st derivative $f^{(M-1)}$ is absolutely continuous on $[-1,1]$, and let $0<x\leqslant C$ for some $C>0$. Then for any $0<\beta\leqslant 1$ and any $N\geqslant 0$, where $X:=\min\{1,x/\beta\}$. Proof. By Lemma 3.2, for $m=n=0$, for any $N\geqslant 0$ we have $|h(x,y)| \lesssim_N (x^N+\beta^N)x^{-1}$ if $|y|\geqslant X$. So the tail integral for $\displaystyle\int fh \,dy$ can be bounded as follows:
As concerns the integral over $\{|y|<X\}$, we take the Taylor expansion of $f(y)$ around zero and get that
$$
\begin{equation}
\begin{aligned} \, \notag &\int_{-X}^X f(y) h(x,y) \, dy \\ &\qquad=\sum_{j=0}^{M-1} \frac{f^{(j)}(0)}{j!}\int_{-X}^X y^j h(x,y)\, dy + O_M\biggl(\frac{X^{M}}{x} \int_{-X}^{X}|f^{(M)}(y)|\,dy\biggr), \end{aligned}
\end{equation}
\tag{3.5}
$$
by Corollary 3.3. Next we use Lemma 3.4 with $N$ replaced by $N+1$ to get that
$$
\begin{equation}
f(0) \int_{-X}^X h(x,y)\, dy=f(0) + O_{N,C}\biggl(\|f\|_{0,0}\biggl(Xx^{N} +\frac{x^{N+1}}{X^{N+1}}\biggr)\biggr),
\end{equation}
\tag{3.6}
$$
while by Lemma 3.5, for any $j>0$ we have
$$
\begin{equation}
\biggl| \frac{f^{(j)}(0)}{j!}\int_{-X}^X y^j h(x,y)\, dy\biggr| \lesssim_{N,j,C} \|f\|_{j,0}X^j\biggl(Xx^{N} +\frac{x^{N+1}}{X^{N+1}}\biggr).
\end{equation}
\tag{3.7}
$$
Putting (3.4)–(3.7) together we obtain the required estimate. Indeed, since ${X\leqslant x/\beta}$, the term $O_M$ in (3.5) is bounded by that in (3.3). Moreover, as $(x/X)^{N+1}=\max(x^{N+1},\beta^{N+1})\lesssim_{C} Cx^N +\beta^N$, the expression in brackets in (3.6) and (3.7) satisfies
$$
\begin{equation*}
Xx^{N} +\frac{x^{N+1}}{X^{N+1}}\lesssim_{C} x^N + \beta^N.
\end{equation*}
\notag
$$
Here we have also used that $X\leqslant 1$.
The lemma is proved. Lemma 3.6 is needed for the proof of Theorem 1.4, while for Theorem 1.3 we only need its simplified version. Corollary 3.7. Let the integrable function $f$ belong to the class $\mathcal{C}^{{M},0}(\mathbb{R})$, ${M}\in \mathbb{N}$, and let $0<x\leqslant C$ for some $C>0$. Then, for any $\delta$, $0<\delta <1$, Proof. This follows from Lemma 3.6 by choosing, for any $0<\delta<1$, $\beta=x^{\delta/(M+1)}$ if $x\leqslant 1$ and $\beta=1$ if $x>1$. Indeed, then for $0<x\leqslant1$ we have $x^M \beta^{-(M+1)} = x^{M-\delta}$ and On the other hand, if $1\leqslant x\leqslant C$, then $x^M \leqslant C^\delta x^{M-\delta}$, and choosing $N=0$ we get that The relations obtained imply the assertion.
The corollary is proved. 3.2. An approximation for $I_q(0)$In what follows it is convenient to write the integrals $I_q(\mathbf{c};A,L^2m)$ as where
$$
\begin{equation}
\widetilde I_q (\mathbf c)=\widetilde I_q (\mathbf c; A,m,L)= \int_{\mathbb R^{d}} w(\mathbf z)h\biggl(\frac qL,F^m(\mathbf z)\biggr) e_q(-\mathbf z\cdot \mathbf c L)\,d\mathbf z.
\end{equation}
\tag{3.9}
$$
The proposition below replaces Lemmas 11, 13 and Theorem 3 in [1]. In contrast to these results, we do not assume that $0\notin\operatorname{supp} w$. Since for $\mathbf{c} = 0$ the exponent $e_q$ in the definition of the integral $I_q(\mathbf{c})$ equals one, we can consider $I_q(0)$ as a function of a real argument $q\in\mathbb{R}$, and we do this in the proposition below; we will use this in § 8.1. Proposition 3.8. Let $q\in\mathbb{R}$, $q\leqslant C L$ for some $C> 0$. a) If $d\geqslant5$ and ${\mathbb{N}}\ni {M}< d/2-1$, then for any $\delta>0$,
$$
\begin{equation}
I_q(0;A, m, L) =L^{d} \sigma_\infty(w;A,m) +O_{m, {M},C,{\delta}}\bigl(q^{M-\delta} L^{d-{M}+{\delta}} \| w\|_{M,d+1}\bigr).
\end{equation}
\tag{3.10}
$$
b) If $d = 4$, $\mathbb{N} \ni {M}\leqslant d/2-1$ and $m=0$, then for any $0<\beta\leqslant 1$ and $N\geqslant 0$,
$$
\begin{equation}
\begin{aligned} \, \notag I_q(0; A,0,L) &=L^{d} \sigma_\infty(w; A,0) + O\biggl(\beta^{-{M}-1}q^{M}L^{d-{M}}\biggl\langle \log\biggl(\frac{q}{L\beta}\biggr)\biggr\rangle \| w\|_{M,d+1}\biggr) \\ &\qquad + O_{C,N}\bigl((q^NL^{d-N}+\beta^N)(\|w\|_{M-1,d+1} + Lq^{-1}\|w\|_{0,d+1})\bigr) . \end{aligned}
\end{equation}
\tag{3.11}
$$
Proof. For $d\geqslant{4}$, applying the co-area formula (see [12], § 3.2.4) we rewrite the integral in (3.9) for $c=0$ in terms of integrals over the hypersurfaces $\Sigma_t$ as follows:
$$
\begin{equation}
\widetilde I_q(0)=\int_\mathbb R \mathcal I(m+t) h\biggl(\frac qL,t\biggr)\, dt, \quad\text{where } \mathcal I(t)=\int_{\Sigma_t} w(\mathbf z)\mu^{\Sigma_t}(d\mathbf z)
\end{equation}
\tag{3.12}
$$
and the measure $\mu^{\Sigma_t}$ is the same as in (1.9). By Theorem 7.3,
$$
\begin{equation}
\| \mathcal I\|_{k, \widetilde K} \lesssim_{k, K, \widetilde K} \| w\|_{k,K} \quad\text{if } \widetilde K < \frac{K+2-d}2, \quad K>d,
\end{equation}
\tag{3.13}
$$
and $ k< d/2-1$. Set $f^m(y) = \mathcal{I}(m+y)$. Then $\|f^m\|_{k,\widetilde K} \lesssim_{m, \widetilde K}\|\mathcal{I}\|_{k,\widetilde K}$, and by (3.13)
To prove a) we apply Corollary 3.7 for $f=f^m$ and $x=q/L$ to the first integral in (3.12). Note that $f^m(0) = \mathcal{I}(m) = \sigma_\infty(w; A,m)$. Then, using (3.13) for $\widetilde K=0$, $K=d+1$ and $k= M$ in combination with (3.14) we get that
$$
\begin{equation*}
\widetilde I_q(0)=\sigma_\infty(w) +O_{M, m,C, \delta} \bigl( q^{M-\delta} L^{-M+\delta}\| w\|_{M, d+1}\bigr).
\end{equation*}
\notag
$$
Now (3.10) follows.
To establish (3.11) we apply Lemma 3.6 to the integral in (3.12) for $m=0$:
$$
\begin{equation*}
\begin{aligned} \, \int_\mathbb R \mathcal I(t) h(x,t)\,dt &=\mathcal I(0) + O_{M}\biggl(\beta^{-M-1}x^{M} \biggl(\frac1X\int_{-X}^{X}|\mathcal I^{(M)} (t)|\,dt\biggr)\biggr) \\ &\qquad +O_{C,N}\bigl((x^N+\beta^N)(\|\mathcal I\|_{M-1,0} +x^{-1} |\mathcal I |_{L_1})\bigr), \end{aligned}
\end{equation*}
\notag
$$
where $x=q/L$ and $ X=\min\{1,x/\beta\}.$ Using Theorem 7.3 for $k=M$ and $M=d+1$ we obtain
$$
\begin{equation*}
\int_{-X}^{X}|\mathcal I^{(M)}(t)|\,dt \lesssim X\langle \log X\rangle \|w\|_{M,d+1}.
\end{equation*}
\notag
$$
Combining this estimate with (3.13) and (3.14) we arrive at (3.11).
Proposition 3.8 is proved. § 4. The term $J_0$In this section we prove the following proposition concerning the term $J_0$ defined in (1.19). Proof. To establish this result we write $J_0$ in the form (1.21). Then the assertion follows from Lemmas 4.2 and 4.3 below, in which we estimate the terms $J_0^+$ and $J_0^-$, once we note that
The proposition is proved. Lemma 4.2. Assume that $w\in L_1(\mathbb{R}^{d})$ and $d\geqslant3$. Then the bound holds for any $\gamma_2\in(0,1)$. Proof. Since, according to Lemma 2.1, $|S_q(0)|\lesssim q^{d/2+1}$, it follows that
$$
\begin{equation*}
|J_0^+|\lesssim\sum_{q>L^{1-\gamma_2}} q^{-d/2+1}I_q(0).
\end{equation*}
\notag
$$
Writing the integral $I_q$ as in (3.8), from Corollary 3.3 we obtain Therefore,
The lemma is proved. Proof. Substituting (3.10) for $C=1$ into the definition of the term $J_0^-$ we obtain $J_0^-=I_A+I_B,$ where
$$
\begin{equation*}
I_A :=L^{d}\sigma_\infty(w)\sum_{q\leqslant L^{1-\gamma_2}} q^{-d}S_q(0)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
|I_B|\lesssim_{M,\delta,m}L^{d-M+\delta}\|w\|_{M,d+1} \sum_{q\leqslant L^{1-\gamma_2}}S_q(0) q^{-d+M}
\end{equation*}
\notag
$$
for $M < d/2-1$ and any $\delta>0$. Lemma 2.3 implies that
$$
\begin{equation*}
\sum_{q\leqslant L^{1-\gamma_2}} q^{-d}S_q(0)=\sigma(A,L^2m) + O(L^{(-d/2+2)(1-\gamma_2)}),
\end{equation*}
\notag
$$
so
$$
\begin{equation*}
I_A=L^{d}\sigma_\infty(w)\sigma(A,L^2m) + O(\sigma_\infty(w) L^{d/2+2 +\gamma_2(d/2-2)}),
\end{equation*}
\notag
$$
whereas $|\sigma_\infty(w)|=|\mathcal I(m)|\leqslant \|\mathcal I\|_{0,0}\leqslant\|w\|_{0,d+1}$ on account of (3.13). As for the term $I_B$, Lemma 2.1 implies that Choosing $M=\lceil d/2\rceil-2$ and $\delta=\gamma_2/2$ we obtain
The lemma is proved. § 5. The term $J_>^{\gamma_1}$We provide here an estimate for the term $J_>^{\gamma_1}$ defined in (1.19). The key point of the proof is an adaptation of Lemma 19 in [1] to our case. We recall the notation (3.8). Proof. Let $f_q(\mathbf z):=w(\mathbf z)h(q/L,F^m(\mathbf z))$. Since
$$
\begin{equation*}
\frac{i}{2\pi}\,\frac{q}{L} |\mathbf c|^{-2}( \mathbf c\cdot \nabla_{\mathbf z})e_q(-\mathbf z\cdot \mathbf c L) =e_q(-\mathbf z\cdot \mathbf c L),
\end{equation*}
\notag
$$
integrating $N$ times by parts in (3.9) we get that
$$
\begin{equation*}
\begin{aligned} \, | \widetilde I_q(\mathbf c)| &\leqslant \biggl(\frac{q}{2\pi L} |\mathbf c|^{-2}\biggr)^N \int_{\mathbb R^{d}} \bigl|( \mathbf c\cdot \nabla_{\mathbf z})^N f_q(\mathbf z)\bigr|\,d\mathbf z \\ &\lesssim_{N} \biggl(\frac qL\biggr)^N |\mathbf c|^{-N} \\ &\qquad\times \sum_{0\leqslant n\leqslant N}\int_{\mathbb R^{d}}\max_{0\leqslant l\leqslant n/2} \biggl|\frac{\partial^{n-l}}{\partial y^{n-l}} h\biggl(\frac qL,F^m(\mathbf z)\biggr) \biggr| |\mathbf z|^{n-2l} \bigl|\nabla_{\mathbf z}^{N-n} w(\mathbf z)\bigr|\,d\mathbf z, \end{aligned}
\end{equation*}
\notag
$$
where $\dfrac{\partial}{\partial y} h$ denotes the derivative of $h$ with respect to the second argument.
First assume that $q\leqslant L$. Then, by Lemma 3.2 for $N=0$,
$$
\begin{equation*}
\begin{aligned} \, &\max_{0\leqslant l\leqslant n/2}\biggl|\frac{\partial^{n-l}}{\partial y^{n-l}} h\biggl(\frac qL, F^m(\mathbf z)\biggr) \biggr|\, |\mathbf z|^{n-2l} \bigl| \nabla_{\mathbf z}^{N-n} w(\mathbf z)\bigr| \\ &\qquad\leqslant \biggl(\frac Lq\biggr)^{n+1}\langle \mathbf z\rangle^{-d-1} \|w\|_{N-n,n+d+1}. \end{aligned}
\end{equation*}
\notag
$$
This implies (5.1) since $n\leqslant N$. Now let $q>L$. Then by part 1) of Lemma 3.1, $h$ is different from zero only if For such $\mathbf{z}$ and $l\leqslant n$ part 3) of Lemma 3.1 implies that
$$
\begin{equation*}
\biggl|\frac{\partial^{n-l}}{\partial y^{n-l}} h\biggl(\frac qL,F^m(\mathbf z)\biggr)\biggr| \lesssim_{n-l} \frac Lq\frac{1}{|F^m(\mathbf z)|^{n-l}} \lesssim_{n-l}\biggl(\frac{L}{q}\biggr)^{n-l+1}.
\end{equation*}
\notag
$$
So
$$
\begin{equation*}
\begin{aligned} \, &\max_{0\leqslant l\leqslant n/2} \biggl|\frac{\partial^{n-l}}{\partial y^{n-l}} h\biggl(\frac qL, F^m(\mathbf z)\biggr) \biggr| \,|\mathbf z|^{n-2l} \bigl|\nabla_{\mathbf z}^{N-n} w(\mathbf z)\bigr| \\ &\qquad\lesssim\max_{0\leqslant l\leqslant n}\frac{(L/q)^{n-l+1}}{\langle \mathbf z\rangle^{2(N-n+l)}} \,\frac{\|w\|_{N-n,2N-n+d+1}}{\langle\mathbf z\rangle^{d+1}}. \end{aligned}
\end{equation*}
\notag
$$
Since from (5.2) we obtain $q/L \lesssim_{{m}} \langle\mathbf{z}\rangle^2$, the first fraction above is bounded by $(L/q)^{N+1}$, and (5.1) follows again.
