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This article is cited in 3 scientific papers (total in 3 papers) Solomyak-type eigenvalue estimates for the Birman-Schwinger operator F. A. Sukochev, D. V. Zanin School of Mathematics and Statistics, University of New South Wales, Sydney, Australia
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     § 1. IntroductionEstimates for the operator $M_fg(\nabla)$ (here $M_f$ is a multiplication operator and $g(\nabla)$ is a function of the gradient) take their origin in the study of bound states1 of Schrödinger operators. The problem of describing the functions $f$ and $g$ such that $M_fg(\nabla)$ belongs to some weak Schatten class $\mathcal{L}_{p,\infty}$ was originally stated by Simon (see Conjecture 1 in [34] and also Ch. 4 in [35]). The first important result in this direction was due to Cwikel [13] (see also Theorem 6.5 in [9]). It states that
$$
\begin{equation*}
\|M_fg(\nabla)\|_{p,\infty}\leqslant c_p\|f\|_p\,\|g\|_{p,\infty}, \qquad f\in L_p(\mathbb{R}^d), \quad g\in L_{p,\infty}(\mathbb{R}^d), \quad 2<p\leqslant\infty.
\end{equation*}
\notag
$$
Here the weak Schatten quasi-norm on the left-hand side is given by the formula
$$
\begin{equation*}
\|T\|_{p,\infty}=\sup_{k\geqslant0}(k+1)^{1/p}\mu(k,T),
\end{equation*}
\notag
$$
where $(\mu(k,T))_{k\geqslant0}$ is the (decreasing) sequence of singular values of the operator $T$ (see [35] and [24]). We refer to estimates of this kind as Cwikel’s estimates (the function $g$ of the gradient is arbitrary). Cwikel’s estimates were strengthened by Weidl [43] as follows:
$$
\begin{equation*}
\|M_fg(\nabla)\|_{p,\infty}\leqslant c_p\|f\otimes g\|_{p,\infty}, \qquad f\otimes g\in L_{p,\infty}(\mathbb{R}^d\times\mathbb{R}^d), \quad 2<p\leqslant\infty.
\end{equation*}
\notag
$$
In [22] a more general version of this estimate, suitable for noncommutative variables $f$ and $g$, was proved. The setting used in [22] comes from quantized calculus and is suitable for treating concrete problems in noncommutative geometry. In particular, Cwikel’s estimates in [22] can be extended to noncommutative Euclidean (Moyal) space and can be used to treat the magnetic Laplacian. In various applications (both to mathematical physics and noncommutative geometry) the following estimates are of primary interest. We fix the function $g$ to be
$$
\begin{equation*}
g(t)=(1+|t|^2)^{-d/(2p)}, \qquad t\in\mathbb{R}^d, \quad p>0,
\end{equation*}
\notag
$$
and investigate the operator $M_fg(\nabla)=M_f(1-\Delta)^{-d/(2p)}$. We are especially interested in the critical exponent, that is, in $p=2$. Physicists would be even happier to consider the function $g(t)=|t|^{-d/p}$; however, the corresponding operator $M_fg(\nabla)$ is known to be unbounded (in the critical case $p=2$; see, for example, the proof of Proposition 7.4 in [35]) and so falls outside the scope of this paper. The best known estimates for the operator $M_f(1-\Delta)^{-d/4}$ (on both $\mathbb{R}^d$ and the $d$-dimensional torus $\mathbb{T}^d$) can be found in Solomyak’s foundational paper [37]. There the estimates were not stated explicitly and only the case of even dimension was treated. The paper [37] was based on the long line of works [5]–[7], [30], [8] by Birman, Solomyak and their collaborators, which were also partly motivated by investigations of the discrete spectra of Schrödinger operators. A general scheme of quasinorm estimates for the operator $M_f(1-\Delta)^{-d/4}$ hatched in those papers was adapted to the case of even dimension and appropriate Orlicz norms in the subsequent papers [38] by Solomyak and [33] by Shargorodsky. The recent preprint of Rozenblum [31] also explores similar ideas. We prove the following estimate for the operator $(1-\Delta_{\mathbb{T}^d})^{-d/4}M_f(1-\Delta_{\mathbb{T}^d})^{-d/4}$ and for the ideal $\mathcal{L}_{1,\infty}$ in the setting of a $d$-dimensional torus $\mathbb{T}^d$. Theorem 1.1 reconstructs Solomyak’s results (see [37], Lemma 2.1 and Theorem 2.1) in a more explicit format and, simultaneously, extends them to an arbitrary dimension. Its proof is modelled after [37], but contains several technical modifications, which should help the reader to digest it more quickly. Throughout this paper, the symbol $c_d$ denotes a constant depending on the dimension $d$ only. Here the Orlicz space $L_{\Phi}(\mathbb{T}^d)$ is defined in terms of the Orlicz function ${\Phi(t)=t\log(e+t)}$, $t>0$, and is frequently denoted by $L\log L(\mathbb{T}^d)$ in the literature; this space was introduced by Zygmund in 1928 (see § 4.6 in [3]). It is interesting to compare the result of Theorem 1.1 with Theorem 1.2 in the recent paper [23] by Lord and these authors. Via tensor multiplier techniques from Banach space theory, it was shown there that if $f \in L_{\Phi}(\mathbb{R}^d)$ then
$$
\begin{equation*}
(1-\Delta_{\mathbb{R}^d})^{-d/4}M_f(1-\Delta_{\mathbb{R}^d})^{-d/4} \in\mathcal{M}_{1,\infty}(L_2(\mathbb{R}^d)),
\end{equation*}
\notag
$$
