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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 Rd, d⩾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∈R the quadric is either an ellipsoid or an empty set, while in the non-sign-definite case Σt is an unbounded hypersurface in Rd, which is smooth if t≠0, while Σ0 is a cone and has a singular point at zero.Let ZdL be the lattice of small period L−1, and let w be a regular real function on Rd, which means that w and its Fourier transform ˆw(ξ) are continuous functions decaying sufficiently rapidly at infinity: for some γ, C>0. Our aim is to study the behaviour of the series where m∈R is such that L2m is an integer.1 Let Then it is obvious that We also write To investigate NL(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′∈Zd, we count (with weights) solutions of the equation F(z′)=mL2, z′∈Zd, while Heath-Brown writes the equation as F(z′)=m, z′∈Zd, so that his m corresponds to our L2m.We start with a key theorem, which expresses the analogue of the Dirac delta function on the integers, that is, the function δ:Z→R such that 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∈N, we denote by eq the exponential function eq(x):=e2πix/q and by ∑∗a(modq) the sum over the residues a such that (a,q)=1, that is, over all integers a∈[1,q−1] that are relatively prime with q.Theorem 1.1. For any Q⩾1 there exists cQ>0 and a smooth function h(x,y): R>0×R→R such that The constant cQ satisfies cQ=1+ON(Q−N) for any N>0, while h is such that h(x,y)⩽c/x and h(x,y)=0 for x>max (so for each n the sum in (1.5) contains finitely many nonzero terms).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},
\begin{equation}
N({\widetilde w}; A, t)=c_QQ^{-2}\sum_{\mathbf c\in \mathbb Z^{d}}\sum_{q=1}^\infty q^{- d} S_q(\mathbf c) I^0_q(\mathbf c),
\end{equation}
\tag{1.6}
where and 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
\begin{equation}
\begin{aligned} \, &\bigl| N_L(w{; A, m})- \sigma_\infty(w)\sigma(A, L^2m) L^{d-2} \bigr| \nonumber \\ &\qquad\qquad \leqslant C L^{d/2+\varepsilon}\bigl(\|w\|_{K_1,K_2}+\|w\|_{0,K_3}\bigr), \end{aligned}
\end{equation}
\tag{1.12}
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)
\begin{equation}
\begin{aligned} \, \notag &\bigl| N_L(w ;A,0 )-\eta(0) \sigma_\infty(w)\sigma^*(A) L^{d-2}\log L - \sigma_1(w;A,L)L^{d-2} \bigr| \\ &\qquad\leqslant C_0 L^{d-2-\varepsilon}\|w\|_{K_1,K_2}, \end{aligned}
\end{equation}
\tag{1.13}
where the constant C_0 depends on \varepsilon and A. The constant \eta(0) is 1 if the determinant \det A is the square of an integer, and it is 0 otherwise. The L-independent constant \sigma^*(A) is finite, while the constant \sigma_1 satisfies uniformly in L. In the case of a perfect-square determinant \det A, when \eta(0)=1, it is given by (1.24). In the case of a nonsquare determinant \det A, when \eta(0)=0 and the term \sigma_1(w;A,L)L^{d-2} gives the asymptotic of the sum N_L, the constant \sigma_1(w;A,L) does not depend on L and has the form
\begin{equation}
\sigma_1(w;A)= \sigma_\infty(w)L(1,\chi)\prod_p(1-\chi(p)p^{-1})\sigma_p(A,0) ,
\end{equation}
\tag{1.14}
where \chi is the Jacobi symbol \biggl(\dfrac{\det A}{*}\biggr) and L(1,\chi) is the Dirichlet L-function.3 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
\begin{equation*}
\bigl| N_L(w; \mathcal F) - \sigma_\infty(w; \mathcal F)\sigma(A, L^2 m) L^{d-2} \bigr| \leqslant C L^{d/2+\varepsilon}\bigl(\|w\|_{K_1,K_2}+\|w\|_{0,K_3}\bigr).
\end{equation*}
\notag
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_qNow 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
\begin{equation*}
\sum_{q\leqslant X}q^{-d} S_q(0)=\sigma(A,L^2m) +O(X^{-d/2+2}).
\end{equation*}
\notag
In particular, \sigma(A,L^2m)=\sum_{q=1}^\infty q^{-d} S_q(0). So |\sigma(A,L^2m)|\lesssim 1 in view of Lemma 2.1. 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_q3.1. 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,
\begin{equation}
\begin{aligned} \, \notag \int_\mathbb R f(y) h(x,y) \, dy &=f(0) + O_{M}\biggl(\frac{x^{M}}{\beta^{M+1}} \frac{1}{X}\int_{-X}^{X}|f^{(M)}(y)|\,dy\biggr) \\ &\qquad+O_{N,C}\bigl((x^N+\beta^N)(\|f\|_{M-1,0}+x^{-1} |f|_{L_1})\bigr), \end{aligned}
\end{equation}
\tag{3.3}
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 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
\begin{equation*}
(x^N+\beta^N) x^{-1} \leqslant 2\beta^N x^{-1} \leqslant 2 x^{M-\delta} \quad\text{if } N\geqslant N_\delta=\frac{(M-\delta +1)(M+1)}{\delta}.