Proposition 5.1 is proved. As a corollary, we obtain an estimate for $J_>^{\gamma_1}$. Corollary 5.2. The term $J_>^{\gamma_1}$ defined as in (1.19) for $\gamma_1\in(0,1)$ and $d\geqslant3$ satisfies where $N_0:=\lceil d+(d+1)/{\gamma_1}\rceil$. Proof. Denoting the $l^1$-norm by $|\,{\cdot}\,|_1$, by the definition of $J_>^{\gamma_1}$ we have
$$
\begin{equation*}
\begin{aligned} \, |J_>^{\gamma_1}| &\lesssim \sum_{s\geqslant L^{\gamma_1}} s^{d-1}\sum_{q=1}^\infty q^{-d}\sup_{|\mathbf c|_1=s}|S_q(\mathbf c)| |I_q(\mathbf c)| \\ &\lesssim \sum_{s\geqslant L^{\gamma_1}} s^{d-1}\sum_{q=1}^\infty q^{1-d/2}L^d\sup_{|\mathbf c |_1=s} |\widetilde I_q(\mathbf c)| \\ &\lesssim_{N,m} \sum_{s\geqslant L^{\gamma_1}} s^{d-1}\sum_{q=1}^\infty q^{-d/2} s^{-N} L^{d+1}\|w\|_{N,2N+d+1}, \end{aligned}
\end{equation*}
\notag
$$
where the second line follows from Lemma 2.1, while the third follows from Proposition 5.1. The sum over $q$ is bounded by a constant. Choosing $N=N_0$ we obtain This completes the proof. § 6. The term $J^{\gamma_1}_<$6.1. The estimateOur next (and final) goal is to estimate the term $J^{\gamma_1}_<$ in (1.18). Proposition 6.1. For any $d\geqslant3$ and $\gamma_1\in(0,1/2)$, where $\overline N=\overline N(d,\gamma_1):= \lceil d^2/\gamma_1\rceil-2d$. Proposition 6.1 will follow from Lemma 6.2, which is a modification of Lemma 22 in [1] and is proved in § 6.2. Lemma 6.2. For any $d\geqslant 3$ and $\mathbf{c}\ne 0$, Proof of Proposition 6.1. According to Lemma 2.1,
$$
\begin{equation*}
\begin{aligned} \, |J^{\gamma_1}_<| &\lesssim \sum_{\mathbf c\ne 0,\,|\mathbf c|\leqslant L^{\gamma_1}} \sum_{q=1}^\infty q^{-d}q^{d/2+1} |I_q(\mathbf c)| \\ &\lesssim L^{d\gamma_1} \max_{\mathbf c\ne0\colon |\mathbf c|\leqslant L^{\gamma_1}} |I_q(\mathbf c)| \sum_{q=1}^\infty q^{-d/2+1} \\ &=L^{d\gamma_1} \biggl(\sum_{q<L} + \sum_{q\geqslant L}\biggr) q^{-d/2+1} \max_{\mathbf c\ne0\colon |\mathbf c|\leqslant L^{\gamma_1}} |I_q(\mathbf c)|=J_{-} + J_{+}, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
J_- :=L^{d\gamma_1} \sum_{q<L} q^{-d/2+1} \max_{\mathbf c\ne0\colon |\mathbf c|\leqslant L^{\gamma_1}} |I_q(\mathbf c)|
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
J_+:=L^{d\gamma_1} \sum_{q\geqslant L} q^{-d/2+1} \max_{\mathbf c\ne0\colon |\mathbf c|\leqslant L^{\gamma_1}}|I_q(\mathbf c)|.
\end{equation*}
\notag
$$
Corollary 3.3, in combination with (3.8) and (3.9), implies that so that
$$
\begin{equation*}
J_{+}\lesssim L^{d\gamma_1} L^{d+1} |w|_{L_1} \sum_{q\geqslant L} q^{-d/2} \lesssim L^{d\gamma_1 + d/2+2} |w|_{L_1}\lesssim L^{d\gamma_1 + d/2+2} \|w\|_{0,d+1}.
\end{equation*}
\notag
$$
On the other hand, since $|\mathbf{c}|\geqslant 1$, from Lemma 6.2 we obtain
$$
\begin{equation*}
\begin{aligned} \, J_{-} &\lesssim_{\gamma_1,m} L^{d\gamma_1} L^{d/2+1+\gamma_1}\bigl(\|w\|_{\overline N,d+5}+\|w\|_{0,\overline N +3d+4}\bigr)\sum_{q< L} q^{-\gamma_1} \\ &\leqslant \bigl(\|w\|_{\overline N,d+5}+\|w\|_{0,\overline N+3d+4}\bigr) L^{\gamma_1(d+1)+d/2+2}. \end{aligned}
\end{equation*}
\notag
$$
The proposition is proved. 6.2. Proof of Lemma 6.26.2.1. An application of the inverse Fourier transformNote that the proof is nontrivial only for $q\lesssim L|\mathbf{c}|$: indeed, for any $\alpha>0$ the bound (6.1) implies that
$$
\begin{equation*}
|I_q(\mathbf c)| \lesssim_\alpha L^d|w|_{L_1} \lesssim_{\alpha} L^d \biggl(\frac{L|\mathbf c|}{q}\biggr)^{-d/2+1+\gamma_1} |w|_{L_1} \quad \text{if } q\geqslant \alpha L |\mathbf c|,
\end{equation*}
\notag
$$
since $|\mathbf{c}|\geqslant 1$ and $-d/2+1+\gamma_1<0$. So it remains to use the inequality $|w|_{L_1}\lesssim \|w\|_{0,d+1}$ again. Let us take $\alpha=\alpha(d,\gamma_1, A) \in(0,1)$ small enough and assume that $q< \alpha L|\mathbf{c}|$. Consider the function $w_2(x)=1/(1+x^2)$ and set
$$
\begin{equation}
\widetilde w(\mathbf z):=\frac{w(\mathbf z)}{w_2(F^m(\mathbf z))}={w(\mathbf z)} (1+F^m(\mathbf z)^2).
\end{equation}
\tag{6.2}
$$
Let
$$
\begin{equation}
p(t):=\int_{-\infty}^{+\infty} w_2(v) h\biggl(\frac{q}{L},v\biggr) e(-tv) \,dv \quad\text{and}\quad e(x):=e_1(x)= e^{2\pi i x}.
\end{equation}
\tag{6.3}
$$
This is the Fourier transform of the function $w_2(\,{\cdot}\,)h(q/L,\cdot\,)$. Then, expressing $w_2 h$ in terms of $p$ by means of the inverse Fourier transform and writing $w(\mathbf{z})=\widetilde w(\mathbf{z})w_2(F^m(\mathbf{z}))$, we find that
$$
\begin{equation*}
w(\mathbf z)h\biggl(\frac qL, F^m(\mathbf z)\biggr) =\widetilde w(\mathbf z)\int_{-\infty}^{+\infty} p(t)e(t F^m(\mathbf z))\,dt.
\end{equation*}
\notag
$$
Substituting this representation into (3.9) we obtain
$$
\begin{equation*}
\widetilde I_q(\mathbf c)=\int_{-\infty}^{+\infty} p(t)e(-tm)\biggl(\int_{\mathbb R^{d}} \widetilde w(\mathbf z) e\bigl(tF(\mathbf z)-\mathbf u\cdot \mathbf z\bigr)\,d\mathbf z \biggr)\,dt, \quad\text{where } \mathbf u:=\frac{\mathbf cL}{q}.
\end{equation*}
\notag
$$
Note that since $q<\alpha|\mathbf{c}|L$. Now set $W_0(x) = c_0^{-d}\prod_{i=1}^{d}w_0(x_i)$ (see (3.1)). Then ${W_0\,{\in}\, C_0^\infty (\mathbb{R}^d)}$, $W_0\geqslant0$ and
$$
\begin{equation}
\operatorname{supp} W_0=[-1, 1]^d \subset \{ x \in \mathbb R^d\colon |x| \leqslant \sqrt{d} \}, \qquad \int_{\mathbb R^d} W_0(x)\,dx=1.
\end{equation}
\tag{6.4}
$$
We set $\delta = |\mathbf{u}|^{-1/2} < \sqrt\alpha$ and express $\widetilde w$ as
$$
\begin{equation*}
\widetilde w(\mathbf z)=\delta^{-d} \int_\mathbb R^d W_0\biggl( \frac{\mathbf z-\mathbf a}{\delta}\biggr) \widetilde w(\mathbf z)\,d\mathbf a.
\end{equation*}
\notag
$$
Then setting $\mathbf b:=(\mathbf z-\mathbf a)/\delta$ we get that
$$
\begin{equation*}
|\widetilde I_q(\mathbf c)| \leqslant \int_{\mathbb R^{d}}\int_{-\infty}^{+\infty} |p(t)|\,|I_{\mathbf a,t}|\, dt\,d\mathbf a,
\end{equation*}
\notag
$$
where in view of (6.4),
$$
\begin{equation*}
I_{\mathbf a,t}:=\int_{\{ |\mathbf b| \leqslant \sqrt{d}\}} W_0(\mathbf b) \widetilde w(\mathbf z)e(tF(\mathbf z) - \mathbf u\cdot\mathbf z)\,d\mathbf b \quad\text{and}\quad \mathbf z:=\mathbf a+\delta\mathbf b.
\end{equation*}
\notag
$$
Consider the exponent in the integral $I_{\mathbf{a},t}$:
$$
\begin{equation*}
f(\mathbf b)=f_{\mathbf a,t}(\mathbf b):=tF(\mathbf a+ \delta\mathbf b) - \mathbf u\cdot (\mathbf a+ \delta\mathbf b).
\end{equation*}
\notag
$$
At the next step we estimate the integral $I_{\mathbf{a},t}$ regarding $(\mathbf{a},t)$ as a parameter. Consider another parameter $R$ satisfying its value will be specified later. Below we distinguish two cases:1) $(\mathbf{a},t)$ belongs to the ‘good’ domain
$$
\begin{equation*}
S_R=\biggl\{ ( \mathbf a,t)\colon |\nabla f(0)|=\delta |t A\mathbf a - \mathbf u| \geqslant R\biggl\langle \frac{t}{|\mathbf u |}\biggr\rangle =R\langle \delta^2t\rangle \biggr\};
\end{equation*}
\notag
$$
2) $(\mathbf{a},t)$ belongs to the ‘bad’ set ${S_R}^c = (\mathbb{R}^d \times \mathbb{R}) \setminus S_R$. 6.2.2. The integral over $S_R$First we consider the integral over the ‘good’ set $S_R$. Proof. Let $\mathbf{l}:=\nabla f(0)/|\nabla f(0)|$ and $\mathcal{L} =\mathbf{l}\cdot \nabla_\mathbf{b}$. Then for $(\mathbf{a},t)\in S_R$ and $|\mathbf{b}|\leqslant \sqrt{d}$,
$$
\begin{equation}
\begin{aligned} \, \notag |\mathcal L f(\mathbf b)| &=\biggl|\mathcal L f(0) +\delta^2 t \nabla f(0)\cdot\frac{A\mathbf b}{|\nabla f(0)|}\biggr| \\ &\geqslant|\nabla f(0)| - \delta^2|t||A\mathbf b| \geqslant R\langle\delta^2 t \rangle - \delta^2|t|\,\|A\| \frac{R}{2\|A\|} \nonumber \\ &\geqslant \frac12 R\langle\delta^2 t \rangle\geqslant \frac R2 \end{aligned}
\end{equation}
\tag{6.6}
$$
(see (6.4)). Since $(2\pi i \mathcal{L} f(\mathbf{b}))^{-1} \mathcal{L} e(f(\mathbf{b})) = e(f(\mathbf{b}))$, integrating $N$ times by parts in $ I_{\mathbf{a},t}$ we obtain
$$
\begin{equation*}
|I_{\mathbf a,t}| \lesssim_{N} \max_{|b_i|\leqslant 1\ \forall\, i}\, \max_{0\leqslant k\leqslant N} \biggl|\mathcal L^{N-k} \widetilde w(\delta \mathbf b +\mathbf a)\frac{\bigl(\mathcal L^2f(\mathbf b)\bigr)^k}{\bigl(\mathcal L f(\mathbf b)\bigr)^{N+k}}\biggr| ,
\end{equation*}
\notag
$$
where we have used that $\mathcal{L}^m f(\mathbf{b})=0$ for $m\geqslant 3$. Since $|\mathcal{L}^2f(\mathbf{b})|\leqslant \delta^2|t||\mathbf{l}\cdot A\mathbf{l}|\leqslant\delta^2|t|\|A\|$, in view of (6.6),
$$
\begin{equation*}
\biggl|\frac{\mathcal L^2 f(\mathbf b)}{\mathcal L f(\mathbf b)}\biggr| \leqslant\frac{\delta^2|t|\,\|A\|}{\frac12 R\langle\delta^2 t \rangle} =\frac{2\|A\|}{R}\leqslant \frac{1}{\sqrt{d}}.
\end{equation*}
\notag
$$
So using that
$$
\begin{equation*}
\biggl|\frac{1}{\mathcal L f(\mathbf b)}\biggr|\leqslant\frac2R
\end{equation*}
\notag
$$
by (6.6), we find that
$$
\begin{equation*}
|I_{\mathbf a,t}| \lesssim_{N} R^{-N} \max_{|b_i|\leqslant 1\ \forall\, i}\, \max_{0\leqslant k\leqslant N}\bigl|\mathcal L^{k} \widetilde w(\delta \mathbf b +\mathbf a)\bigr|.
\end{equation*}
\notag
$$
Thus, denoting the indicator function of the set $S_R$ by $\mathbf 1_{S_R}$ we have
$$
\begin{equation*}
\begin{aligned} \, \int_{\mathbb R^{d}}|I_{\mathbf a,t}|\mathbf 1_{S_R}\,d\mathbf a &\lesssim_{N} R^{-N}\int_{\mathbb R^{d}} \Bigl(\langle\mathbf a\rangle^{d+1} \max_{|b_i|\leqslant 1\ \forall\, i}\, \max_{0\leqslant k\leqslant N}\bigl|\mathcal L^k\widetilde w(\delta \mathbf b +\mathbf a)\bigr| \Bigr)\, \frac{d\mathbf a}{\langle\mathbf a\rangle^{d+1}} \\ & \lesssim_{N} R^{-N}\|\widetilde w\|_{N,d+1} \lesssim_{N,m} R^{-N}\|w\|_{N,d+5} \end{aligned}
\end{equation*}
\notag
$$
for every $t$. Hence the left-hand side of (6.5) satisfies the relation
To prove (6.5) it remains to show that By virtue of Lemma 3.2 for $N=2$,
$$
\begin{equation*}
\biggl|\frac{\partial^k}{\partial v^k} h(x,v)\biggr| \lesssim_k x^{-k-1}\min\biggl\{1,\frac{x^2}{v^2}\biggr\}, \qquad k\geqslant 1,
\end{equation*}
\notag
$$
and by Corollary 3.3, $|h(x,v)|\lesssim x^{-1}$. Then integration by parts in (6.3) shows that, for any $M\geqslant 0$,
$$
\begin{equation*}
\begin{aligned} \, |p(t)| & \lesssim_M |t^{-M}| \biggl(\int_{-\infty}^\infty|w_2^{(M)}(v)|x^{-1}\,dv \\ &\qquad+ \max_{1\leqslant k \leqslant M}\int_{-\infty}^\infty|w_2^{(M-k)}(v)|x^{-k-1}\min\biggl\{1,\frac{x^2}{v^2}\biggr\}\,dv\biggr), \end{aligned}
\end{equation*}
\notag
$$
where $x:=q/L$. Writing the latter integral as the sum $\displaystyle \int_{|v|\leqslant x} + \int_{|v|>x}$ we see that
$$
\begin{equation*}
\int_{|v|\leqslant x}=x^{-k-1}\int_{|v|\leqslant x} |w_2^{(M-k)}(v)|\,dv \lesssim_{M} x^{-k}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\int_{|v|>x}=x^{-k+1}\int_{|v|>x} \frac{ |w_2^{(M-k)}(v)|}{v^2}\,dv \lesssim_{ M} x^{-k}.