where the (Dixmier-Macaev) ideal $\mathcal{M}_{1,\infty}$, the submajorization closure of $\mathcal{L}_{1,\infty}$, is strictly larger than $\mathcal{L}_{1,\infty}$ (see [12], for example). In the current paper we propose a different approach in order to derive, in Theorem 1.1, a stronger estimate for the smaller ideal $\mathcal{L}_{1,\infty}$. Our approach is based on Solomyak’s ideas from [36] and [37], which were employed there in the case of even dimension. Rozenblum (private communication) asked whether it is possible to extend the result of Theorem 1.1 to Euclidean space. We show there is a stark contrast between bounds for the Dixmier-Macaev ideal $\mathcal{M}_{1,\infty}$ and the weak Schatten-von Neumann ideal $\mathcal{L}_{1,\infty}$. The statement of Theorem 1.1 is false if $\mathbb{T}^d$ is replaced by $\mathbb{R}^d$, for any symmetric function space on $\mathbb{R}^d$. This surprising fact is established in Theorem 1.2 below. Theorem 1.2. For every symmetric quasi-Banach function space $E(\mathbb{R}^d)$ on $\mathbb{R}^d$ there exists $f\in E(\mathbb{R}^d)$, $f\geqslant0$, such that As our third main result we derive an alternative estimate. It is presented in Theorem 1.3 below, which yields a suitable extension of Theorem 1.1 to Euclidean space $\mathbb{R}^d$. This estimate captures the results known in the literature and concerning Euclidean-space estimates for weak ideals in the critical case. It also delivers the best known (to date) Solomyak-type estimate on $\mathbb{R}^d$ in the case of the weak Schatten class $\mathcal{L}_{1,\infty}$. Theorem 1.3. Let $d\in\mathbb{N}$. Let $\Phi(t)=t\log(e+t)$, $t>0$, and $f\in L_{\Phi}(\mathbb{R}^d)$. Then provided that the integral on the right-hand side is finite. It was already proved in § 2.5 in [23] that the operator featuring in Theorem 1.3 is bounded whenever $f$ belongs to the Orlicz space2 $L_{\Phi}(\mathbb{R}^d)$. In the special case when $d=2$ and $f\geqslant0$ the expression on the right-hand side of the inequality in Theorem 1.3 can be glimpsed in Shargorodsky’s work [33], where it was used to obtain sharp estimates for the number of negative eigenvalues of a Schrödinger operator (the fruitful idea to use inversion, employed in [33], originates from [4]). Note, however, that Solomyak-type estimates were not considered in [33]. The proof of Theorem 1.3 reveals the conformal invariance of Solomyak-type estimates. In the pre-critical case, this idea can be traced back to [16]. Frank [14] investigated conformal invariance in the pre-critical case (for Rumin’s inequality, which happens to be equivalent to Solomyak-type estimates). We prove the invariance of a Solomyak-type estimate in the critical case with respect to inversion (which is in fact the only nonlinear conformal transform for $d>2$). Theorem 1.3 is new for dimension $d\neq 2$. For $d=2$ it can be deduced with some effort from results of Solomyak [36] and Shargorodsky [33]. The proof is presented in § 8 and is due to Frank. In § 7, we present an alternative description of the quantity on the right-hand side of Theorem 1.3 (see Proposition 5.1). 1.1. The strategy of the proofOur approach to the proof of Theorem 1.1 is based on Sobolev’s embedding theorem and follows the pattern elaborated in the papers by Birman and Solomyak cited above, with crucial improvements due to Solomyak [36], [37]. One should note that even the boundness (in the uniform norm) of the operator
$$
\begin{equation*}
(1-\Delta_{\mathbb{T}^d})^{-d/4}M_f(1-\Delta_{\mathbb{T}^d})^{-d/4}
\end{equation*}
\notag
$$
is nontrivial (for an unbounded measurable function $f$ on $\mathbb{T}^d$). Indeed, an estimate for the operator norm of this operator is equivalent to the critical case of Sobolev’s embedding theorem (see, for example, Theorem 2.3 in [23]). In § 4 we restate Theorem 1.1 as
$$
\begin{equation*}
\bigl\|M_{f^{1/2}}(1-\Delta_{\mathbb{T}^d})^{-d/4}\bigr\|_{2,\infty}\leqslant c_d\|f\|_{L_{\Phi}(\mathbb{T}^d)}^{1/2}, \qquad 0\leqslant f\in L_{\Phi}(\mathbb{T}^d).
\end{equation*}
\notag
$$
Note that $(1-\Delta_{\mathbb{T}^d})^{-d/4}$ sends $L_2(\mathbb{T}^d)$ to the Sobolev space $W^{d/2,2}(\mathbb{T}^d)$. It is easily verified that the embedding $\operatorname{id}\colon W^{d/2,2}(\mathbb{T}^d)\to L_2(\mathbb{T}^d)$ is a compact operator. Hence, at least for a bounded function $f$, the multiplication mapping $M_{f^{1/2}}$ from $W^{d/2,2}(\mathbb{T}^d)$ to $L_2(\mathbb{T}^d)$ is also compact. We adopt Solomyak’s viewpoint on Theorem 1.1 as an estimate for the approximation numbers of the operator $M_{f^{1/2}}$ from $W^{d/2,2}(\mathbb{T}^d)$ to $L_2(\mathbb{T}^d)$ (this viewpoint is made clear in Lemma 4.3 below). Solomyak used some methods developed by Birman and Solomyak and presented, for example, in [8] (see Theorems 1.1–1.4 there and the explanations that follow). The key tools in our proof are the homogeneous Sobolev inequality on the cube (Theorem 2.3) and Besicovitch’s covering lemma (Lemma 3.4). The use of coverings instead of partitions, which had previously been used to construct approximating finite-rank operators, was pioneered by Rozenblum; see also the comments preceding the proof of Theorem 3.1. The crucial importance of Theorem 2.3 becomes apparent in the proof of Lemma 4.1. Then Besicovitch’s covering lemma is used to choose a linear operator of prescribed rank $n$ that approximates $M_{f^{1/2}}$ with required accuracy. Sobolev’s embedding theorem in the critical case was proved by Hansson, Brezis and Wainger, and Cwikel and Pustylnik and was examined further in [41], where it was proposed to replace norm estimates with distributional ones. This approach allows one to compute the operator norm of the operator ${(1-\Delta_{\mathbb{R}^d})^{-d/4}M_f(1-\Delta_{\mathbb{R}^d})^{-d/4}}$ and becomes an indispensable tool in the proof of Theorem 1.2. The technique of the proof of Theorem 1.3 relies on the inversion trick (attributed in [33] to Grigoryan and Nadirashvili [18]; see also [4] and [16]). This technique allows one to compare the operators
$$
\begin{equation*}