\end{equation*}
\notag
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_0In 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 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
\begin{equation*}
|I_B|\lesssim_{M,\delta,m}L^{d-M+\delta}\|w\|_{M,d+1} \sum_{q\leqslant L^{1-\gamma_2}}q^{-d/2+1+M}.
\end{equation*}
\notag
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),
\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*}
\notag
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,
\begin{equation*}
|I_q(\mathbf c)|\lesssim_{\gamma_1,m}L^{d/2+1+\gamma_1} \biggl(\frac{q}{|\mathbf c|}\biggr)^{d/2-1-\gamma_1} \bigl(\|w\|_{\overline N,d+5}+\|w\|_{0,\overline N+3d+4}\bigr),
\end{equation*}
\notag
where \overline N and \gamma_1 are the same as in Proposition 6.1. 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_RFirst 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}^cNow 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
\begin{equation*}
\int_{{S_R}^c} |p(t)|\, |I_{\mathbf a, t}|\, d\mathbf a \,dt \lesssim_m R^{d}|\mathbf u|^{-d/2+1+\beta}\|w\|_{0,K(d,\beta)},
\end{equation*}
\notag
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
\begin{equation}
|I_{\mathbf a, t}|\lesssim\max_{|b_i|\leqslant 1\ \forall\, i}|\widetilde w(\delta \mathbf b +\mathbf a)|\leqslant \|\widetilde w\|_{0,0}.
\end{equation}
\tag{6.10}
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
\begin{equation}
\begin{alignedat}{2} &| \partial^k\mathcal I(t)|\lesssim_{k,M}\|f\|_{k,M} &\quad &\textit{if }\ k< \frac{d}2-1, \\ &| \partial^k\mathcal I(t)| \lesssim_{k,M}\|f\|_{k,M}(1-\log |t|) &\quad &\textit{if }\ k\leqslant \frac{d}2-1. \end{alignedat}
\end{equation}
\tag{7.5}
On the other hand, for |t|\geqslant 1, setting \kappa=(M+2-d)/2, we have
\begin{equation}
\begin{alignedat}{2} |\partial^k\mathcal I(t)| &\lesssim_{k,M}\|f\|_{k,M}\langle t\rangle^{-\kappa} &\quad &\textit{if }\ 1\leqslant k\leqslant \frac{d}2-1, \ \ k\leqslant k_*, \\ |\mathcal I(t)| &\lesssim_{M,\kappa'}\|f\|_{0,M}\langle t\rangle^{-\kappa'} &\quad &\forall\, \kappa'<\kappa. \end{alignedat}
\end{equation}
\tag{7.6}
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:
\begin{equation}
dz|_{\Sigma_t^x}=du\,|x|^{-1}\,dx\,|z|\,d_{\sigma_t^x(u,x)}y.
\end{equation}
\tag{7.11}
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 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=4In 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},
\begin{equation}
\sum_{q\leqslant X} S_q(\mathbf c; A,0)=\eta(\mathbf c)\sigma_\mathbf c^*(A)\sum_{q\leqslant X}q^{d-1} + O_{\varepsilon}(X^{\alpha+\varepsilon} (1+|\mathbf c|)) ,
\end{equation}
\tag{8.2}
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},
\begin{equation*}
\sum_{q\leqslant X} q^{-d}S_q(0; A,0)=\sigma^*(A)\log X + \widehat C_A + O_{\varepsilon} (X^{\alpha + \varepsilon -d}),
\end{equation*}
\notag
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}
\begin{equation*}
\sum_{q\leqslant X} q^{-d}S_q(0; A,0)=L(1,\chi)\prod_p (1-\chi(p)p^{-1})\sigma_p(A,0) + O_{\varepsilon}(X^{-1/2+\varepsilon}),
\end{equation*}
\notag
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_0The 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
\begin{equation*}
|I_B|\lesssim\sum_{q\leqslant \rho L}q^{-1}(|f_q|+|g_q|) \lesssim_{N} L^d\bigl(\rho\log L + \rho^{N-1}+L^{1-d}\bigr)\|w\|_{d/2-1,d+1}
\end{equation*}
\notag
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
\begin{equation*}
S_p (k)=\bigcup^k_{j=0} {S}_p (k, j)\quad\text{and} \quad N_p (k)=\sum^{k}_{j=0} {N}_p(k,j).
\end{equation*}
\notag
In particular, {N}_p(1,0)={\mathcal N}_p-1 since {N}_p(1,1)=1. We claim that and thus
\begin{equation}
{N}_p(k,0)={N}_p(1,0) p^{(d-1)(k-1)} =({\mathcal N}_p-1) p^ {(d-1)(k-1)}.
\end{equation}
\tag{8.19}
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) 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, We proceed now by induction, supposing that, for k\geqslant 1, 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].
Citation in format AMSBIB
\Bibitem{VlaDymKuk23} |
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