\end{equation*}
\notag
$$
Then for any $M\geqslant 0$,
$$
\begin{equation}
\begin{alignedat}{2} |p(t)| &\lesssim_M \biggl(\frac{q}{L}|t|\biggr)^{-M} &\quad &\text{if } \frac qL <1 , \\ |p(t)| &\lesssim_M \biggl(\frac{q}{L}\biggr)^{-1} |t|^{-M} &\quad &\text{if } \frac qL \geqslant 1. \end{alignedat}
\end{equation}
\tag{6.9}
$$
Choosing $M=2$ for $|t|>\langle L/q \rangle$ and $M=0$ for $|t|\leqslant \langle L/q \rangle$ we obtain (6.8).
Lemma 6.3 is proved. 6.2.3. The integral over ${S_R}^c$Now we consider the integral over the ‘bad’ set ${S_R}^c$. Lemma 6.4. For any $d\geqslant 1$, $1\leqslant R\leqslant |\mathbf{u}|^{1/3}$ and $0<\beta<1$ we have where $K(d,\beta)= d+\lceil d^2/2\beta\rceil+4$. Proof. On ${S_R}^c$ we use for $I_{\mathbf{a}, t}$ the easy upper bound The fact that $(\mathbf{a},t)\in {S_R}^c$ implies that the integration against $d\mathbf{a}$ for fixed $t$ is limited to the region where $|A\mathbf a - t^{-1}\mathbf u| \leqslant ({R}/{\delta |t|}) \langle t / |\mathbf u| \rangle$ or
First we consider the case when $|t|\geqslant |\mathbf{u}|^{1-\beta/d}$. Since $|\mathbf{u}| >1$, considering the cases $|t| \leqslant |\mathbf{u}|$ and $|t| \geqslant |\mathbf{u}|$ separately we see that
$$
\begin{equation}
\frac{R}{\delta|t|}\biggl\langle \frac{t}{|\mathbf u|} \biggr\rangle \leqslant R|\mathbf u|^{-1/2+\beta/d}.
\end{equation}
\tag{6.12}
$$
In view of (6.10)–(6.12)
$$
\begin{equation}
\biggl|\int_{\mathbb R^{d}} |I_{\mathbf a,t}|\mathbf 1_{{S_R}^c}(\mathbf a,t)\, d\mathbf a\biggr| \lesssim R^{d}|\mathbf u|^{-d/2+\beta} \|\widetilde w\|_{0,0}.
\end{equation}
\tag{6.13}
$$
Since $|F^m(\mathbf{z})|\lesssim_m \langle \mathbf{z} \rangle^2$, by the definition (6.2) of the function $\widetilde w$ we have ${\|\widetilde w\|_{0,0}\lesssim_m \|w\|_{0,4}}$. Then the left-hand side of (6.13) satisfies
$$
\begin{equation*}
\biggl|\int_{\mathbb R^{d}} |I_{\mathbf a,t}|\mathbf 1_{{S_R}^c}(\mathbf a,t)\, d\mathbf a\biggr|\lesssim_m {R^{d}}|\mathbf u|^{-d/2+\beta} \| w\|_{0,4}.
\end{equation*}
\notag
$$
Taking into account that, by (6.8),
$$
\begin{equation*}
\int_{|t|\geqslant |\mathbf u|^{1-\beta/d}} |p(t)|\,dt \lesssim \frac{L}{q}\leqslant|\mathbf u|,
\end{equation*}
\notag
$$
we obtain
$$
\begin{equation}
\begin{aligned} \, &\int_{|t|\geqslant |\mathbf u|^{1-\beta/d}}\biggl(\int_{\mathbb R^{d}} |p(t)|\, |I_{\mathbf a, t}|\mathbf 1_{{S_R}^c}(\mathbf a,t) \,d\mathbf a \biggr) dt \nonumber \\ &\qquad\qquad \lesssim_mR^{d}|\mathbf u|^{-d/2+1+\beta} \| w\|_{0,4}. \end{aligned}
\end{equation}
\tag{6.14}
$$
Now let $|t| \leqslant |\mathbf{u}|^{1-\beta/d}$. Then the right-hand side of (6.11) is bounded by the quantity $\|A^{-1}\|R/(\delta|t|)$, so that $|\mathbf{a}|\gtrsim |A^{-1}\mathbf{u}|/|t|-\|A^{-1}\|R/(\delta|t|)$. Since $|A^{-1}\mathbf{u}|\geqslant C_A|\mathbf{u}|$ and $R\leqslant |\mathbf{u}|^{1/3}$, we have
$$
\begin{equation*}
|\mathbf a|\gtrsim_A \frac{|\mathbf u|- R C'_A\sqrt{|\mathbf u|}}{|t|}\geqslant (1-C'_A|\mathbf u|^{-1/6})\frac{|\mathbf u|}{|t|} \geqslant \frac12\,\frac{|\mathbf u|}{|t|}\geqslant \frac 12 |\mathbf u|^{\beta/d},
\end{equation*}
\notag
$$
where $C_A'=C_A^{-1}\|A^{-1}\|$, since $|\mathbf{u}|^{-1}\leqslant \alpha$, provided that $\alpha$ is sufficiently small so that $1-C'_A\alpha^{1/6}\geqslant 1/2$. Then $1\lesssim |\mathbf{a}|/|\mathbf{u}|^{\beta/d}$ on ${S_R}^c$, and therefore
$$
\begin{equation*}
\mathbf 1_{{S_R}^c}(\mathbf a,t) \lesssim |\mathbf u|^{-d/2+\beta/d} |\mathbf a|^{d^2/(2\beta)-1},
\end{equation*}
\notag
$$
and we deduce from (6.10) that for such values of $t$ we have
$$
\begin{equation*}
\begin{aligned} \, \biggl|\int_{\mathbb R^d} |I_{\mathbf a,t}|\mathbf 1_{{S_R}^c}(\mathbf a,t) \,d\mathbf a\biggr| &\lesssim |\mathbf u|^{-d/2+\beta/d} \int_{\mathbb R^{d}} |\mathbf a|^{d^2/(2\beta)-1} \max_{|b_i|\leqslant 1\ \forall\, i}|\widetilde w(\delta \mathbf b +\mathbf a)|\, d\mathbf a \\ &\lesssim_m|\mathbf u|^{-d/2+\beta/d} \|w\|_{0,K(d,\beta)}, \end{aligned}
\end{equation*}
\notag
$$
where $K(d,\beta)=d+\lceil d^2/2\beta\rceil+4$. On the other hand, by inequality (6.9) for $M=0$ we have
$$
\begin{equation*}
\int_{|t|\leqslant |\mathbf u|^{1-\beta/d}}|p(t)|\,dt \lesssim |\mathbf u|^{1-\beta/d},
\end{equation*}
\notag
$$
from which we obtain
$$
\begin{equation}
\int_{|t|\leqslant |\mathbf u|^{1-\beta/d}} \biggl(\int_{\mathbb R^{d}}|p(t)| |I_{\mathbf a, t}| \mathbf 1_{{S_R}^c}(\mathbf a,t)\,d\mathbf a\biggr)\,dt \lesssim_m|\mathbf u|^{-d/2+1} \| w\|_{0,K(d,\beta)}.
\end{equation}
\tag{6.15}
$$
Putting (6.14) and (6.15) together we obtain the required assertion. Lemma 6.4 is proved. 6.2.4. The end of the proofIn order to complete the proof of Lemma 6.2 we combine Lemmas 6.3 and 6.4 to get that
$$
\begin{equation*}
|\widetilde I_q(\mathbf c)|\lesssim_{N,m} \biggl(\frac Lq R^{-N} +R^{d} |\mathbf u|^{-d/2+1+\beta}\biggr)\bigl( \|w\|_{N,d+5}+\|w\|_{0,K(d,\beta)}\bigr).
\end{equation*}
\notag
$$
We fix here $\gamma_1\in (0,1/2)$, $\beta = {\gamma_1}/2$ and $R=|\mathbf u|^{\gamma_1/(2d)}\leqslant|\mathbf u|^{1/3}$ and take $N=\lceil d^2/\gamma_1\rceil-2d>0$ (notice that $R \geqslant \alpha^{-\gamma_1/(2d)}\geqslant 2\|A\|\sqrt{d}$ if $\alpha$ is sufficiently small, so that the assumptions of Lemma 6.3 are satisfied). Then
$$
\begin{equation*}
K(d,\beta)=N+3d+4 \quad\text{and}\quad R^{-N}\leqslant|\mathbf u|^{-d/2+\gamma_1}\leqslant |\mathbf c| \biggl(\frac{L|\mathbf c|}{q}\biggr)^{-d/2+\gamma_1}
\end{equation*}
\notag
$$
since $|\mathbf{c}|\geqslant 1$. Moreover,
$$
\begin{equation*}
R^d |\mathbf u|^{-d/2+1+\beta}=|\mathbf u|^{-d/2+1+\gamma_1} =\biggl(\frac{L|\mathbf c|}q\biggr)^{-d/2+1+\gamma_1}.
\end{equation*}
\notag
$$
Lemma 6.2 is proved. § 7. Integrals over quadricsOur goal in this section is to study the integrals $\mathcal{I}(t;w)$ over the quadrics $\Sigma_t$. We begin with the case of quadratic forms $F$ written in a convenient normal form (Theorem 7.1), and then we show (Theorem 7.3) how to reduce the general integrals $\mathcal{I}(t;w)$ to the integrals corresponding to such quadratic forms. In this section we assume that and we do not use boldface to denote vectors since most of the variables we use are vectors.7.1. Quadratic forms in the normal formOn the space
$$
\begin{equation*}
\mathbb R^{d}=\mathbb R^n_u\times \mathbb R_x^{d_1}\times \mathbb R_y^{d_1}=\{z=(u,x,y)\},\quad\text{where } d\geqslant3,\quad n\geqslant 0 \quad\text{and}\quad d_1\geqslant 1,
\end{equation*}
\notag
$$
consider the quadratic form
$$
\begin{equation}
F(z)=\frac12|u|^2+x\cdot y=\frac12 Az\cdot z, \qquad A(u,x,y)=(u,y,x).
\end{equation}
\tag{7.1}
$$
Note that $A$ is an orthogonal operator, $|Az|=|z|$. As in § 1.1, we consider the quadrics $\Sigma_t=\{z\colon F(z)=t\}$, $t\in \mathbb{R}$. Note that, for $t\neq 0$, $\Sigma_t$ is a smooth hypersurface, while $\Sigma_0$ is a cone with singularity at the origin. We denote the volume element on $\Sigma_t$ (on $\Sigma_0\setminus\{0\}$ if $t=0$) induced from $\mathbb{R}^{d}$ by $dz|_{\Sigma_t}$ and set (see below as concerns this measure for $t=0$). For $k_*\in \mathbb{N}\cup\{0\}$ and a function $f$ on $\mathbb{R}^{d}$ satisfying we consider the integrals
$$
\begin{equation}
\mathcal I(t)=\mathcal I(t;f)=\int_{\Sigma_t} f(z)\mu^{\Sigma_t}(dz).
\end{equation}
\tag{7.4}
$$
Our first goal is to establish the following result. Theorem 7.1. Given a quadratic form $F(z)$ as in (7.1) and a function $f\in\mathcal{C}^{k_*,M}(\mathbb{R}^d)$, $M>{d}$, consider the integral $\mathcal{I}(t;f)$ defined in (7.4). Then the function $\mathcal{I}(t)$ defined by (7.4) is $C^k$-smooth if $k<d/2-1$ and $k\leqslant k_*$, and it is $C^k$-smooth outside zero if $k\leqslant \min(d/2-1,k_*)$. For $ 0<|t|\leqslant 1$ Example A.3 in [13], shows that, in general, the log-factor cannot be removed from the right-hand side of (7.5). Proof of Theorem 7.1. The theorem is proved below in several steps. In the proof, for a fixed vector $x\in \mathbb{R}^{d_1}$ we consider its orthogonal complement in $\mathbb{R}^{d_1}$, a hyperspace $x^\perp$. We denote its elements by $\overline x$, and provide $x^\perp$ with the Lebesgue measure $d\overline x$. If $d_1=1$, then $x^\perp$ degenerates to the space $\mathbb{R}^0= \{0\}$, and $d\overline x$ to the $\delta$-measure at $0$. In practice, this means that for $d_1=1$ the spaces $x^\perp$ and $y^\perp$ (and integrals over them) disappear from our construction. It makes the case $d_1=1$ easier but notationally different from $d_1\geqslant2$. For example, in formula (7.8) for $d_1=1$ the affine space $\sigma_t^x(u', x')$ becomes the point $(u', x', (t- \frac12 |u'|^2) |x'|^{-2} x')$, the measure $d\mu^{\Sigma_t}|_{\Sigma_t^x}$ in (7.13) becomes $du\, |x|^{-1}\,dx$, and so on. Accordingly, below we write the proof only for $d_1\geqslant2$, leaving the case $d_1=1$ to the reader as an easy exercise. 7.2. Disintegration of the two measuresOur goal in this subsection is to find a convenient disintegration of the measures $dz|_{\Sigma_t}$ and $\mu^{\Sigma_t}$, following the proof of Theorem 3.6 in [11]. Recall that we write elements $z\in\mathbb{R}^d$ as $z=(u,x,y)$, where $u\in\mathbb{R}^d$ and ${x,y\in \mathbb{R}^{d_1}}$. Set (if $t<0$, then $\Sigma_t^x=\Sigma_t$). Then for any $t$ $\Sigma_t^x$ is a smooth hypersurface in $\mathbb{R}^d$, and the mapping
$$
\begin{equation}
\Pi_t^x\colon\Sigma_t^x\to\mathbb R^n\times \mathbb R^{d_1}\setminus\{0\}, \qquad (u,x,y)\mapsto (u,x),
\end{equation}
\tag{7.7}
$$
is a smooth affine Euclidean vector bundle. Its fibres are
$$
\begin{equation}
\sigma^x_t(u',x'):=(\Pi_t^x)^{-1}(u',x')=\biggl( u', x', {x'}^\perp + \frac{t-\frac12|u'|^2}{|x'|^2} x'\biggr),
\end{equation}
\tag{7.8}
$$
where ${x'}^\perp$ is the orthogonal complement to $x'$ in $\mathbb{R}^{d_1}$. For any $x'\neq 0$ set
$$
\begin{equation*}
U_{x'}=\biggl\{x\colon |x-x'|\leqslant \frac12 |x'|\biggr\}, \qquad U=\mathbb R^n\times U_{x'} \times \mathbb R^{d_1}.
\end{equation*}
\notag
$$
Now we construct a trivialisation of the bundle $\Pi_t^x$ over $U$. To do this we fix any orthonormal frame $(e_1,\dots,e_{d_1})$ in $\mathbb{R}^{d_1}$ such that the ray $\mathbb{R}_+ e_1$ intersects $U_{x'}$. Then
$$
\begin{equation*}
x_1>0 \quad \forall\, x=(x_1,\dots,x_{d_1})=:(x_1,\overline x) \in U_{x'}.
\end{equation*}
\notag
$$
We wish to construct a diffeomorphism
$$
\begin{equation*}
\Phi_t\colon \mathbb R^n\times U_{x'}\times \mathbb R^{d_1-1}\to U\cap \Sigma_t,
\end{equation*}
\notag
$$
which is affine in the third argument and has the form
$$
\begin{equation}
\Phi_t(u,x, \overline \eta)=(u,x,\Phi_t^{u,x}(\overline \eta)), \qquad \Phi_t^{u,x}(\overline \eta)=(\varphi_t(u,x, \overline \eta),\overline \eta) \in \mathbb R^{d_1}, \quad \overline\eta\in \mathbb R^{d_1-1}.
\end{equation}
\tag{7.9}
$$
We easily see that $\Phi_t(u,x,\overline \eta)\in \Sigma_t$ if and only if
$$
\begin{equation}
\varphi_t(u,x,\overline \eta)=\frac{t-\frac12|u|^2-\overline x\cdot\overline \eta}{x_1}.