(1-\Delta_{\mathbb{R}^d})^{-d/4}M_f(1-\Delta_{\mathbb{R}^d})^{-d/4} \quad\text{and}\quad (1-\Delta_{\mathbb{R}^d})^{-d/4}M_{Vf}(1-\Delta_{\mathbb{R}^d})^{-d/4},
\end{equation*}
\notag
$$
where the function $Vf$ is defined in Lemma 5.3. It is crucial that, whenever the function $f$ has support outside the unit ball, the function $Vf$ has support in the unit ball. This idea allows us to reduce the problem to the case when $f$ has support in the unit ball, that is, to Solomyak-type estimates on a torus $\mathbb{T}^d$. To the best of our knowledge, this is the first case when the inversion trick is used in investigations of Solomyak-type estimates. AcknowledgementsThe authors thank Professors N. S. Trudinger and E. Valdinoci for useful discussions about Sobolev’s embedding theorem, and Professor G. Rozenblum for his interest in this paper and for discussions with him, which led to numerous improvements (both mathematical and historical) in the exposition. We thank Professor R. L. Frank for communicating to us the result presented in § 8 and for drawing our attention to [14]. § 2. PreliminariesThroughout what follows the constants $c_{x,y}$ depend only on the choice of $x$ and $y$. The exact values of such constants can change from line to line. The notation $A\approx B$ means that there is a constant $c>1$ such that $c^{-1}A\leqslant B\leqslant cA$. An integral without an explicitly indicated measure is assumed to be taken with respect to the Lebesgue measure. 2.1. Symmetric function spacesLet $(\Omega,\omega)$ be a $\sigma$-finite measure space. Let $S(\Omega,\omega)$ be the collection of all $\omega$-measurable functions on $\Omega$ such that, for some ${n\in\mathbb{N}}$, the function $|f|\chi_{\{|f|>n\}}$ has support on a set of finite measure. For every ${f\in S(\Omega,\omega)}$ the distribution function is finite for all sufficiently large $t$. For $f\in S(\Omega,\omega)$ one can define the notion of the decreasing rearrangement of $f$ (denoted by $\mu(f)$). This is the positive decreasing function on $\mathbb{R}_+$ that is equimeasurable with $|f|$. We refer to [21] for the properties of the decreasing rearrangement.Let $E(\Omega,\omega)\subset S(\Omega,\omega)$, and let $\|\cdot\|_E$ be a quasi-Banach norm on $E(\Omega,\omega)$ such that
We say that $(E(\Omega,\omega),\|\cdot\|_E)$ (or simply $E$) is a symmetric quasi-Banach function space (symmetric space for brevity). If $\Omega=\mathbb{R}_+$, then the function is called the fundamental function of $E$. The fundamental function can be defined similarly when $\Omega$ is an interval or an arbitrary $\sigma$-finite atomless measure space (though we use it only in the simplest cases like $\mathbb{T}^d$). The concrete examples of measure spaces $(\Omega,\omega)$ considered in this paper are $\mathbb{T}^d=\mathbb{R}^d/(2\pi\mathbb{Z})^d$ (equipped with the normalised Haar measure $\nu$), $\mathbb{R}_+$, $\mathbb{R}^d$ (equipped with the Lebesgue measure $\nu$), measurable subsets of these spaces and compact $d$-dimensional Riemannian manifolds $(X,g)$.Among concrete symmetric spaces used in this paper are the $L_p$-spaces and Orlicz spaces. Given a convex function $\Phi$ on $[0,\infty)$ such that $\Phi(0)=0$, the Orlicz space $L_{\Phi}(\Omega,\omega)$ is defined by
$$
\begin{equation*}
L_{\Phi}(\Omega,\omega)=\bigl\{f\in S(\Omega,\omega)\colon \Phi(\lambda|f|)\in L_1(\Omega,\omega) \text{ for each } \lambda>0\bigr\}.
\end{equation*}
\notag
$$
We equip it with the norm
$$
\begin{equation}
\|f\|_{L_{\Phi}}=\inf\biggl\{\lambda>0\colon \biggl\|\Phi\biggl(\frac{|f|}{\lambda}\biggr)\biggr\|_1\leqslant 1\biggr\}.
\end{equation}
\tag{2.1}
$$
We refer the reader to [20] for further information about Orlicz spaces. For the particular function $\Phi(t)=t\log(e+t)$, $t>0$, we have $f\in L_{\Phi}(\mathbb{R}^d)$ if and only if $\mu(f)\chi_{(0,1)}\in L_{\Phi}(0,1)$ and $f\in L_1(\mathbb{R}^d)$. We also define a dilation operator $\sigma_r$, $r>0$, which acts on $S(\mathbb{R},\nu)$ (or $S(\mathbb{R}^d,\nu)$) by the formula
$$
\begin{equation*}
(\sigma_rf)(t)=f\biggl(\frac{t}{r}\biggr), \qquad f\in S(\mathbb{R},\nu).
\end{equation*}
\notag
$$
It is sometimes convenient to consider dilations of functions which are a priori defined only on some subset (typically, an interval or a cube) of $\mathbb{R}$ or $\mathbb{R}^d$. In this case, first of all we extend $f$ to a function on $\mathbb{R}$ (or $\mathbb{R}^d$) by setting $f=0$ outside the original domain of $f$. 2.2. Trace idealsThe following material is standard; for more details we refer the reader to [24] and [35]. Let $H$ be a complex separable infinite-dimensional Hilbert space, let $B(H)$ denote the set of all bounded operators on $H$, and let $K(H)$ denote the ideal of compact operators on $H$. Given $T\in K(H)$, the sequence of singular values $\mu(T) = \{\mu(k,T)\}_{k=0}^\infty$ is defined by
$$
\begin{equation*}
\mu(k,T) = \inf\bigl\{\|T-R\|_{\infty}\colon \mathrm{rank}(R) \leqslant k\bigr\}.
\end{equation*}
\notag
$$
Here $\|\cdot\|_{\infty}$ denotes the operator norm. It is often convenient to identify the sequence $(\mu(k,T))_{k\geqslant0}$ with the step function $\sum_{k\geqslant0}\mu(k,T)\chi_{[k,k+1)}$ on the semiaxis $(0,\infty)$. Below inequalities of the form $\mu(S)\leqslant\mu(T)$ should be understood pointwise, that is, $\mu(k,S)\leqslant\mu(k,T)$ for every $k\geqslant0$. Let $p \in (0,\infty)$. The weak Schatten class $\mathcal{L}_{p,\infty}$ is the set of operators $T$ such that $\mu(T)$ belongs to the weak $L_p$-space $l_{p,\infty}$, with the quasinorm
$$
\begin{equation*}
\|T\|_{p,\infty} = \sup_{k\geqslant 0} (k+1)^{1/p}\mu(k,T) < \infty.
\end{equation*}
\notag
$$
Clearly, $\mathcal{L}_{p,\infty}$ is an ideal in $B(H)$. We also have the following form of Hölder’s inequality:
$$
\begin{equation}
\|TS\|_{r,\infty} \leqslant c_{p,q}\|T\|_{p,\infty}\|S\|_{q,\infty}
\end{equation}
\tag{2.2}
$$
for some constant $c_{p,q}$, where $1/r=1/p+1/q$. Indeed, this follows from the definition of the above quasinorms and the inequality
$$
\begin{equation*}
\mu(2n,TS)\leqslant \mu(n,T)\mu(n,S), \qquad n\geqslant 0,
\end{equation*}
\notag
$$
(see, for example, [17], Corollary 2.2). One ideal of particular interest is $\mathcal{L}_{1,\infty}$. 2.3. Sobolev spaces on cubesLet $m\in\mathbb{Z}_+$. For every open cube $\Pi\subset\mathbb{R}^d$ we define the Sobolev space $W^{m,2}(\Pi)$ as follows:
$$
\begin{equation*}
W^{m,2}(\Pi)=\bigl\{ u\in L_2(\Pi)\colon \nabla^{\alpha}u\in L_2(\Pi),\ |\alpha|_1\leqslant m \bigr\}.
\end{equation*}
\notag
$$
Here $|\alpha|_1=\sum_{k=1}^d|\alpha_k|$ for $\alpha=(\alpha_1,\dots,\alpha_d)\in\mathbb{Z}^d_+$ and $\nabla^{\alpha}f$ is understood as a distributional derivative. We equip $W^{m,2}(\Pi)$ with the (nonhomogeneous) Sobolev norm given by the formula
$$
\begin{equation*}
\|u\|_{W^{m,2}(\Pi)}^2=\sum_{|\alpha|_1\leqslant m}\|\nabla^{\alpha}u\|_{L_2(\Pi)}^2
\end{equation*}
\notag
$$
for every $u\in W^{m,2}(\Pi)$ (see p. 44 in [1]). It is a standard fact (see, for instance, Theorem 3.5 in [1]) that $(W^{m,2}(\Pi),\|\cdot\|_{W^{m,2}(\Pi)})$ is a Hilbert space. Let $s>0$ and let $m=\lfloor s\rfloor$. If $s\neq m$, then we define the Sobolev space $W^{s,2}(\Pi)$ as follows:
$$
\begin{equation*}
W^{s,2}(\Pi)=\biggl\{ u\in W^{m,2}(\Pi)\colon \int_{\Pi}\int_{\Pi}\frac{|(\nabla^{\alpha}u)(x) -(\nabla^{\alpha}u)(y)|^2}{|x-y|_2^{d+2(s-m)}}\,d\nu(x)\,d\nu(y)<\infty \biggr\}.
\end{equation*}
\notag
$$
Here $|x|_2=(\sum_{k=1}^dx_k^2)^{1/2}$ for every $x\in\mathbb{R}^d$. We equip $W^{s,2}(\Pi)$ with the (nonhomogeneous) Sobolev norm given by the formula
$$
\begin{equation*}
\|u\|_{W^{s,2}(\Pi)}^2=\|u\|_{W^{m,2}(\Pi)}^2+\sum_{|\alpha|_1\leqslant m}\int_{\Pi}\int_{\Pi} \frac{|(\nabla^{\alpha}u)(x)-(\nabla^{\alpha}u)(y)|^2}{|x-y|_2^{d+2(s-m)}}\,d\nu(x)\,d\nu(y)
\end{equation*}
\notag
$$
for every $u\in W^{s,2}(\Pi)$ (see Theorem 7.48 in [1]). It is known that $(W^{s,2}(\Pi), \|\,{\cdot}\,\|_{W^{s,2}(\Pi)})$ is a Hilbert space (see, for example, p. 205 and Theorem 7.48 in [1] for the proof of completeness; the parallelogram identity can be verified directly). 2.4. Sobolev spaces on $ {\mathbb{R}}^d$ and on $ {\mathbb{T}}^d$Recall that the Sobolev space $W^{s,2}(\mathbb{R}^d)$, $ s>0$, admits an easier description (see, for example, Theorem 7.63 in [1]):
$$
\begin{equation*}
W^{s,2}(\mathbb{R}^d)=\bigl\{ u\in L_2(\mathbb{R}^d)\colon (1-\Delta_{\mathbb{R}^d})^{s/2}u\in L_2(\mathbb{R}^d)\bigr\},
\end{equation*}
\notag
$$
with an equivalent norm
$$
\begin{equation*}
\|u\|_{W^{s,2}(\mathbb{R}^d)}=\|(1-\Delta_{\mathbb{R}^d})^{s/2}u\|_2,\qquad u\in W^{s,2}(\mathbb{R}^d).
\end{equation*}
\notag
$$
Here $\Delta_{\mathbb{R}^d}$ is the Laplace operator on $\mathbb{R}^d$. We also need the notion of Sobolev spaces on the torus:
$$
\begin{equation*}
W^{s,2}(\mathbb{T}^d)=\bigl\{ u\in L_2(\mathbb{T}^d)\colon (1-\Delta_{\mathbb{T}^d})^{s/2}u\in L_2(\mathbb{T}^d)\bigr\}
\end{equation*}
\notag
$$
with the norm
$$
\begin{equation*}
\|u\|_{W^{s,2}(\mathbb{T}^d)}=\|(1-\Delta_{\mathbb{T}^d})^{s/2}u\|_2, \qquad u\in W^{s,2}(\mathbb{T}^d).