\end{equation}
\tag{7.10}
$$
The mapping $\overline\eta\to\Phi_t^{u,x}(\overline\eta)$ with this function $ \varphi_t$ is affine, and the range of $\Phi_t$ equals $U\cap \Sigma_t$. In the coordinates $(u,x,\eta_1,\overline\eta)\in \mathbb{R}^n\times U_{x'}\times \mathbb{R}\times \mathbb{R}^{{d_1}-1}$ on the domain $U \subset \mathbb{R}^{d}$ the hypersurface $\Sigma_t^x$ is embedded in $\mathbb{R}^{d}$ as the graph of the function $(u,x,\overline \eta)\mapsto \eta_1 = \varphi_t$. Accordingly, in the coordinates $(u,x,\overline\eta)$ on $U\cap \Sigma_t$ the volume element on $\Sigma_t$ reads
$$
\begin{equation*}
\overline \rho_t(u,x,\overline \eta)\,du\,dx\,d\overline \eta,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\overline \rho_t=(1+|\nabla \varphi_t|^2)^{1/2} =\biggl(1+\frac{|u|^2+|\overline \eta|^2+|\overline x|^2 + x_1^{-2}(t-\frac12 |u|^2 -\overline x\cdot \overline \eta)^2}{x_1^2}\biggr)^{1/2}.
\end{equation*}
\notag
$$
Going over from the variable $\overline \eta\in \mathbb{R}^{{d_1}-1}$ to $y=\Phi_t^{u,x}(\overline \eta) \in \sigma_t^x(u,x)$ we replace $d\overline \eta$ by $|{\det\Phi_t^{u,x}(\overline \eta)}|\,d_{\sigma_t^x(u,x)}y$. Here $d_{\sigma_t^x(u,x)}y$ is the Lebesgue measure on the ${(d_1-1)}$-dimensional affine Euclidean space $\sigma_t^x(u,x)$, while $\det\Phi_t^{u,x}$ denotes the determinant of the linear mapping $\Phi_t^{u,x}$, viewed as a linear isomorphism of Euclidean space $\mathbb{R}^{d_1-1} = \{\overline\eta\}$ and the tangent space to $\sigma_t^x(u,x)$, identified with the Euclidean space $x^\perp\subset \mathbb{R}^{d_1}$. Accordingly, we write the volume element on ${\Sigma_t \cap U}$ as where
$$
\begin{equation*}
\rho_t(u,x,y)=\overline \rho_t(u,x,\overline \eta) |{\det\Phi_t^{u,x}(\overline \eta)}|, \qquad (u,x,y) \in \Sigma_t, \quad \Phi_t^{u,x}(\overline \eta)=y.
\end{equation*}
\notag
$$
Now we calculate the density $\rho_t$. We fix a point $z_*=(u_*,x_*, y_*)\in U\cap \Sigma_t $ and choose a frame $(e_1,\dots,e_{d_1})$ such that $e_1=x_*/|x_*|$. Then
$$
\begin{equation*}
x_*=(|x_*|,0)\quad\text{and} \quad y_*=\bigl(y_{*1}, \overline y_*\bigr), \qquad y_{*1}=\biggl(\frac{t-\frac12|u_*|^2}{|x_*|} \biggr), \quad \overline y_*\in \mathbb R^{d_1-1}.
\end{equation*}
\notag
$$
So (see (7.9) and (7.10)) $\Phi_t^{u_*,x_*}(\overline \eta)=(y_{*1},\overline \eta) =\widetilde y\in\sigma_t^x(u_*,x_*)$ (that is, ${\varphi_t(z_*) = y_{*1}}$). In these coordinates
$$
\begin{equation*}
\rho_t(u_*,x_*,y_{*1}, \overline y_*)=\overline \rho_t(u_*,x_*, \overline y_*),
\end{equation*}
\notag
$$
which equals
$$
\begin{equation*}
\bigl(1+|x_*|^{-2}(|u_*|^2+|\overline y_*|^2 +|y_{*1}|^2)\bigr)^{1/2} =\frac{(|x_*|^2 +|u_*|^2+|\overline y_*|^2 +|y_{*1}|^2)^{1/2}}{|x_*|}.
\end{equation*}
\notag
$$
That is, $\rho_t(z_*)=|z_*|/|x_*|$. Since $z_*$ is any point in $U\cap \Sigma_t$, we have proved the following. Proposition 7.2. The volume element $dz|_{\Sigma_t^x}$ with respect to the projection $\Pi_t^x$ disintegrates as follows: That is, for any function $f\in C_0^0(\Sigma_t^x)$, Similarly, if we set $\Sigma_t^y= \{(u,x,y)\in \Sigma_t\colon y\neq 0\}$ and consider the projection
$$
\begin{equation*}
\Pi_t^y\colon\Sigma_t^y\to\mathbb R^n\times \mathbb R^{d_1}\setminus\{0\}, \qquad (u,x,y)\mapsto (u,y),
\end{equation*}
\notag
$$
then
$$
\begin{equation}
dz|_{\Sigma_t^y}=du\,|y|^{-1}\,dy\,|z|\,d_{\sigma_t^y(u,y)}x.
\end{equation}
\tag{7.12}
$$
Set $\Sigma_t^0= \{(u,x,y)\in \Sigma_t\colon x=y= 0\}$. Then $\Sigma_t\setminus \Sigma_t^0$ is a smooth manifold and $dz|_{\Sigma_t}$ defines a smooth measure on it. By (7.11) and (7.12) the function $|z|^{-1}$ is locally integrable on $\Sigma_t$ with respect to the measure $dz|_{\Sigma_t}$. So $\mu^{\Sigma_t}$ (see (7.2)) is a well-defined Borel measure on $\Sigma_t$. Since $|Az|=|z|$, in view of (7.11) and (7.12) we have
$$
\begin{equation}
d\mu^{\Sigma_t}|_{\Sigma_t^x}=du\,|x|^{-1}\,dx\,d_{\sigma_t^x(u,x)}y \quad\text{and}\quad d\mu^{\Sigma_t}|_{\Sigma_t^y}=du\,|y|^{-1}\,dy\,d_{\sigma_t^y(u,y)}x.
\end{equation}
\tag{7.13}
$$
The measure $\mu^{\Sigma_t}$ defines a Borel measure on $\mathbb{R}^{d}$ with support on $\Sigma_t$. It will also be denoted by $\mu^{\Sigma_t}$. 7.3. An analysis of the integral $\mathcal{I}(t;f)$Note that for any $t$ the mapping
$$
\begin{equation*}
L_t\colon\Sigma_0^x\to \Sigma_t^x, \qquad (u,x,y)\mapsto (u,x,y+t|x|^{-2}x),
\end{equation*}
\notag
$$
defines an affine isomorphism of the bundles $\Pi_0|_{\Sigma_0^x}$ and $\Pi_t|_{\Sigma_t^x}$. Since $L_t$ preserves the Lebesgue measure on fibres, in view of (7.11) it takes the measure $\mu^{\Sigma_0}$ to $\mu^{\Sigma_t}$. From (7.13) we see that for any $t$ the integral $ \mathcal{I}(t)$ defined in (7.4) can be written as
$$
\begin{equation}
\begin{aligned} \, \notag &\mathcal I(t;f)\int_{\Sigma_0}f(L_t(z))\mu^{\Sigma_0}(dz) \\ &\qquad=\int_{\mathbb R^n\times \mathbb R^{d_1}} |x|^{-1}\biggl(\int_{\sigma(u,x)} f(u,x,y+t|x|^{-2}x)\,d_{\sigma^x(u,x)}y \biggr)\,du\,dx. \end{aligned}
\end{equation}
\tag{7.14}
$$
Here $\sigma(u,x):=\sigma_0^x(u,x)=x^\perp-\frac12|u|^2|x|^{-2}x$. We recall that $f(u,x,y)$ satisfies (7.3). Taking any smooth function $\varphi(t) \geqslant 0$ on $\mathbb{R}$ which vanishes for $|t|\geqslant 2$ and equals $1$ for $|t|\leqslant 1$, we write
$$
\begin{equation*}
f=f_{00}+f_1,\quad \text{where } f_{00}=\varphi(|(x,y)|^2)f\text{ and } f_{1}=(1- \varphi(|(x,y)|^2))f.
\end{equation*}
\notag
$$
Setting $B_r(\mathbb{R}^m)= \{\xi\in \mathbb{R}^m\colon |\xi|\leqslant r\}$ and $B^r(\mathbb{R}^m)= \{\xi\in \mathbb{R}^m\colon |\xi|\geqslant r\}$ we see that
$$
\begin{equation}
\operatorname{supp} f_{00}\subset \mathbb R^n\times B_{\sqrt2}(\mathbb R^{2{d_1}}) \quad\text{and}\quad \operatorname{supp} f_1\subset \mathbb R^n\times B^1(\mathbb R^{2{d_1}}).
\end{equation}
\tag{7.15}
$$
Setting next $f_{11}(z) = f_1(z)(1-\varphi(4|x|^2))$ and $f_{10}(z)=f_1(z)\varphi(4|x|^2)$ we write Since for $(x,y)\in B^1(\mathbb{R}^{2{d_1}})$ we have $|x|\geqslant 1/\sqrt{2}$ or $|y|\geqslant 1/\sqrt{2}$, in view of (7.15) we obtain
$$
\begin{equation}
\begin{gathered} \, \operatorname{supp} f_{11}\subset \mathbb R^n\times B^{1/2}(\mathbb R^{d_1}_x)\times \mathbb R^{d_1}_y, \\ \operatorname{supp}f_{10}\subset \mathbb R^n\times \mathbb R^{d_1}_x\times B^{1/\sqrt2}(\mathbb R^{d_1}_y). \end{gathered}
\end{equation}
\tag{7.16}
$$
Obviously, for $i,j=0,1$ we have $\|f_{ij}\|_{k,m}\leqslant C_{k,m} \|f\|_{k,m}$ for all $k\leqslant k_*$ and $m\leqslant M$. Setting $\mathcal{I}_{ij}(t)= \mathcal{I}(t;f_{ij})$ we obtain
$$
\begin{equation*}
\mathcal I(t;f)=\mathcal I_{00}(t)+\mathcal I_{10}(t) +\mathcal I_{11}(t).
\end{equation*}
\notag
$$
7.3.1. The integral $\mathcal{I}_{00}(t)$By (7.14) $\mathcal{I}_{00}(t)$ is a continuous function, and for ${1\leqslant k\leqslant k_*}$,
$$
\begin{equation}
\begin{aligned} \, \notag \partial^k \mathcal I_{00}(t) &=\int_{\mathbb R^n}\biggl(\int_{B_{\sqrt2}(\mathbb R^{d_1})}|x|^{-1}\,dx \biggr)\,du \\ \notag &\qquad\times \int_{y\in \sigma(u,x)}\frac{d^k}{dt^k}f_{00}(u,x,y+t|x|^{-2} x) \, d_{\sigma(u,x)}y \\ \notag &=\int_{\mathbb R^n}\int_{B_{\sqrt2}(\mathbb R^{d_1})} |x|^{-1} \\ &\qquad\times\biggl( \int_{y\in \sigma(u,x)}d_y^k f_{00}(u,x,y+t|x|^{-2} x)[|x|^{-2}x]\,d_{\sigma(u,x)}y \biggr)\,dx\,du , \end{aligned}
\end{equation}
\tag{7.17}
$$
where $d_y^k f_{00}[|x|^{-2}x]$ denotes the action of the differential $d_y^k f_{00}$ on the set of $k$ vectors, each of which is equal to $|x|^{-2}x$. Setting $\tau=t-\frac12|u|^2$, for $y\in\sigma(u,x)$ we have
$$
\begin{equation}
y+t|x|^{-2} x=\overline y+\tau|x|^{-2}x \quad \text{for some } \overline y\in x^\perp.
\end{equation}
\tag{7.18}
$$
Then we write the integral over $y$ in (7.17) as
$$
\begin{equation}
\int_{x^\perp} d_y^k f_{00}(u,x,\overline y+\tau|x|^{-2}x) [|x|^{-2}x] \,d\overline y.
\end{equation}
\tag{7.19}
$$
Since $|\overline y+\tau x|x|^{-2}|^2=|\overline y|^2 + \tau^2|x|^{-2}$, on the support of the integrand we have
$$
\begin{equation}
|x|\leqslant \sqrt2 \quad\text{and}\quad |\overline y|^2+ \tau^2 |x|^{-2} \leqslant 2.
\end{equation}
\tag{7.20}
$$
In particular,
$$
\begin{equation}
|\tau|=\biggl|t-\frac12|u|^2\biggr|\leqslant \sqrt 2\, |x|\leqslant 2
\end{equation}
\tag{7.21}
$$
in (7.19). By (7.15) the diameter of the domain of integration in (7.19) is bounded by $\sqrt2$. So for any $m\geqslant 0$ the integral (7.19) is bounded by $C_{k,m} |x|^{-k}\langle u\rangle^{-m}\|f\|_{k,m}$. Setting $R=|u|$ and $r=|x|$ we get that
$$
\begin{equation}
|\partial^k \mathcal I_{00}(t)| \lesssim_{k,M} \|f\|_{k,M} \int_{0}^{\sqrt2} r^{d_1-k-2} \biggl(\int_0^\infty R^{n-1}\langle R\rangle^{-M}\chi_{|\tau|\leqslant \sqrt 2\, r} \, dR\biggr)\,dr .
\end{equation}
\tag{7.22}
$$
If $n=0$, then the integral against $dR$ must be removed from the right-hand side. Below we estimate the integral $\partial^k \mathcal{I}_{00}(t)$ separately for $n=0$ and $n\geqslant 1$. a) If $n=0$, then $\tau=t$, from (7.21) we obtain $|x|\geqslant t/\sqrt2 $, and we see from (7.15) that $\mathcal{I}_{00}(t)$ is $C^{k_*}$-smooth for $t\ne 0$ (since $f\in C^{k_*}$). Then from (7.22) we obtain
$$
\begin{equation}
|\partial^k \mathcal I_{00}(t)| \lesssim_{k,M} \|f\|_{k,M} \int_{|t|/\sqrt2}^{\sqrt2}r^{d_1-k-2} \chi_{|t|\leqslant 2}\, dr .
\end{equation}
\tag{7.23}
$$
From here it follows that
$$
\begin{equation}
\begin{alignedat}{2} |\partial^k \mathcal I_{00}(t)| &\lesssim_{k} \|f\|_{k,M} &\quad &\text{if } k\leqslant \min({d_1}-2, k_*), \\ |\partial^k \mathcal I_{00}(t)| &\lesssim_{k} \|f\|_{k,M}\bigl(1+\bigl|\log |t|\bigr|\bigr) &\quad &\text{if } k\leqslant \min({d_1}-1, k_*) , \end{alignedat}
\end{equation}
\tag{7.24}
$$
while $\mathcal{I}_{00}(t)=0$ for $|t|\geqslant 2$. b) If $n\geqslant 1$, then to estimate $\partial^k \mathcal{I}_{00}(t)$ we split the integral for $ \mathcal{I}_{00}(t)$ into a sum of two. Namely, for fixed $t\neq 0$ we write $f_{00}$ as
$$
\begin{equation*}
f_{00}=f_{00<}+f_{00>},\quad \text{where } f_{00<}=f_{00}\varphi\biggl(\frac{8|x|^2}{t^2}\biggr),
\end{equation*}
\notag
$$
and $\varphi$ is the function used to define the functions $f_{ij}$, $0\leqslant i,j\leqslant1$. Then
$$
\begin{equation}
\operatorname{supp} f_{00<} \subset \{2|x| \leqslant |t|\} \quad\text{and}\quad \operatorname{supp} f_{00>} \subset \{2\sqrt2\, |x| \geqslant |t|\}.
\end{equation}
\tag{7.25}
$$
Using obvious notation we have
$$
\begin{equation*}
\mathcal I_{00}(t)=\mathcal I_{00<}(t)+\mathcal I_{00>}(t),
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{aligned} \, \mathcal I_{00<}(t) &= \int_{\mathbb R^n}\int_{B_{\sqrt2}(\mathbb R^{d_1})\cap B_{|t|/2}(\mathbb R^{d_1})} |x|^{-1} \\ &\qquad\times \biggl(\int_{\substack{y\in \sigma(u,x)\\|x|^2+|y+t|x|^{-2}x|^2\leqslant2}} f_{00<}(u,x,y+t|x|^{-2} x) \, d_{\sigma(u,x)}y \biggr)\, dx\,du \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \mathcal I_{00>}(t) &=\int_{\mathbb R^n}\int_{B_{\sqrt2}(\mathbb R^{d_1})\cap B^{|t|/2\sqrt2}(\mathbb R^{d_1})}|x|^{-1} \\ &\qquad\times\biggl(\int_{\substack{y\in \sigma(u,x)\\|x|^2+|y+t|x|^{-2}x|^2\leqslant2}} f_{00>}(u,x,y+t|x|^{-2} x) \, d_{\sigma(u,x)}y \biggr)\,dx \,du . \end{aligned}
\end{equation*}
\notag
$$
Consider the function $\mathcal{I}_{00<}(t)$ first. We observe that by (7.18), for $y\in \sigma(u,x)$ and $|x|\leqslant |t|/2$ (cf. (7.25))
$$
\begin{equation*}
\bigl|y+t|x|^{-2}x\bigr|\geqslant |\tau|\,|x|^{-1}= \biggl|t-\frac12|u|^2\biggr|\,|x|^{-1}\geqslant-t|x|^{-1}>\sqrt 2 \quad \text{for } t<0 ,
\end{equation*}
\notag
$$
so that $\mathcal{I}_{00<}(t)= 0$ for $t<0$. For $t>0$, performing the change of variables $\sqrt{t} u'=u$, $tx'=x$ we obtain
$$
\begin{equation*}
\begin{aligned} \, \mathcal I_{00<}(t) &= t^{d/2-1}\int_{\mathbb R^n}\int_{B_{\sqrt2/t}(\mathbb R^{d_1})\cap B_{1/2}(\mathbb R^{d_1})} |x'|^{-1} \varphi(8|x'|^2) \\ &\quad\times \biggl(\int_{\substack{y\in \sigma(u',x')\\|x'|^2t^2+|y+|x'|^{-2}x'|^2\leqslant2}} f_{00}(\sqrt{t}\, u',tx',y+|x'|^{-2} x')\, d_{\sigma(u',x')}y \biggr)\,dx'\, du', \end{aligned}
\end{equation*}
\notag
$$
where we notice that $\sigma(u',x') =\sigma(u,x)$. We differentiate with respect to $t$, observing (using induction on $k$) that for any $l$ and $k$ we have
$$
\begin{equation*}
\begin{aligned} \, &\frac{d^k}{dt^k} t^lg(\sqrt t\, u',tx') \\ &\qquad=\sum_{l_1+l_2+l_3=k}c_{l_1,l_2,l_3} t^{l-l_1-l_2/2}({u'}^{l_2}\cdot\nabla_u)^{l_2} ({x'}^{l_3}\cdot \nabla_x)^{l_3}g(\sqrt t u',tx') \end{aligned}
\end{equation*}
\notag
$$
for any sufficiently regular function $g$ and suitable constants $c_{l_1,l_2,l_3}$. From this we obtain
$$
\begin{equation*}
\begin{aligned} \, &|\partial^k \mathcal I_{00<}(t)| \lesssim_{k,M}\max_{l_1+l_2+l_3=k}t^{d/2-1-l_1-l_2/2} \|f\|_{k,M} \int_{\mathbb R^n}{|u'|}^{l_2} \langle u'\sqrt t\rangle^{-M} \\ &\qquad \times \int_{B_{\sqrt2/t}(\mathbb R^{d_1})\cap B_{1/2}(\mathbb R^{d_1})} |x'|^{l_3-1} \biggl( \int_{\substack{y\in \sigma(u',x')\\|x'|^2t^2+|y+|x'|^{-2}x'|^2\leqslant2}}\,d_{\sigma(u',x')}y \biggr)\,dx'\, du'. \end{aligned}
\end{equation*}
\notag
$$
Denoting points in $x^\perp$ by $\overline y$ we see that the integral against $d_{\sigma(u',x')}y$ is bounded by
$$
\begin{equation}
\int_{\substack{\overline y\in x^\perp\\|x'|^2t^2+|\overline y+\tau'|x'|^{-2}x'|^2\leqslant2}} 1\,d\overline y, \qquad \tau'=1-\frac12|u'|^2.
\end{equation}
\tag{7.26}
$$
By (7.21) $|\tau'|\leqslant\sqrt2\,|x'|$ on the support of the integrand. Hence we have there
$$
\begin{equation}
1-\sqrt2\, |x'|\leqslant\frac{|u'|^2}2\leqslant 1+\sqrt2 \, |x'| .
\end{equation}
\tag{7.27}
$$
As the domain of integration with respect to $\overline y$ is bounded, the integral (7.26) is bounded by a constant. So putting $|x'|=r'$ and $|u'|=R'$ and using (7.27) we have
$$
\begin{equation*}
\begin{aligned} \, \biggl|\partial^k\mathcal I_{00<}(t)\biggr| &\lesssim_{k,M} \max_{l_1+l_2+l_3=k}\|f\|_{k,M}t^{d/2-l_1-l_2/2-1}\int_0^{1/2} {r'}^{d_1-2+l_3} \\ &\qquad\times \biggl(\int_{\sqrt2\sqrt{1-\sqrt2\, r'}}^{\sqrt2\sqrt{1+\sqrt2\, r'}} {R'}^{n-1+l_2}\langle{R'}^2t\rangle^{-M/2} \, dR' \biggr)\, dr'. \end{aligned}
\end{equation*}
\notag
$$
Since $r'\leqslant 1/ 2$, on the domain of integration we have
$$
\begin{equation*}
\sqrt{2-\sqrt2}\leqslant R'\leqslant \sqrt{2+\sqrt2},\quad\text{while}\quad \sqrt2\sqrt{1+\sqrt2\, r'}-\sqrt2\sqrt{1-\sqrt2 \, r'}\lesssim r'.
\end{equation*}
\notag
$$
So the integral in $dR'$ is bounded by $C \langle t\rangle^{-M/2} r'$. Therefore,
$$
\begin{equation*}
|\partial^k\mathcal I_{00<}(t)| \lesssim_{k,M} \max_{l_1+l_2+l_3=k}\|f\|_{k,M}\, t^{d/2-l_1-l_2/2-1} \langle t\rangle^{-M/2} \int_0^{1/2}{r'}^{d_1-1+l_3}\, dr'.
\end{equation*}
\notag
$$
This implies that for $0< t\leqslant4$, for any $k\leqslant k_*$ and any $d_1\geqslant 1$ we have
$$
\begin{equation}
|\partial^k\mathcal I_{00<}(t)|\lesssim_{k}\|f\|_{k,0}\, t^{d/2-k-1}.
\end{equation}
\tag{7.28}
$$
On the other hand, for any $t \geqslant4$ and any $k\leqslant k_*$
$$
\begin{equation}
\begin{aligned} \, \notag |\partial^k\mathcal I_{00<}(t)| &\lesssim_{k,M} \max_{l_1+l_2+l_3=k}\|f\|_{k,M,{d}}\, t^{d/2- M/2-l_1-l_2/2-1} \\ &\qquad\times\int_0^{\sqrt2/t} {r'}^{d_1-1+l_3} \, dr' \lesssim_{k,M}\|f\|_{k,M}\, t^{-(M+2+k +2d_1-{d})/2} . \end{aligned}
\end{equation}
\tag{7.29}
$$
We recall that $\mathcal{I}_{00<}(t)$ vanishes for $t<0$. For $\mathcal{I}_{00>}(t)$ we note first that by (7.20) and (7.25) the function $\mathcal{I}_{00>}(t)$ vanishes for $|t|> 4$. Next, using induction on $k$ we observe that
$$
\begin{equation}
\begin{aligned} \, \notag &\frac{d^k}{dt^k} g(tx|x|^{-2})\biggl(1-\varphi\biggl(\frac{8|x|^2}{t^2}\biggr)\biggr) \\ &\qquad=\sum_{l_1+l_2+l_3=k} c_{l_1,l_2,l_3}|x|^{2(l_2-l_1)}t^{-3l_2-l_3} ((x\cdot \nabla)^{l_1}g)\,\frac{d^{l_2}}{dy^{l_2}}(1-\varphi) , \end{aligned}
\end{equation}
\tag{7.30}
$$
where $c_{l_1,l_2,l_3}= 0$ if $l_3>0$ and $l_2=0$. Since $\varphi'\neq 0$ only for $|t|/2\sqrt2\leqslant|x|\leqslant |t|/2$, we have
$$
\begin{equation*}
\frac{d^{l_2}}{dy^{l_2}}(1-\varphi)t^{-3l_2-l_3}\lesssim_{l_2,l_3} |x|^{-3l_2-l_3} , \qquad l_2>0 ,
\end{equation*}
\notag
$$
so that
$$
\begin{equation*}
\biggl|\frac{d^k}{dt^k} g(tx|x|^{-2}) \biggl(1-\varphi\biggl(\frac{8|x|^2}{t^2}\biggr)\biggr) \biggr| \lesssim_k |x|^{-k}\|g\|_{k,0} .
\end{equation*}
\notag
$$
From here, in a way similar to (7.22), setting $|x|=r$ and $|u|=R$ again we get that
$$
\begin{equation*}
|\partial^k\mathcal I_{00>}(t)| \lesssim_{k,M} \|f\|_{k,M} \int_{|t|/(2\sqrt 2)}^{\sqrt2}r^{d_1-k-2} \biggl(\int_0^\infty R^{n-1}\langle R\rangle^{-M}\chi_{|\tau|\leqslant \sqrt 2\, r} \, dR \biggr)\,dr;
\end{equation*}
\notag
$$
here and below $\displaystyle\int_a^b dr=0$ if $b\leqslant a$. Since on the domain of integration, in view of (7.25) and the coefficient $\chi_{|\tau|\leqslant \sqrt 2\, r}$ we have $R^2\leqslant 6\sqrt 2\,r$, it follows that
$$
\begin{equation}
\begin{aligned} \, \notag |\partial^k \mathcal I_{00>}(t)| &\lesssim_{k,M,n}\|f\|_{k,M}\int_{|t|/(2\sqrt2)}^{\sqrt2}r^{d/2-k-2}\, dr \\ &\lesssim_{k,M} \begin{cases} \|f\|_{k,M} ,& k<\dfrac{d}2-1, \\ \|f\|_{k,M}(1+|{\log|t|}|) ,& k\leqslant \dfrac{d}2-1. \end{cases} \end{aligned}
\end{equation}
\tag{7.31}
$$
If $k<d/2-1$, then by the above $\partial^k \mathcal{I}_{00}(t)$ is bounded for all $t$. In this case, modifying the integrand in (7.17) by the coefficient $\chi_{|x|\geqslant\varepsilon}$, we see that the functions $\mathcal{I}_{00>}^\varepsilon$ and $ \mathcal{I}_{00<}^\varepsilon $ thus obtained satisfy the same estimates as the above functions $\mathcal{I}_{00>}$ and $ \mathcal{I}_{00<}$, so these estimates also hold for the function $\mathcal{I}_{00}^\varepsilon$. For $\varepsilon>0$ the functions $\partial^k \mathcal{I}_{00}^\varepsilon(t)$ are obviously continuous in $t$ and converge to $\partial^k \mathcal{I}_{00}(t)$ uniformly on bounded intervals. So the latter function is also continuous. In a similar way, for $k=d/2-1$, $\partial^k \mathcal{I}_{00}(t)$ is continuous on any set $|t| \geqslant\varepsilon>0$, so it is continuous for ${t\ne0}$. 7.3.2. The integral $\mathcal{I}_{11}(t)$By (7.16) and similarly to (7.17) and (7.19), for any $k\leqslant k_*$ we have
$$
\begin{equation*}
\partial^k\mathcal I_{11}(t)=\int_{\mathbb R^n}\int_{|x|\geqslant 1/2}|x|^{-1}\biggl(\int_{x^\perp} d_y^k\, f_{11}(u,x,\overline y+\tau x |x|^{-2})[x|x|^{-2}]\,d\overline y \biggr)\,dx \,du.
\end{equation*}
\notag
$$
We easily see that $\mathcal{I}_{11}(t)$ is a $C^k$-smooth function and, since $M>{d}$ and ${\bigl|\overline y+\tau x|x|^{-2}\bigr|\geqslant |\overline y|}$, we have
$$
\begin{equation}
|\partial^{k}\mathcal I_{11} (t)|\lesssim_{k,M}\|f\|_{k,M} \quad \forall \, t.
\end{equation}
\tag{7.32}
$$
Now let $|t|\geqslant 1$. Let us write $\partial^k\mathcal{I}_{11}$ as
$$
\begin{equation}
\partial^k\mathcal I_{11}(t)=\int_{\mathbb R^n} \int_{|x|\geqslant 1/2}|x|^{-k-1} \int_{x^\perp} \Phi_k(\overline z)\,d\overline y \,dx\, du,
\end{equation}
\tag{7.33}
$$
where $\overline z=(u,x,\overline y)$, $\overline y\in x^\perp$, and
$$
\begin{equation}
|\Phi_k(\overline z)| \lesssim_{k} \|f\|_{k,M} \langle \widehat z\rangle^{-M}, \quad\text{where }\widehat z=(u,x,\overline y+\tau x|x|^{-2}).
\end{equation}
\tag{7.34}
$$
Clearly,
$$
\begin{equation}
|\widehat z|\geqslant |\overline z|, \qquad |\widehat z|\geqslant 2^{-1/2} \bigl( |\overline z| + |\tau| |x|^{-1}\bigr).
\end{equation}
\tag{7.35}
$$
Below we distinguish the cases $n\geqslant 1$ and $n=0$. 1) Let $n\geqslant 1$. (a) First we integrate in (7.33) with respect to $u$ in the spherical shell
$$
\begin{equation*}
O:=\biggl\{u\colon |\tau|=\biggl|t-\frac12|u|^2\biggr|\leqslant\frac12 t\biggr\}.
\end{equation*}
\notag
$$
It is empty if $t<0$, while for $t\geqslant 0$ we have $O=\{u\colon t \leqslant |u|^2\leqslant 3t\}$. By (7.34) and the first relation in (7.35), for $t\geqslant0$ the part of the integral in (7.33) corresponding to $u\in O$ is bounded by
$$
\begin{equation*}
K :=C_k\|f\|_{k,M} \int_O \int_{|x|\geqslant 1/2}|x|^{-k-1} \int_{x^\perp} \bigl( |t|+|x|^2+|\overline y|^2\bigr)^{-M/2}\,d\overline y\, dx\, du.
\end{equation*}
\notag
$$
Since $\displaystyle\int_O 1\,du\leqslant Ct^{n/2}$, putting $r=|x|$, $|t|+r^2=T^2$ and $R=|\overline y|/T$ we find that
$$
\begin{equation*}
K \lesssim_{k}\|f\|_{k,M}t^{n/2} \int_{1/2}^{\infty}r^{d_1-2-k}T^{d_1-1-M} \int_0^\infty R^{d_1-2}(1+R^2)^{-M/2}\,dR\, dr.
\end{equation*}
\notag
$$
The integral against $dR$ is bounded since $M>d_1$, so that
$$
\begin{equation*}
K \lesssim_{k,M}\|f\|_{k,M}t^{n/2}\int_{1/2}^{\infty}r^{d_1-2-k} (|t|+r^2)^{(d_1-1-M)/2}\,dr.
\end{equation*}
\notag
$$
Recalling that we are considering the case $t\geqslant 1$, we put $r= \sqrt {t}\,l$. Then
$$
\begin{equation*}
K\lesssim_{kM}\|f\|_{k,M}t^{(n+1+d_1-2-k+d_1-1-M)/2} \int_{t^{-1/2}/2}^\infty l^{d_1-2-k} (1+l^2)^{(d_1-1-M)/2}\,dl .