\end{equation*}
\notag
$$
Here $\Delta_{\mathbb{T}^d}$ is the Laplace operator on $\mathbb{T}^d$. We identify $\mathbb{T}^d$ with the cube $(-\pi,\pi)^d$ whose opposite faces are ‘glued’. We equip $\mathbb{T}^d$ with the normalised Haar measure $\nu$. The distance between two points $x,y\in\mathbb{T}^d$ is given by
$$
\begin{equation*}
\operatorname{dist}(x,y)=|x-y|_2, \qquad x,y\in\mathbb{T}^d,
\end{equation*}
\notag
$$
where $x-y$ is treated as an element of $(-\pi,\pi)^d$. Proof. Let $(e_n)_{n\in\mathbb{Z}^d}$ be the Fourier basis of $L_2(\mathbb{T}^d)$. Then
$$
\begin{equation*}
\begin{aligned} \, &\int_{\mathbb{T}^d}\int_{\mathbb{T}^d} \frac{|u(x)-u(y)|^2}{\operatorname{dist}(x,y)^{d+2s}}\,d\nu(x)\,d\nu(y) \\ &\qquad=\sum_{m,n\in\mathbb{Z}^d}\widehat{u}(n)\overline{\widehat{u}(m)} \int_{\mathbb{T}^d}\int_{\mathbb{T}^d} \frac{(e_n(x)-e_n(y))\overline{(e_m(x)-e_m(y))}}{\operatorname{dist}(x,y)^{d+2s}}\,d\nu(x)\,d\nu(y). \end{aligned}
\end{equation*}
\notag
$$
Clearly,
$$
\begin{equation*}
\frac{(e_n(x)-e_n(y))\overline{(e_m(x)-e_m(y))}}{\operatorname{dist}(x,y)^{d+2s}} =e_{n-m}(y)\frac{(e_n(x-y)-1)\overline{(e_m(x-y)-1)}}{\operatorname{dist}(x-y,0)^{d+2s}}.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\begin{aligned} \, &\int_{\mathbb{T}^d}\int_{\mathbb{T}^d} \frac{(e_n(x)-e_n(y))\overline{(e_m(x)-e_m(y))}}{\operatorname{dist}(x,y)^{d+2s}}\,d\nu(x)\,d\nu(y) \\ &\qquad=\delta_{n,m}\int_{\mathbb{T}^d} \frac{(e_n(x)-1)\overline{(e_m(x)-1)}}{\operatorname{dist}(x,0)^{d+2s}}\,d\nu(x). \end{aligned}
\end{equation*}
\notag
$$
Consequently, In the above sum the term corresponding to $n=0$ vanishes. Thus, we estimate (from above and below) the terms corresponding to $n\neq0$.
It is immediate that
$$
\begin{equation*}
\begin{aligned} \, \int_{\mathbb{T}^d}\frac{|e_n(x)-1|^2}{\operatorname{dist}(x,0)^{d+2s}}\,d\nu(x) &=(2\pi)^{-d}\int_{(-\pi,\pi)^d}\frac{|e_n(x)-1|^2}{|x|_2^{d+2s}}\,d\nu(x) \\ &=(2\pi)^{-d}\int_{\mathbb{R}^d}\frac{|e_n(x)-1|^2}{|x|_2^{d+2s}}\,d\nu(x)+O(1). \end{aligned}
\end{equation*}
\notag
$$
Observe that here we have used the assumption that $0<s<1$ (otherwise the integral is divergent). Set $k=(1,0,\dots,0)$. Since $n\neq0$, there exists $U\in \operatorname{SO}(d)$ such that $n=|n|_2\cdot Uk$. Substituting into the integral $x=\dfrac1{|n|_2}Uy$ we obtain
$$
\begin{equation*}
(2\pi)^{-d}\int_{\mathbb{R}^d}\frac{|e_n(x)-1|^2}{|x|_2^{d+2s}}\,d\nu(x)=c_{d,s}|n|_2^{2s}, \qquad 0\neq n\in\mathbb{Z}^d.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\int_{\mathbb{T}^d}\frac{|e_n(x)-1|^2}{\operatorname{dist}(x,0)^{d+2s}}\,d\nu(x)\in (c_{d,s}'|n|_2^{2s},c_{d,s}''|n|_2^{2s}), \qquad 0\neq n\in\mathbb{Z}^d,
\end{equation*}
\notag
$$
and therefore
$$
\begin{equation*}
\int_{\mathbb{T}^d}\int_{\mathbb{T}^d} \frac{|u(x)-u(y)|^2}{\operatorname{dist}(x,y)^{d+2s}}\,d\nu(x)\,d\nu(y) \in \biggl(c_{d,s}'\sum_{n\in\mathbb{Z}^d}|n|_2^{2s}|\widehat{u}(n)|^2,c_{d,s}'' \sum_{n\in\mathbb{Z}^d}|n|_2^{2s}|\widehat{u}(n)|^2\biggr).
\end{equation*}
\notag
$$
This suffices to complete the proof.
Theorem 2.1 is proved. 2.5. Sobolev spaces on toric cubesWe recall that by a toric cube we always mean a Cartesian product of (open) arcs3 of equal length. It follows from this formulation that different cubes are always parallel to each other. When the torus is identified with $(-\pi,\pi)^d$, a cube is often4 the same as a (properly oriented) Euclidean cube in $(-\pi,\pi)^d$. If a toric cube $\Pi$ is a Euclidean cube under this identification, then $W^{s,2}(\Pi)$ is defined as in § 2.3. Otherwise, take some $t\in\mathbb{T}^d$ such that $t+\Pi$ is a Euclidean cube under this identification. Then
$$
\begin{equation*}
W^{s,2}(\Pi)=\{u\in L_2(\Pi)\colon u(\cdot-t)\in W^{s,2}(t+\Pi)\}.
\end{equation*}
\notag
$$
Of course, this notion does not depend on the choice of $t\in\mathbb{T}^d$. In other words, the last equality holds for every $t\in\mathbb{T}^d$. 2.6. Sobolev’s embedding theorem for $s=d/2$The following result is a variant of the well-known Moser-Trudinger inequality [42] (this result was independently established by Yudovich [44] and, more explicitly, by Pohozaev [28]). Trudinger’s result suggests that the Sobolev space $W^{d/2,2}(\mathbb{T}^d)$ is embedded in the Orlicz space $\exp(L_2)(\mathbb{T}^d)$ (see also Theorem 2.2 below). For detailed proof we refer the reader to [23], Lemma 2.2, and [27]. In what follows $\exp(L_2)$ denotes the Orlicz space associated with the Orlicz function $t\to e^{t^2}-1$, $t>0$. 2.7. Homogeneous semi-norms on Sobolev spacesIn what follows we need the notion of the homogeneous Sobolev seminorm: for $s=m\in\mathbb{Z}_+$, it is defined by
$$
\begin{equation*}
\|u\|_{W^{m,2}_{\mathrm{hom}}(\Pi)}^2 =\sum_{|\alpha|_1=m}\|\nabla^{\alpha}u\|_{L_2(\Pi)}^2.
\end{equation*}
\notag
$$
For $s\notin\mathbb{Z}_+$, $m=\lfloor s\rfloor$, it is defined by
$$
\begin{equation*}
\|u\|_{W^{s,2}_{\mathrm{hom}}(\Pi)}^2 =\sum_{|\alpha|_1=m}\int_{\Pi}\int_{\Pi}\frac{|(\nabla^{\alpha}u)(x) -(\nabla^{\alpha}u)(y)|^2}{|x-y|_2^{d+2(s-m)}}\,d\nu(x)\,d\nu(y).
\end{equation*}
\notag
$$
It is immediate that
$$
\begin{equation*}
\|u\|_{W^{s,2}_{\mathrm{hom}}(\Pi)}\leqslant \|u\|_{W^{s,2}(\Pi)}, \qquad u\in W^{s,2}(\Pi).
\end{equation*}
\notag
$$
In the case when $s$ is an integer, the following assertion is Theorem 1.1.16 in [25]. In [36] Solomyak used it (for even dimension $d$ and $s=d/2$) without a proof or reference. The following proof was provided to us by Rozenblum (according to him, this is a folklore result in the St Petersburg school). Rozenblum’s proof is simpler than our original argument, and we include it here with his kind permission. Theorem 2.3. Let $d\in\mathbb{N}$ and let $\Pi=(-\pi,\pi)^d$. If $u\in W^{s,2}(\Pi)$, $s>0$, is orthogonal (in $L_2(\Pi)$) to every polynomial of degree strictly less than $s$, then Proof. We only prove the assertion for noninteger $s$. Set $m=\lfloor s\rfloor$.
Assume the contrary and choose a sequence $(u_k)_{k\geqslant0}\subset W^{s,2}(\Pi)$ such that
In particular, for every $\alpha$ with $|\alpha|_1=m$ we have
$$
\begin{equation}
\|\nabla^{\alpha}u_k\|_{W^{s-m,2}_{\mathrm{hom}}(\Pi)}\to0, \qquad k\to\infty.
\end{equation}
\tag{2.3}
$$
It is crucial that $W^{s,2}(\Pi)$ is compactly embedded in $W^{m,2}(\Pi)$ (this fundamental fact is available, for instance, in Theorem 3.27 in [26]). Passing to a subsequence if needed, we can assume that $u_k\to u_\infty$ in $W^{m,2}(\Pi)$. For every $\alpha$ with $|\alpha|_1=m$, $\nabla^{\alpha}u_k\to \nabla^{\alpha}u_\infty$ in $L_2(\Pi)$. Passing to a subsequence if needed, we can assume that $\nabla^{\alpha}u_k\to \nabla^{\alpha}u_\infty$ almost everywhere. Fix $\alpha$ with $|\alpha|_1=m$. Set
$$
\begin{equation*}
v_k(x,y)=\frac{(\nabla^{\alpha}u_k)(x)-(\nabla^{\alpha}u_k)(y)}{|x-y|_2^{d+2(s-m)}}, \qquad x,y\in\Pi,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
v_{\infty}(x,y)=\frac{(\nabla^{\alpha}u_{\infty})(x) -(\nabla^{\alpha}u_{\infty})(y)}{|x-y|_2^{d+2(s-m)}}, \qquad x,y\in\Pi.
\end{equation*}
\notag
$$
It follows that $v_k\to v_\infty$ almost everywhere. On the other hand (2.3) means that $v_k\to0$ in $L_2(\Pi\times\Pi)$. Hence $v_\infty=0$. Equivalently, $\nabla^{\alpha}u_\infty$ is a constant.
Since $\nabla^{\alpha}u_\infty$ is a constant for every $\alpha$ such that $|\alpha|_1=m$, it follows that $u_\infty$ is a polynomial of degree $m$ (or less). Let $p$ be any polynomial of degree $m$ (or less). Since the mapping
$$
\begin{equation*}
f\to \langle f,p\rangle_{L_2(\Pi)}, \qquad f\in W^{m,2}(\Pi),
\end{equation*}
\notag
$$
is a continuous linear functional on $W^{m,2}(\Pi)$, it follows that
$$
\begin{equation*}
\langle u_k,p\rangle_{L_2(\Pi)}\to \langle u_{\infty},p\rangle_{L_2(\Pi)}, \qquad k\to\infty.
\end{equation*}
\notag
$$
On the other hand the choice of $u_k$ is such that
$$
\begin{equation*}
\langle u_k,p\rangle_{L_2(\Pi)}=0, \qquad k\geqslant0.
\end{equation*}
\notag
$$
Thus, for every polynomial $p$ of degree $m$ (or less). Since $u_\infty$ is itself a polynomial of degree $m$, it follows that $u_\infty=0$. Therefore, $u_k\to0$ in $W^{m,2}(\Pi)$, which contradicts the condition that $\|u_k\|_{W^{m,2}(\Pi)}=1$ for every $k\geqslant0$.
Theorem 2.3 is proved. § 3. Solomyak-type theorem on coveringsFormally, Theorem 3.1 below is new. However, its result is stated in [36]. Recall that the torus $\mathbb{T}^d$ is equipped with the normalised Haar measure $\nu$. For an Orlicz function $\Phi$ and $f\in L_{\Phi}(\mathbb{T}^d)$ set
$$
\begin{equation*}
J_f^{\Phi}(A)=\nu(A)\bigl\|\sigma_{1/\nu(A)}\mu(f|_A)\bigr\|_{L_{\Phi}}, \qquad A\subset\mathbb{T}^d, \quad \nu(A)>0.