\end{equation*}
\notag
$$
Since $M>2d_1$, the integral with respect to $l$ is convergent and we obtain
$$
\begin{equation*}
K \lesssim_{k,M}\|f\|_{k,M}|t|^{-(M+2-{d}+k)/2}|t|^{\max(0, k+1-d_1)/2} Y(t),
\end{equation*}
\notag
$$
where $ Y=\log t$ if $k=d_1-1$ and $Y=1$ otherwise. Then, in the case when $Y=1$ the component of (7.33) corresponding to $u\in O$ is bounded by
$$
\begin{equation}
C(k, M,d)\|f\|_{k,M}|t|^{- \kappa}, \quad\text{where } \kappa=\frac{M+2-{d}}2,
\end{equation}
\tag{7.36}
$$
for all $ |t|\geqslant 1$, since $\max(0, k+1-d_1)\leqslant k$. If $ Y=\log t$, then the same estimate holds for $d_1\geqslant 2$ since $\max(0, k+1-d_1)< k$. In the case when $d_1=1$ and $ Y=\log t$ (that is, $k=0$) we obtain (7.36) with $\kappa$ replaced by any $\kappa'<\kappa$ (and with a constant $C$ depending on $\kappa'$). (b) Now consider the integral with respect to $u\in O^c= \mathbb{R}^n\setminus O$. In this domain
$$
\begin{equation*}
|\tau|=\biggl|t-\frac12|u|^2\biggr|\geqslant \frac12 |t|.
\end{equation*}
\notag
$$
So, by inequalities (7.34) and (7.35),
$$
\begin{equation*}
|\Phi_k(\overline z) | \lesssim_{k}\langle(u,\overline y)\rangle^{-M}\quad\text{and}\quad |\Phi_k(\overline z) |\lesssim_{k}(|t||x|^{-1} +|x|)^{-M}.
\end{equation*}
\notag
$$
Let $M=M_1+M_2$, $M_j\geqslant 0$. Then the part of the integral (7.33) corresponding to $u\in O^c$ is bounded by
$$
\begin{equation*}
C\|f\|_{k,M} \int_{|x|\geqslant 1/2} |x|^{-1-k}(t|x|^{-1}+|x|)^{-M_1} \biggl(\int_{\mathbb R^n}\int_{x^\perp} \langle(u,\overline y)\rangle^{-M_2}\,d\overline y \,du \biggr)\,dx.
\end{equation*}
\notag
$$
Choosing $M_2=n+{d_1}-1+\gamma$, where $0<\gamma<1$ (then $M_1, M_2>0$ since $M>d$), we achieve that the integral against $du\,d\overline y$ is bounded by $C(\gamma)$ for any $\gamma$. Since by Young’s inequality6 for any $A,B>0$, we have
$$
\begin{equation*}
(t|x|^{-1}+|x|)^{-M_1}\leqslant C_a|x|^{(2a-1)M_1}|t|^{-aM_1}\quad\text{for } 0<a<1.
\end{equation*}
\notag
$$
So the above integral is bounded by
$$
\begin{equation*}
C(\gamma)\|f\|_{k,M}|t|^{-aM_1}\int_{|x|\geqslant 1/2}|x|^{-1-k+bM_1}\,dx, \quad\text{where } b=2a-1 \in (-1, 1) .
\end{equation*}
\notag
$$
Set $b_*=(1+k-d_1)/{M_1}$. Then for $b=b_*$ the exponent of $|x|$ in the above formula is $-{d_1}$, and $b_*>-1$ if $\gamma$ is sufficiently small, since $M>d$. Noting that
$$
\begin{equation*}
a(b_*)M_1=\frac{b_*+1}2M_1=\frac{M+2+k-{d} -\gamma}2= \kappa +\frac{k}2 - \frac\gamma2
\end{equation*}
\notag
$$
($\kappa$ was defined in (7.36)) we see that
$$
\begin{equation}
\begin{aligned} \, &\text{for } k\geqslant1 \text{ the part of integral (7.33) corresponding}\notag \\ \notag &\text{to }u\in O^c \text{ is bounded by the quantity (7.36)}, \\ &\text{while for }k=0\text{ it is bounded by the quantity }(7.36)\notag \\ &\text{with }\kappa\text{ replaced by any }\kappa'<\kappa. \end{aligned}
\end{equation}
\tag{7.37}
$$
2) Now let $n=0$. Then
$$
\begin{equation}
|\partial^k\mathcal I_{11}(t)|\leqslant \int_{|x|\geqslant 1/2} |x|^{-1-k} \int_{x^\perp} \Phi_k(\overline z) \, d\overline y\, dx , \qquad \overline z=(x,\overline y),
\end{equation}
\tag{7.38}
$$
where $|\Phi_k(\overline z) |{ \lesssim_k}\langle \widehat z\rangle^{-M}$ for $\widehat z = (x,\overline y+ tx|x|^{-2})$. Repeating literally the above step 1), (b), for $n=0$ we see that for $|t|\geqslant 1$ the integral in (7.38) can also be bounded by (7.36). We recall that for $|t|\leqslant 1$ the derivative $\partial^k\mathcal{I}_{11}(t)$ was estimated in (7.32). 7.3.3. The integral $\mathcal{I}_{10}(t)$Now we use the second disintegration in (7.13) instead of the first. Since by (7.16) $|y|\geqslant 1/\sqrt 2$ on the support of the integrand, repeating the above argument for $x$ and $y$ swapped we see that $\mathcal{I}_{10}(t)$ meets the same estimates as $\mathcal{I}_{11}(t)$. 7.3.4. The end of the proof of Theorem 7.1Finally,
For the reason explained at the end of § 7.3.1, the derivatives involved are continuous functions. This proves the theorem. Theorem 7.1 is proved. 7.4. Linear transformations of quadricsIn this subsection we denote by $C_0$ spaces of continuous functions with compact support. In $\mathbb{R}^{d}=\{z\}$ we consider a quadratic form with real coefficients7 $F(z)=\frac12 Az \cdot z$ of signature $(n_0, n_+, n_-)$ such that $n_0=0$ and $n_+ \geqslant n_- =: d_1\geqslant1$. Set $n=n_+ - n_-$. Using the standard diagonal normal form of a symmetric quadratic form, we construct a linear transformation
$$
\begin{equation*}
L\colon \mathbb R^{d} \to \mathbb R^{d}, \qquad z \mapsto Z=(u,x,y), \quad u\in \mathbb R^n, \quad x, y\in \mathbb R^{d_1},
\end{equation*}
\notag
$$
such that $Q(L(z)) = F(z)$, where $Q(Z)=\frac 12 |u|^2 + x\cdot y$. Consider the corresponding quadrics $\Sigma^Q_t=\{Z\colon Q(Z)=t\}$ and $\Sigma^F_t=\{z\colon F(z)=t\}$ and the $\delta$-measures $\mu^Q_t$ and $\mu^F_t$ on them (for example, see [14], § II.7):
$$
\begin{equation}
\langle \mu^Q_t, f^Q \rangle =\lim_{\varepsilon\to 0} \frac1{2\varepsilon} \int_{t-\varepsilon \leqslant Q(Z) \leqslant t+\varepsilon} f^Q(Z)\, dZ
\end{equation}
\tag{7.39}
$$
and
$$
\begin{equation*}
\langle \mu^F_t, f^F \rangle =\lim_{\varepsilon\to 0} \frac1{2\varepsilon} \int_{t-\varepsilon \leqslant F(z) \leqslant t+\varepsilon} f^F(z)\, dz,
\end{equation*}
\notag
$$
where $f^Q, f^F \in C_0(\mathbb{R}^d)$ and $\langle \mu, f\rangle$ denotes the integral of the function $f$ against the measure $\mu$. Then $\mu^Q_t$ and $\mu^F_t$ are Borel measures in $\mathbb{R}^d$ with supports in $\Sigma_t^Q$ and $\Sigma_t^F$, respectively, and for $f^Q \in C_0(\Sigma_t^Q\setminus\{0\})$ and $f^F \in C_0(\Sigma_t^F\setminus \{0\})$ we have
$$
\begin{equation*}
\langle \mu^Q_t, f^Q \rangle=\int_{\Sigma^Q_t} \frac{f^Q(Z)}{|\nabla Q(Z)|}\, dZ|_{\Sigma^Q_t} \quad\text{and}\quad \langle \mu^F_t, f^F \rangle=\int_{\Sigma^F_t} \frac{f^F(z)}{|\nabla F(z)|}\, dz|_{\Sigma^F_t},
\end{equation*}
\notag
$$
where $dZ|_{\Sigma^{Q}_t}$ ($dZ|_{\Sigma^{F}_t}$) is the volume element on $\Sigma^{Q}_t\setminus\{0\}$ (on $\Sigma^{F}_t\setminus\{0\}$, respectively), induced from $\mathbb{R}^{d}$; see [14]. Now let $f^F = f^Q \circ L$. Then the integral in (7.39) equals
$$
\begin{equation*}
\int_{t-\varepsilon \leqslant Q(Z) \leqslant t+\varepsilon} f^Q(Z)\, dZ=|{\det L}| \int_{t-\varepsilon \leqslant F(z) \leqslant t+\varepsilon} f^F(z)\, dz,
\end{equation*}
\notag
$$
so passing to the limit we get that Thus, to examine the function
$$
\begin{equation}
t\mapsto \mathcal I^F(t;f)=\langle \mu^F_t, f\rangle, \quad\text{where } \mu^F_t=| \nabla F(z)|^{-1}dz|_{\Sigma^F_t},
\end{equation}
\tag{7.41}
$$
we are free to use any linear coordinate system in $\mathbb{R}^{d}$, since by changing the coordinates we only modify $\mathcal{I}^F$ by a constant factor. 7.5. Sign-definite formsFinally, consider the case when $n_0=0$ and $\min(n_+, n_-)=0$, that is, when the form $F(z)=\frac12 Az \cdot z$ is sign definite and nondegenerate. Suppose for definiteness that $n_-=0$. Then there exists a linear transformation $L$ such that $F(z)=Q(L(z))$, where $Q(Z)=\frac12|Z|^2$, $Z\in \mathbb{R}^d$. The quadric $\Sigma_t$ reduces to the empty set for $t<0$, so the function $\mathcal{I}^F(t)$ (see (7.41)) vanishes for $t<0$. The calculation in § 7.4 remains true in this case, so (7.40) and the change of coordinates $Z=\sqrt{2t}\,Z'$ show that
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, \mathcal I^F(t;f) &=C(d,L) t^{-1}\int_{|Z|=\sqrt{2t} } f^Q( Z)\mu_{S^{d-1}_{\sqrt{2t}} }(dZ) \\ &=C(d,L) t^{d/2-1}\int_{|Z'|=1} f^Q(\sqrt{2 t}\, Z')\mu_{S_1^{d-1}}(dZ'), \end{aligned} \\ t > 0, \quad f^Q=f\circ L^{-1}, \end{gathered}
\end{equation*}
\notag
$$
where $\mu_{S_r^{d-1}}$ is the volume element on the $(d-1)$-sphere of radius $r$. From this relation we immediately see that, for any $k\leqslant \min(d/2-1,k_*)$,
$$
\begin{equation*}
\begin{alignedat}{2} |\partial^k\mathcal I^F(t)| &\lesssim_{k} \|f\|_{k,0} &\quad &\text{if } 0\leqslant t\leqslant1, \\ |\partial^k\mathcal I^F(t)| &\lesssim_{k,M} \|f\|_{k,M} t^{-(M+2+k-d)/2} &\quad &\text{if } t\geqslant1. \end{alignedat}
\end{equation*}
\notag
$$
7.6. The general resultWe sum up the results obtained in the following statement. Theorem 7.3. Consider an arbitrary nondegenerate quadratic form $F(z)=\frac12 Az \cdot z$ on $\mathbb{R}^{d}$, $d\geqslant 3$, and a function $f\in\mathcal{C}^{k_*,M}(\mathbb{R}^d)$, $M>d$. Then the assertions of Theorem 7.1 hold for the corresponding integral (see (7.41)). Proof. i) If $n_+\geqslant n_-$, then by means of a linear change of variable $F$ can be reduced to the normal form (7.1), where $d_1\geqslant0$. Now the assertion follows from the argument in §§ 7.4 and 7.5 and Theorem 7.1.
ii) If $n_- >n_+$, then the quadratic form $-F$ is as in i), and the assertion follows again since, obviously, The theorem is proved. § 8. Appendix8.1. The term $J_0$: the case $d=4$In this section we find the asymptotic behaviour of the term $J_0$ in (1.19) in the case when Throughout this section we always assume that (8.1) holds.8.1.1. Preliminary results and definitionsWe will need Lemmas 30 and 31 from [1] in the case $m=0$, $d=4$, which we state below without proofs. Recall that the constants $\sigma^*_\mathbf{c}(A)$ are defined in (1.10) and $\sigma^*(A)=\sigma^*_{\mathbf 0}(A)$. Set $\alpha:=7/2$ and recall (8.1). Lemma 8.1 (Lemma 30 in [1]). For any $\varepsilon>0$ and $X\in\mathbb{N}$, where $\eta(\mathbf{c})=1$ if $\mathbf{c}\cdot A^{-1}\mathbf{c} = 0$ and, at the same time, $\det A$ is a square of an integer, and $\eta(\mathbf{c}) = 0$ otherwise. Moreover, $|\sigma_\mathbf{c}^*(A)|\lesssim_{\varepsilon} 1+|\mathbf{c}|^\varepsilon$ when $\eta(\mathbf{c})\neq 0$. Lemma 8.2 (Lemma 31 in [1]). Assume that the determinant $\det A$ is a square of an integer. Then for any $\varepsilon>0$ and $X\in\mathbb{N}$, where $\widehat C_A$ is a constant depending only on $A$. Otherwise, if $\det A$ is not a square of an integer, then for any $\varepsilon>0$ and $X\in\mathbb{N}$ where $\chi$ is the Jacobi symbol $\biggl(\dfrac{\det A}{*}\biggr)$ and $L(1,\chi)$ is the Dirichlet $L$-function. We also need the following construction. For $r\in\mathbb{R}_{>0}$ set
$$
\begin{equation}
I^*(r) :=\widetilde I_{rL}(0)=\int_{\mathbb R^{d}} w(\mathbf z)h(r,F^0(\mathbf z))\,d\mathbf z.
\end{equation}
\tag{8.3}
$$
Consider the function $K(\rho;w,A)$, $\rho\in\mathbb{R}_{>0}$ defined by
$$
\begin{equation}
\begin{aligned} \, K(\rho) &:=\eta(0)\sigma^*(A)\biggl( \sigma_\infty(w;A,0)\log \rho \nonumber \\ &\qquad +\int_{\rho}^\infty r^{-1}I^*(r)\,dr\biggr) +\sigma_\infty(w;A,0)\widehat C_A, \end{aligned}
\end{equation}
\tag{8.4}
$$
where the constant $\eta(0)$ is defined in accordance with Lemma 8.1 and $\widehat C_A$ is defined in accordance with Lemma 8.2. Note that the functions $I^*(r)$ and $K(\rho)$ do not depend on $L$. We claim that the function $K(\rho)$, $\rho>0$, can be extended to $\rho=0$ by continuity. Indeed, for $0<\rho_1<\rho_2\leqslant 1$
$$
\begin{equation}
K(\rho_2)-K(\rho_1)=\eta(0)\sigma^*(A)\biggl( \sigma_\infty(w;A,0) \log\frac{\rho_2}{\rho_1} - \int_{\rho_1}^{\rho_2}r^{-1}I^*(r)\,dr\biggr).