\end{equation*}
\notag
$$
This definition is technically simpler (though, eventually, equivalent) than the one given in [36] (see formulae (4) and (13) there). Throughout, we view the torus $\mathbb{T}^d$ as a Cartesian product of $d$ copies of the one-dimensional torus $\mathbb{T}^1$, and a cube in $\mathbb{T}^d$ is defined as a Cartesian product of arcs of equal length. Theorem 3.1. Let $L_{\Phi}$ be a separable Orlicz space on $(0,1)$. For every $f\in L_{\Phi}(\mathbb{T}^d)$ and $n\in\mathbb{N}$ there exist $m(n)\leqslant c_dn$ and a collection $(\Pi_k)_{k=1}^{m(n)}$ of toric cubes in $\mathbb{T}^d$ such that (i) each point in $\mathbb{T}^d$ belongs to at least one $\Pi_k$, $1\leqslant k\leqslant m(n)$; (ii) each point in $\mathbb{T}^d$ belongs to at most $c_d$ cubes $\Pi_k$, $1\leqslant k\leqslant m(n)$; (iii) for every $1\leqslant k\leqslant m(n)$ we have $J_f^{\Phi}(\Pi_k)=\frac1n\|f\|_{L_{\Phi}}$. The lemma below manifests the fact that every Orlicz space is distributionally concave (see [2] for a detailed discussion of this notion). The use of this concept distinguishes our proof from the proof in [36]. We write $\bigoplus_{i\in\mathbb{I}}x_i$ for the disjoint sum of the functions $(x_i)_{i\in\mathbb{I}}$. Lemma 3.1. Let $\Phi$ be an Orlicz function, and let $L_{\Phi}$ be the corresponding Orlicz space, either on $(0,1)$ or on $(0,\infty)$. Then for every sequence $(f_k)_{k\geqslant1}\subset L_{\Phi}$ and every sequence of scalars $(\lambda_k)_{k\geqslant1}\subset(0,1)$ such that $\sum_{k\geqslant1}\lambda_k=1$. Proof. For definiteness we consider spaces on $(0,\infty)$. Let $\Psi$ be the complementary Orlicz function. Then
$$
\begin{equation*}
\|x\|_{L_{\Phi}}\leqslant\sup_{\|y\|_{L_\Psi}\leqslant 1}|\langle x,y\rangle|\leqslant 2\|x\|_{L_{\Phi}}
\end{equation*}
\notag
$$
(see equation (9.24) in [20]). Here
Choose $g_k\in L_{\Psi}$ such that $\|g_k\|_{L_{\Psi}}\leqslant 1$ and such that
$$
\begin{equation*}
\langle f_k,g_k\rangle\geqslant\frac12\|f_k\|_{L_{\Phi}}.
\end{equation*}
\notag
$$
Then
$$
\begin{equation*}
\begin{aligned} \, \sum_{k\geqslant1}\lambda_k\|f_k\|_{L_{\Phi}} & \leqslant 2\sum_{k\geqslant1}\lambda_k\langle f_k,g_k\rangle=2\sum_{k\geqslant1}\langle \sigma_{\lambda_k}f_k,\sigma_{\lambda_k}g_k\rangle \\ &=2\biggl\langle \bigoplus_{k\geqslant1}\sigma_{\lambda_k}f_k,\bigoplus_{k\geqslant1}\sigma_{\lambda_k}g_k\biggr\rangle\leqslant 4\biggl\|\bigoplus_{k\geqslant1}\sigma_{\lambda_k}f_k\|_{L_{\Phi}}\| \bigoplus_{k\geqslant1}\sigma_{\lambda_k}g_k\biggr\|_{L_{\Psi}}. \end{aligned}
\end{equation*}
\notag
$$
Since $\|g_k\|_{L_{\Psi}}\leqslant1$, it follows that $\|\Psi(g_k)\|_1\leqslant1$. Thus,
$$
\begin{equation*}
\biggl\|\Psi\biggl(\bigoplus_{k\geqslant1}\sigma_{\lambda_k}g_k\biggr)\biggr\|_1 =\sum_{k\geqslant1}\|\Psi(\sigma_{\lambda_k}g_k)\|_1 =\sum_{k\geqslant1}\lambda_k\|\Psi(g_k)\|_1\leqslant1
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\biggl\|\bigoplus_{k\geqslant1}\sigma_{\lambda_k}g_k\biggr\|_{L_{\Psi}}\leqslant 1.
\end{equation*}
\notag
$$
A combination of these estimates yields the assertion.
Lemma 3.1 is proved. The next lemma delivers the subadditivity of the functional $J_f^{\Phi}$ and is an easy consequence of Lemma 3.1. Lemma 3.2. Let $\Phi$ and $f$ be as in Theorem 3.1. If $(A_k)_{k=0}^n$ is an arbitrary Lebesgue measurable partition of $\mathbb{T}^d$, then Proof. Set $\lambda_k=\nu(A_k)$, $1\leqslant k\leqslant n$, so that $\sum _{k=1}^n \lambda_k=1$, and let
$$
\begin{equation*}
f_k=\sigma_{\lambda_k^{-1}}\mu(f|_{A_k}), \qquad 1\leqslant k\leqslant n.
\end{equation*}
\notag
$$
It is immediate that
$$
\begin{equation*}
\mu(f)=\mu\biggl(\bigoplus_{k=1}^n\sigma_{\lambda_k}f_k\biggr).
\end{equation*}
\notag
$$
By Lemma 3.1 we have
The lemma is proved. We equip the $\sigma$-algebra of Lebesgue measurable sets in $\mathbb{T}^d$ with the usual metric
$$
\begin{equation*}
\operatorname{dist}(A_1,A_2)=\nu(A_1\bigtriangleup A_2), \qquad A_1, A_2\subset \mathbb{T}^d.
\end{equation*}
\notag
$$
Here the symmetric difference is defined by the usual formula
$$
\begin{equation*}
A_1\bigtriangleup A_2=(A_1\setminus A_2)\cup(A_2\setminus A_1).
\end{equation*}
\notag
$$
Given $f\in L_{\Phi}(\mathbb{T}^d)$, define a function $F_f\colon [0,1]\to\mathbb{R}_+$ by
$$
\begin{equation*}
F_f(t)=2\|\mu(f)\chi_{(0,t)}\|_{L_{\Phi}}+2t^{1/2}\|f\|_{L_{\Phi}} +4t^{1/2}\|\sigma_{1/(2t^{1/2})}\mu(f)\|_{L_{\Phi}}, \qquad t\in[0,1].