\end{equation}
\tag{8.5}
$$
Using that $I^*(r)=L^{-d} I_{rL}(0)$ (see (3.8)) we write the term $I^*(r)$ from (8.5) in the form given by Proposition 3.8, b). Then $I^*(r)$ takes the form of the right-hand side of (3.11) divided by $L^d$ for ${q=rL}$. The leading term in the resulting formula for $I^*(r)$ is $\sigma_\infty(w;A,0)$, and the corresponding integral $\displaystyle\int_{\rho_1}^{\rho_2}r^{-1}\sigma_\infty\,dr$ in (8.5) annihilates the first term in brackets in (8.5). Then, setting $M=d/2-1,$ $\beta=r^{\overline \gamma}$, $\overline\gamma =\gamma/d$ and $0<\gamma<1$ in the formula for $I^*(r)$ obtained from (3.11) as just mentioned, we obtain the estimate
$$
\begin{equation*}
\begin{aligned} \, |K(\rho_2)-K(\rho_1)| &\lesssim_{N} \|w\|_{d/2-1,d+1}\int_{\rho_1}^{\rho_2} \bigl(r^{(1-\overline \gamma)d/2-2}\langle \log r\rangle+ r^{N-2}+r^{\overline \gamma N-2}\bigr)\,dr \\ &\lesssim_\gamma\rho_2^{d/2-1-\gamma}\|w\|_{d/2-1,d+1} . \end{aligned}
\end{equation*}
\notag
$$
The last inequality here has been obtained by choosing $N=N(\gamma)$ to be sufficiently large and writing
$$
\begin{equation*}
r^{d/2(1-\overline \gamma)-2}\langle \log r\rangle \lesssim_\gamma r^{d/2(1-\overline \gamma)-2 - \gamma/2}=r^{d/2 -2 - \gamma}.
\end{equation*}
\notag
$$
Therefore, $K(\rho)$ extends to $\rho=0$ by continuity and
$$
\begin{equation}
| K(\rho)- K(0)|\lesssim_{\gamma}\rho^{d/2-1-\gamma} \|w\|_{d/2-1,d+1},
\end{equation}
\tag{8.6}
$$
so the function $K$ is $(d/2-1-\gamma)$-Hölder continuous at zero for any $\gamma>0$. 8.1.2. An estimate for $J_0$The argument in this subsection is related to § 13 of [1]. Here we restrict ourselves to the case when the determinant $\det A$ is the square of an integer, so that, in particular, $\eta(0)=1$. We use this specification only in the proof of Lemma 8.5, when we use Lemma 8.2. The case of a nonsquare determinant is easier and can be treated similarly, using the second assertion of Lemma 8.2. Proposition 8.3. Assume that the determinant $\det A$ is the square of an integer. Then for any $0<\varepsilon<1/5$, Proof. To establish Proposition 8.3 we write $J_0$ in the form (1.21), $J_0=J^+_0+J^-_0$, where
$$
\begin{equation*}
J^+_0:=\sum_{q>\rho L} q^{-d}S_q(0) I_q(0) \quad\text{and}\quad J_0^-:=\sum_{q\leqslant \rho L} q^{-d}S_q(0) I_q(0),
\end{equation*}
\notag
$$
for some $\rho\leqslant 1$. Then the assertion follows from Lemmas 8.4 and 8.5 below. Recall that $\alpha=7/2$.
Lemma 8.4. Let $w\in L_1(\mathbb{R}^{d})$. Then for any $\gamma>0$, any $\rho\leqslant 1$ and $L$ satisfying $\rho L>1$, Proof. To simplify the notation, in this proof we set $I_q:=I_q(0)$ and $S_q:=S_q(0)$. We recall the formula of summation by parts for sequences $(f_q)$ and $(g_q)$:
$$
\begin{equation*}
\sum_{m<q\leqslant n} f_q (g_q-g_{q-1})=f_ng_n-f_{m+1}g_{m} - \sum_{m<q<n} (f_{q+1}-f_q)g_q.
\end{equation*}
\notag
$$
We fix an arbitrary $R\in\mathbb{N}$ and use this formula for $m=R$, $n=2R$, $f_q=q^{-d}I_q$ and $g_q=\sum_{R< q'\leqslant q} S_{q'}$, so that $g_R=0$ and $S_q=g_q-g_{q-1}$ for $q>R$. We find that
$$
\begin{equation}
\sum_{R< q\leqslant 2R} q^{-d}S_qI_q=(2R)^{-d}I_{2R} \sum_{R< q\leqslant 2R} S_q- \sum_{R< q< 2R}\widetilde\partial_q(q^{-d}I_q) \sum_{R< q'\leqslant q} S_{q'} ,
\end{equation}
\tag{8.7}
$$
where, given a sequence $(a_q)$, we set $\widetilde\partial_q a_q:=a_{q+1}-a_q$. By (3.8) and (3.9),
$$
\begin{equation*}
I_q=L^{d}\int_{\mathbb R^{d}} w(\mathbf z) h\biggl(\frac qL,F^0(\mathbf z)\biggr)\,d\mathbf z.
\end{equation*}
\notag
$$
So
$$
\begin{equation}
|I_q|\lesssim \frac{L^{d+1}}{q} |w|_{L_1} \quad\text{and}\quad |\partial_q I_q|\lesssim \frac{L^{d+1}}{q^2} |w|_{L_1} ,
\end{equation}
\tag{8.8}
$$
where the first estimate above follows from Corollary 3.3 while the second follows from Lemma 3.2 for $m=1$ and $n=N=0$. Then $|\widetilde\partial_q(q^{-d}I_q)|\lesssim L^{d+1}q^{-d-2} |w|_{L_1}$. According to (8.2) for $\varepsilon$ replaced by $\gamma$, for $R'\leqslant 2R$ we have
$$
\begin{equation}
\sum_{R< q\leqslant R'} S_{q}=\eta(0)\sigma^*(A)\sum_{R< q\leqslant R'}q^{d-1} + O_{\gamma}(R^{\alpha+\gamma}),
\end{equation}
\tag{8.9}
$$
where we recall that $\sigma_{\mathbf 0}^*(A)=\sigma^*(A)$. Let us view the right-hand side of (8.7) as a linear functional $G((S_q))$ on the space of sequences $(S_q)$. Then substituting (8.9) into the right-hand side of (8.7) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\sum_{R< q\leqslant 2R}q^{-d}S_qI_q=\eta(0)\sigma^*(A) G\bigl((q^{d-1})\bigr) \\ &\qquad\qquad+ O_\gamma\biggl( L^{d+1}|w|_{L_1}\biggl(R^{-d-1+\alpha +\gamma} + \sum_{R< q\leqslant 2R} q^{-d-2+\alpha+\gamma}\biggr)\biggr), \end{aligned}
\end{equation}
\tag{8.10}
$$
where the $O_\gamma$-term is obtained by applying (8.8), together with the above estimate for $\widetilde\partial_q(q^{-d}I_q)$, and replacing the sums $\sum S_q$ and $\sum S_{q'}$ on the right-hand side of (8.7) by $O_{\gamma}(R^{\alpha+\gamma})$. According to formula (8.7) of summation by parts in which $S_q$ is replaced by $q^{d-1}$, we have $\sum_{R< q\leqslant 2R}q^{-d}q^{d-1}I_q=G((q^{d-1}))$. Thus, by (8.10),
$$
\begin{equation*}
\sum_{R< q\leqslant 2R} q^{-d}S_qI_q=\eta(0)\sigma^*(A)\sum_{R< q\leqslant 2R} q^{-1}I_q +O_\gamma\bigl(L^{d+1}R^{-d-1+\alpha +\gamma} |w|_{L_1}\bigr) .
\end{equation*}
\notag
$$
Then, setting $R_l=\lfloor 2^l\rho L \rfloor$ we obtain
$$
\begin{equation*}
\begin{aligned} \, J_0^+ &=\sum_{l=0}^\infty\sum_{R_l<q \leqslant R_{l+1}}q^{-d}I_qS_q \\ &=\eta(0)\sigma^*(A)\sum_{q> \rho L} q^{-1}I_q + O_\gamma\biggl( \rho^{\alpha+\gamma-d-1}L^{\alpha+\gamma}|w|_{L_1}\sum_{l=0}^\infty 2^{-l(d+1-\alpha-\gamma)}\biggr) \\ &=\eta(0)\sigma^*(A)\sum_{q> \rho L} q^{-1}I_q +O_\gamma\bigl(\rho^{\alpha+\gamma-d-1}L^{\alpha+\gamma}|w|_{L_1}\bigr) . \end{aligned}
\end{equation*}
\notag
$$
It remains to compare the sum $A:=\sum_{q> \rho L} q^{-1}I_q$ with the integral Since $L^dI^*(r)=I_{rL}$, after changing the variable of integration $r$ to $q=rL$, $B$ takes the form Then
$$
\begin{equation}
|A-B|\leqslant \biggl|\sum_{q> \rho L} q^{-1}I_q - \int_{\lfloor \rho L\rfloor +1 }^\infty q^{-1}I_{q}\,dq\biggr| + \biggl|\int_{\rho L}^{\lfloor \rho L\rfloor +1 } q^{-1}I_{q}\,dq\biggr|.
\end{equation}
\tag{8.11}
$$
By (8.8), $|q^{-1}I_{q}|\lesssim q^{-2}L^{d+1}|w|_{L^1}$ and $|\partial_q (q^{-1}I_{q})|\lesssim q^{-3}L^{d+1}|w|_{L^1}$. Thus, both terms on the right-hand side of (8.11) are bounded by
Lemma 8.4 is proved. Recall that $\widehat C_A$ is the constant arising in Lemma 8.2. Lemma 8.5. Assume that the determinant $\det A$ is the square of an integer. Then for any $\gamma>0$ and $N>1$, any $\rho\leqslant 1$ and $L$ satisfying $\rho L>1$, Proof. Substituting Proposition 3.8, b), for $M=d/2-1=1$ and $\beta=1$ into the definition of the term $J_0^-$, we obtain for
$$
\begin{equation*}
I_A :=L^{d}\sigma_\infty(w)\sum_{q\leqslant \rho L} q^{-d}S_q(0) \quad\text{and}\quad I_B:=\sum_{q\leqslant\rho L}S_q(0) q^{-d} (f_q+g_q) ,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
|f_q|\lesssim q L^{d-1}\biggl\langle\log\frac{q}{L}\biggr\rangle \|w\|_{d/2-1,d+1}
\end{equation*}
\notag
$$
and By Lemma 8.2,
$$
\begin{equation*}
\sum_{q\leqslant \rho L} q^{-d}S_q(0)=\sigma^*(A)\log(\rho L) +\widehat C_A+O_{\gamma}((\rho L)^{\alpha+\gamma-d}).
\end{equation*}
\notag
$$
So
$$
\begin{equation*}
I_A=L^{d}\sigma_\infty(w)\bigl(\sigma^*(A)\log(\rho L) +\widehat C_A\bigr) + O_{\gamma}(\sigma_\infty(w)L^{\alpha+\gamma}\rho^{\alpha+\gamma-d}),
\end{equation*}
\notag
$$
while
$$
\begin{equation}
|\sigma_\infty(w)|=|\sigma_\infty(w;A,0)|=|\mathcal I(0)| \leqslant \|\mathcal I\|_{0,0}\lesssim_A\|w\|_{0,d+1}
\end{equation}
\tag{8.12}
$$
on account of (3.13). As for the term $I_B$, since $d=4$, Lemma 2.1 implies that for $N\geqslant 2$. The estimates obtained for $I_A$ and $I_B$ imply the required assertion.
The lemma is proved. Now we complete the proof of Proposition 8.3. The leading term of $J_0$ is the sum of the leading terms in the formulae for $J_0^+$ and $J_0^-$ in Lemmas 8.4 and 8.5. Since $\eta(0)=1$, it takes the form
$$
\begin{equation*}
\begin{aligned} \, & L^d\sigma^*(A)\biggl(\int_\rho^\infty r^{-1}I^*(r)\,dr + \sigma_\infty(w)\log(\rho L)\biggr) + L^d\sigma_\infty(w) \widehat C_A \\ &\qquad=\sigma_\infty(w)\sigma^*(A) L^d\log L +K(0) L^{d} + O_\gamma\bigl(L^{d}\rho^{d/2-1-\gamma}\|w\|_{d/2-1,d+1}\bigr), \end{aligned}
\end{equation*}
\notag
$$
where we used (8.4) and (8.6) in the last equality. Then we find that
$$
\begin{equation*}
\begin{aligned} \, J_0 &=\sigma_\infty(w)\sigma^*(A)L^d\log L + K(0) L^d+ O_{\gamma,N}\bigl((\rho^{\alpha+\gamma-d-1}L^{\alpha+\gamma} + \rho^{-2}L^{d-1} \\ &\qquad+L^d(\rho^{d/2-1-\gamma}+\rho\log L +\rho^{N-1}+L^{1-d}))\|w\|_{d/2-1,d+1}\bigr) , \end{aligned}
\end{equation*}
\notag
$$
since $|w|_{L_1}\lesssim\|w\|_{0,d+1}$. Now we take $\rho = L^{-1/5}$ and $N=2$, and, using that $d=4$, we obtain the assertion of the proposition. Proposition 8.3 is proved. 8.1.3. An estimate for $\sigma_1(w;A,L)$In this section we obtain an upper bound for the subleading-order term $\sigma_1$ of the asymptotics in Theorem 1.4. In the case when the determinant $\det A$ is not a square of an integer, $\sigma_1$ is given by (1.14) and the task is not complicated. Indeed, according to Lemma 8.2, the product $\prod_p(1-\chi(p)p^{-1})\sigma_p(A,0)$ is finite (and independent of $L$). On the other hand $|\sigma_\infty(w;A,0)|\lesssim \|w\|_{0,d+1}$ by (8.12). Thus, In the case when $\det A$ is a full square, $\sigma_1$ is given by (1.24) and the required estimate is less trivial. Proof. Since $\eta(\mathbf{c})$ takes value $0$ or $1$, according to the definition (1.24) of $\sigma_1$, we have
First we estimate the term $K(0)$. According to (8.6), On the other hand $\sigma^*(A)$ is independent of $L$, and in view of Lemma 8.2 it is finite. Then, by the definition (8.4) of $K(\rho)$,
$$
\begin{equation*}
|K(1)|\lesssim \int_1^\infty r^{-1} |I^*(r)|\, dr + |\sigma_\infty(w;A,0)\widehat C_A|.
\end{equation*}
\notag
$$
By the definition (8.3) of the integral $I^*(r)$ and Corollary 3.3,
$$
\begin{equation*}
|I^*(r)|\lesssim r^{-1 }|w|_{L_1}\lesssim r^{-1}\|w\|_{0,d+1}.
\end{equation*}
\notag
$$
Then, in view of (8.12), $|K(1)|\lesssim \|w\|_{0,d+1}$, so that, by (8.14),
Let us now estimate the terms $\sigma_\infty^\mathbf{c}(w)$, which are given by (1.23):
$$
\begin{equation*}
\sigma_\infty^\mathbf c(w)=L^{-d}\sum_{q=1}^\infty q^{-1} I_q(\mathbf c;A,0,L)=Y_1(\mathbf c) + Y_2(\mathbf c),
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
Y_1=L^{-d} \sum_{q=1}^{L|\mathbf c|^{-M}}q^{-1}I_q(\mathbf c) \quad\text{and}\quad Y_2=L^{-d} \sum_{q>L|\mathbf c|^{-M}}q^{-1}I_q(\mathbf c),
\end{equation*}
\notag
$$
and $M\in\mathbb{N}$ will be chosen in what follows. Using that $d=4$, according to Lemma 6.2, we have
$$
\begin{equation*}
|Y_1(\mathbf c)|\lesssim_\gamma L^{-1+\gamma}|\mathbf c|^{-1+\gamma}C(w) \sum_{q=1}^{L|\mathbf c|^{-M}} q^{-\gamma}\lesssim|\mathbf c|^{-(1-\gamma)(M+1)}C(w),
\end{equation*}
\notag
$$
where we set $C(w):=\|w\|_{\overline N,d+5} + \|w\|_{0,\overline N+3d+4}.$ On the other hand, by Proposition 5.1,
$$
\begin{equation*}
|I_q(\mathbf c)|\lesssim_N L^{d+1}q^{-1}|\mathbf c|^{-N}\| w\|_{N,2N+d+1}
\end{equation*}
\notag
$$
for every $N\in\mathbb{N}$. Hence
$$
\begin{equation*}
|Y_2(\mathbf c)|\lesssim_N L|\mathbf c|^{-N}\|w\|_{N,2N+d+1} \sum_{q>L|\mathbf c|^{-M}} q^{-2} \lesssim |\mathbf c|^{-N+M} \|w\|_{N,2N+d+1}.
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
|\sigma_\infty^\mathbf c(w)|\lesssim_{\gamma,N} \bigl(|\mathbf c|^{-(1-\gamma)(M+1)} + |\mathbf c|^{-N+M}\bigr) \bigl(\|w\|_{\overline N,\overline N+3d+4} + \|w\|_{N,2N+d+1}\bigr).
\end{equation*}
\notag
$$
By Lemma 8.1, $|\sigma_\mathbf{c}^*(A)|\lesssim_\gamma 1+ |\mathbf{c}|^\gamma$ if $\eta(\mathbf{c})=1$, so that
$$
\begin{equation*}
\sum_{\mathbf c \ne 0\colon \eta(\mathbf c)=1} |\sigma_\mathbf c^*(A) \sigma_\infty^\mathbf c(w)|\lesssim_{\gamma,N} \|w\|_{\overline N,\overline N+3d+4} + \|w\|_{N,2N+d+1},
\end{equation*}
\notag
$$
provided that $M$ and $N-M$ are sufficiently large and $\gamma$ is sufficiently small. Choosing $M=d$, $N=2d+1$ and $\gamma=1/(d+3)$ we obtain Together with (8.13) and (8.15), this implies the assertion of the proposition.
Proposition 8.6 is proved. 8.2. The constants $\sigma(A,0)$ and $\sigma^*(A)$It is clear that Theorem 1.3 (or 1.4) provides an approximation to the series $N_L(w;A,m)$ in terms of the singular integral $\sigma_\infty(w)$ only if the singular series $\sigma(A,m)$ (or $\sigma^*(A)$) is strictly positive. In fact, this singular series is known to be strictly positive under very general assumptions, namely, for nonsingular forms of any degree that have nonsingular solutions in $\mathbb{R}$ and in every $p$-adic field (provided the singular series is absolutely convergent); see § 7 in [15], for example. However, since the most interesting case in applications to mathematical physics is the case of the quadratic form $F_d(x,y)$ below, we present here a direct elementary evaluation of the constants $\sigma(A, 0)$ for $d\geqslant 5$ and $\sigma^*(A)$ for $d=4$ in this case, independently of the general theory. In this subsection we consider the case when the quadratic form reads
$$
\begin{equation}
F(x,y)=\sum_{i=1}^{d/2} x_iy_i=: F_d(x,y), \quad \text{where } d=2s\geqslant 4,
\end{equation}
\tag{8.16}
$$
and $x=(x_1,\dots,x_{s})$ and $y=(y_1,\dots,y_{s})$. Our goal is to evaluate the constants $\sigma(A, 0)$ for $d\geqslant 5$ and $\sigma^*(A)$ for $d=4$. Below we use the usual notation to indicate that an integer $m$ divides or does not divide an integer vector $s$ (for example, $2\mid (8,6)$ and $2\nmid(8,7)$). In view of the definitions (1.10) and (1.11), our first aim is to compute the constants $\sigma_p(A,0)$. For a prime number $p$ and $k\in\mathbb{N}$ consider the set
$$
\begin{equation*}
S_p(k)=\{(x,y)\ \operatorname{mod} p^{k}\colon F_d(x,y)=0\ \operatorname{mod} p^{k} \}
\end{equation*}
\notag
$$
and let $ N_p(k):=\sharp S_p(k)$. Note that the set $S_p(k)$ and the constant $N_p(k)$ depend on $d$. Then the constants $\sigma_p$ can be expressed as
$$
\begin{equation}
\sigma_p(d):=\sigma_p(A,0)=\lim_{k\to \infty}\frac{N_p(k)}{p^{(d-1)k}}.
\end{equation}
\tag{8.17}
$$
This relation was mentioned in [1], p. 199, without a proof; we sketch its rigorous derivation at the end of this section. Let ${\mathcal N}_p(d):= N_p(1)$ be the number of $\mathbb{F}_p$-rational points on the hypersurface $\{F_d=0 \ \operatorname{mod} p\}$. Proof. For $j=0,1, \dots, k$ we define $S_p(k,j)$ as a set of $(x,y) \in S_p(k)$ such that
$$
\begin{equation*}
(x,y)=p^j (x',y')\ \operatorname{mod}p^k, \quad\text{where } p \nmid (x',y').
\end{equation*}
\notag
$$
So $S_p(k,0)=\{(x,y) \in S_p(k)\colon p \nmid (x,y) \}$ and $S_p(k,k)=\{(0,0)\}$. Two sets $S_p(k,j) $ and $S_p(k,j') $ with $j\ne j'$ do not intersect, and putting $ {N}_p(k,j)=\sharp {S}_p(k,j)$ we obtain In particular, ${N}_p(1,0)={\mathcal N}_p-1$ since $ {N}_p(1,1)=1$. We claim that and thus Indeed, we argue by induction on $k$. Let $k=2$ and $(x,y) \in S_p(2,0)$. Let us express $(x,y)$ as $(x_0+pa, y_0+pb)$, where $(x_0, y_0)$, $(a,b) \in \mathbb{F}_p^d$. Then $p \nmid (x_0,y_0) $, so that $(x_0, y_0)\in S_p(1,0)$.
Now we fix some $(x_0, y_0)\in S_p(1,0)$ and look for $(a,b)\in \mathbb{F}_p^{d}$ such that $(x_0+ pa, y_0+pb)\in S_p(2,0)$. Since $p^2 F(a,b) =0 \ \operatorname{mod} p^2$ and $p \nmid (x_0,y_0)$, the relation $F(x,y) =0$ mod $ p^2$ implies a nontrivial linear equation on $(a,b) \in \mathbb{F}_p^d$. So each $(x_0,y_0) \in S_p(1,0)$ generates exactly $p^{d-1}$ vectors $(x,y)\in {S}_p(2,0)$, which proves the formula for $k=2$. This argument remains valid for any $k\geqslant 2$, provided that we represent $(x,y) \ \operatorname{mod} p^{k}$ in the form $(x_0+p^{k-1}a, y_0+p^{k-1}b)$, where $(x_0,y_0)\in\mathbb{F}_{p^{k-1}}^d$ and $(a,b) \in \mathbb{F}_p^d$. Now let $(x,y)\in {S}_p(k,j)$ with $j\geqslant 1$. Then $(x,y)=p^{j}(x',y') \ \operatorname{mod} p^k$, where ${p \nmid (x',y')}$ and $(x',y')$ satisfies $p^{2j}F(x',y')=0 \ \operatorname{mod} p^k$. Thus, $(x',y')\,{\in}\, {S}_p(k- 2j,0)$, if $j\leqslant (k-1)/2$, that is, $j\leqslant \lfloor(k-1)/2\rfloor=:j_k$. The correspondence $(x,y) \mapsto (x',y')$ is a well-defined map from $S_p(k,j)$ to $S_p(k- 2j,0)$. Indeed, if $(x_1,y_1)\sim(x,y)$ in $S_p(k,j)$, then $p^{k-j} \mid((x'_1, y'_1) - (x',y'))$, so $(x'_1,y'_1) \sim (x',y')$ in $S_p(k-2j,0)$. Since this map is obviously surjective, it is a bijection of $S_p(k,j)$ onto $S_p(k-2j,0)$, which implies in view of (8.19) that
$$
\begin{equation*}
{N}_p(k,j)={N}_p(k-2j,0)=({\mathcal N}_p-1) p^{(d-1)(k-2j-1)}.
\end{equation*}
\notag
$$
By (8.19) this formula also holds for $j=0$. Any $(x,y)$ such that $p^{j} \mid (x,y)$ for some $j\geqslant j_k+1$ satisfies $F(x,y) = 0 \ \operatorname{mod} p^{k}$. Thus,
$$
\begin{equation*}
\sum_{j={j_k+1}}^{k} {N}_p(k,j)=\sharp\{(x,y)\ \operatorname{mod}p^k\colon (x,y)=0\ \operatorname{mod} p^{j_k+1}\}=p^{d(k-j_k-1)}\leqslant p^{dk/2}.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
N_p(k)=(\mathcal N_p-1p^{(d-1)(k-1)} \sum_{j=0}^{j_k}p^{-2 j (d-1)}+O(p^{dk/2}).
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\sigma_p=\lim_{k\to \infty}\frac{N_p(k)}{p^{(d-1)k}}=({\mathcal N}_p-1) p^{1-d } \sum^{\infty}_{j=0} p^{-2 j(d-1)} =\frac{p^{1-d } ({\mathcal N}_p-1 ) }{1-p^{2 -2d}} ,
\end{equation*}
\notag
$$
which proves (8.18).
Lemma 8.7 is proved. Now we deduce a formula for ${\mathcal N}_p(d)$ using induction on $d/2=s$. For $d=2$ we have ${\mathcal N}_p(2)=\sharp\{(x,y) \in \mathbb{F}_p^2\colon xy=0 \ \operatorname{mod} p\}=2p-1$. Next,
$$
\begin{equation*}
\begin{aligned} \, {\mathcal N}_p(d+2) &=\sharp \{\text{solutions with } x_{s+1}=0\}+\sharp \{ \text{solutions with } x_{s+1}\ne 0\} \\ &=p{\mathcal N}_p(d )+(p-1)p^{d}. \end{aligned}
\end{equation*}
\notag
$$
Therefore, for any even $d=2s\geqslant2$, and thus
$$
\begin{equation*}
\sigma_p(d)=\frac{1+p^{1-s}-p^{-s}-p^{1-d}}{1-p^{2-2d}} =\frac{(1+p^{1-s})(1-p^{-s})}{1-p^{2-2d}}.
\end{equation*}
\notag
$$
Since by Euler’s formula $\prod_p (1-p^{-l}) = 1/\zeta(l)$ for any $l>1$, in the case $d=4$, from (1.11) and the formula obtained for $\sigma_p(d)$ we get that
$$
\begin{equation*}
\sigma(A,0;d=4)=\prod_{p}\sigma_p(4)=\frac{\zeta(6)}{\zeta(2) }\prod_{p}(1+p^{-1}).
\end{equation*}
\notag
$$
This product does not converge, but
$$
\begin{equation*}
\sigma^*{(A;d=4)}=\prod_{p}(1-p^{-1})\sigma_p(4) =\frac{\zeta(6)}{\zeta(2)^2}=\frac{4\pi^2}{105}\simeq 0.376
\end{equation*}
\notag
$$
converges. Further,
$$
\begin{equation*}
\sigma{(A,0;d=6)} =\frac{\zeta(2)\zeta(10) }{\zeta(3)\zeta(4)}\simeq 1.265
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\sigma{(A,0;d=8)} =\frac{\zeta(3)\zeta(14) }{\zeta(4)\zeta(6)}\simeq 1.092,
\end{equation*}
\notag
$$
while
$$
\begin{equation*}
\begin{aligned} \, 1 <\sigma(A,0;d) &=\frac{\zeta(s-1)\zeta(2d-2)}{\zeta(s)\zeta(d-2)} \\ &=\frac{(1+2^{1-s})(1+2^{2-4s})}{(1+2^{-s})(1+2^{2-2s})}+ o(1)=1+ o(1) \end{aligned}
\end{equation*}
\notag
$$
tends to $1$ as $d=2s\geqslant 10$ grows. It remains to prove (8.17). By the definition (1.10)
$$
\begin{equation*}
\sigma_p=\sum_{t=0}^\infty p^{-dt}S_{p^t}(\mathbf 0),
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
S_{p^t}(\mathbf 0)=\mathop{{\sum}^*}_{a \,(\operatorname{mod}p^t)}\, \sum_{\mathbf b\,(\operatorname{mod}p^t)} e_{p^t}(aF(\mathbf b)) .
\end{equation*}
\notag
$$
Note that $p^{-dt}S_{p^t}(\mathbf 0)=1$ for $t=0$, while for $t=1$ we have
$$
\begin{equation*}
\begin{aligned} \, p^{-d}S_{p}(\mathbf 0) &=p^{-d}\sum_{a=1 }^{p-1}\, \sum_{\mathbf b\, (\operatorname{mod}p)} e_{p}(aF({\mathbf b})) \\ &=p^{-d}\sum_{a=1 }^{p-1}\, \sum_{\mathbf b\, (\operatorname{mod}p),\, p|F({\mathbf b}) }1+p^{-d}\sum_{a=1 }^{p-1}\, \sum_{\mathbf b\, (\operatorname{mod}p),\,p\nmid F({\mathbf b}) }e_{p}(aF({\mathbf b})) \\ &=p^{-d}(p-1){\mathcal N}_p(d)+p^{-d}(-1)(p^d-{\mathcal N}_p(d)) \\ &=p^{1-d} {\mathcal N}_p(d)-1, \end{aligned}
\end{equation*}
\notag
$$
since for any $n,m\neq 0$ such that $(m,n)=1$. Therefore,
$$
\begin{equation*}
\sum_{t=0}^1 p^{-dt}S_{p^t}(\mathbf 0)=p^{1-d} N_p(1).
\end{equation*}
\notag
$$
We proceed now by induction, supposing that, for $k\geqslant 1$,
$$
\begin{equation*}
\sum_{t=0}^k p^{-dt}S_{p^t}(\mathbf 0)=p^{(1-d)k}N_{p}(k).
\end{equation*}
\notag
$$
Then we write
$$
\begin{equation*}
S_{p^{k+1}}(\mathbf 0)= \mathop{{\sum}^*}_{a\, (\operatorname{mod}p^{k+1})}\, \sum_{\mathbf b\, (\operatorname{mod}p^{k+1})} e_{p^{k+1}}(aF(\mathbf b))=\Sigma_1 + \Sigma_2+ \Sigma_3,
\end{equation*}
\notag
$$
where we have defined
$$
\begin{equation}
\begin{aligned} \, \Sigma_1 &:=\mathop{{\sum}^*}_{a\, (\operatorname{mod} p^{k+1})}\, \sum_{p^{k+1}|F({\mathbf b})}1=p^{k}(p-1)N_{p}(k+1), \\ \Sigma_2&:= \mathop{{\sum}^*}_{a\, (\operatorname{mod} p^{k+1})}\, \sum_{F({\mathbf b})=lp^k}e_{p}(al) =-p^{k}(p^dN_{p}(k)-N_{p}(k+1)), \\ \Sigma_3&:= \mathop{{\sum}^*}_{a\, (\operatorname{mod}p^{k+1})}\, \sum_{s=0}^{k-1} \, \sum_{F({\mathbf b})=lp^s}e_{p^{k+1-s}}(al)=0 \end{aligned}
\end{equation}
\tag{8.21}
$$
for a nonzero integer $l=l(b)$ such that $p\nmid l$. The above equalities essentially follow by a repeated application of (8.20). In this way we obtain
$$
\begin{equation*}
\frac{S_{p^{k+1}}(\mathbf 0)}{p^{d(k+1)}} =\frac{p^{k+1}N_p(k+1)- p^{d+k}N_p(k)}{p^{d(k+1)}} =\frac{N_p(k+1)}{p^{(d-1)(k+1)}} - \frac{N_p(k)}{p^{(d-1)k}},
\end{equation*}
\notag
$$
which completes the induction step, thus proving (8.17). AcknowledgementThe authors thank Professor Heath-Brown for advising them concerning the paper [1].
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\Bibitem{VlaDymKuk23} |
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