\end{equation*}
\notag
$$
The following assertion improves Lemma 4 in [36] slightly and adjusts it to the case of $\mathbb{T}^d$. Lemma 3.3. Let $L_{\Phi}$ be a separable5 Orlicz space on $(0,1)$. Then for every ${f\in L_{\Phi}(\mathbb{T}^d)}$ the mapping $A\to J_f^{\Phi}(A)$ is continuous with respect to the metric $\operatorname{dist}$. More precisely, for all measurable sets $A_1,A_2\subset\mathbb{T}^d$, Proof. Fix $\epsilon\in(0,1)$ and suppose $\nu(A_1\bigtriangleup A_2)<\epsilon^2$. We consider the following two logically possible cases separately.
Case 1. Let $\nu(A_1)>\epsilon$ and $\nu(A_2)>\epsilon$. Set $A_3=A_1\cup A_2$. Note that
$$
\begin{equation*}
\nu(A_1)\leqslant\nu(A_3)\leqslant (1+\epsilon)\nu(A_1)\quad\text{and} \quad \nu(A_2)\leqslant \nu(A_3)\leqslant (1+\epsilon)\nu(A_2).
\end{equation*}
\notag
$$
By the triangle inequality we have
$$
\begin{equation*}
\begin{aligned} \, J_f^{\Phi}(A_3) &=\nu(A_3)\|\sigma_{1/\nu(A_3)}\mu(f|_{A_3})\|_{L_{\Phi}} \\ &\leqslant \nu(A_3)\|\sigma_{1/\nu(A_3)}\mu(f|_{A_2\setminus A_1})\|_{L_{\Phi}}+\nu(A_3)\|\sigma_{1/\nu(A_3)}\mu(f|_{A_1})\|_{L_{\Phi}}. \end{aligned}
\end{equation*}
\notag
$$
Clearly,
$$
\begin{equation*}
\nu(A_3)\|\sigma_{1/\nu(A_3)}\mu(f|_{A_2\setminus A_1})\|_{L_{\Phi}}\leqslant \|f|_{A_2\setminus A_1}\|_{L_{\Phi}}\leqslant \|\mu(f)\chi_{(0,\epsilon^2)}\|_{L_{\Phi}}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\nu(A_3)\|\sigma_{1/\nu(A_3)}\mu(f|_{A_1})\|_{L_{\Phi}}\leqslant \nu(A_3)\|\sigma_{1/\nu(A_1)}\mu(f|_{A_1})\|_{L_{\Phi}}=\frac{\nu(A_3)}{\nu(A_1)}\cdot J_f^{\Phi}(A_1).
\end{equation*}
\notag
$$
Since $\nu(A_3)<(1+\epsilon)\nu(A_1)$, it follows that
$$
\begin{equation*}
0\leqslant J_f^{\Phi}(A_3)-J_f^{\Phi}(A_1)\leqslant \|\mu(f)\chi_{(0,\epsilon^2)}\|_{L_{\Phi}} +\epsilon \cdot J_f^{\Phi}(A_1)\leqslant \|\mu(f)\chi_{(0,\epsilon^2)}\|_{L_{\Phi}}+\epsilon\|f\|_{L_{\Phi}}.
\end{equation*}
\notag
$$
Similarly,
$$
\begin{equation*}
0\leqslant J_f^{\Phi}(A_3)-J_f^{\Phi}(A_2)\leqslant \|\mu(f)\chi_{(0,\epsilon^2)}\|_{L_{\Phi}}+\epsilon\|f\|_{L_{\Phi}}.
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
|J_f^{\Phi}(A_1)-J_f^{\Phi}(A_2)|\leqslant 2\|\mu(f)\chi_{(0,\epsilon^2)}\|_E+2\epsilon\|f\|_{L_{\Phi}}\leqslant F_f(\epsilon^2),
\end{equation*}
\notag
$$
where the last estimate follows immediately from the definition of $F_f$. This completes the proof in Case 1.
Case 2. Let $\nu(A_1)\leqslant\epsilon$ or $\nu(A_2)\leqslant\epsilon$. Since $\nu(A_1\bigtriangleup A_2)<\epsilon^2$, we have simultaneously $\nu(A_1)\leqslant2\epsilon$ and $\nu(A_2)\leqslant2\epsilon$. From the definition of $J_f^{\Phi}$ we obtain
$$
\begin{equation*}
J_f^{\Phi}(A_k)\leqslant 2\epsilon\|\sigma_{1/(2\epsilon)}\mu(f)\|_{L_{\Phi}}, \qquad k=1,2.
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
|J_f^{\Phi}(A_1)-J_f^{\Phi}(A_2)|\leqslant 4\epsilon\|\sigma_{1/(2\epsilon)}\mu(f)\|_{L_{\Phi}}\leqslant F_f(\epsilon^2).
\end{equation*}
\notag
$$
This completes the proof in Case 2.
Lemma 3.3 is proved. The following assertion is well known (see, for instance, Appendix B in [15] or Theorem II.18.1 in [11] for a similar result on coverings by closed cubes). Observe that the argument in [11] extends practically verbatim to the coverings considered in this paper. Lemma 3.4 (Besicovitch covering lemma). For every $x\in\mathbb{T}^d$ let $\Pi_x\subset \mathbb{T}^d$ be an open nonempty toric cube with centre $x$. Then there exists $c_d\in\mathbb{N}$ and subsets $(S_l)_{l=1}^{c_d}$ of $\mathbb{T}^d$ such that (i) $\mathbb{T}^d=\bigcup_{l=1}^{c_d}\bigcup_{x\in S_l}\Pi_x$; (ii) $\Pi_{x_1}\cap\Pi_{x_2}=\varnothing$ for $x_1,x_2\in S_l$, $x_1\neq x_2$. Here the constant $c_d$ depends only on $d$, but not on the system $(\Pi_x)_{x\in\mathbb{T}^d}$. The proof of Theorem 3.1 follows a pattern established in [36], but covers the case of an arbitrary dimension $d$. According to Rozenblum, the idea to use coverings instead of partitions (unlike in earlier papers of Birman and Solomyak) belongs to him. In [36] (see also the earlier book [8]) a handcrafted covering lemma of Rozenblum’s was replaced by Besicovitch’s covering lemma. Proof of Theorem 3.1. Fix $f\in L_{\Phi}(\mathbb{T}^d)$. Let $\Pi_{x,t}$ be the open cube with centre $x\in \mathbb{T}^d$ and side $t\in (0,1)$. By Lemma 3.3 the function is continuous. By the intermediate value theorem there exists $t=t(x)$ such that Set $\Pi_x=\Pi_{x,t(x)}$, $x\in\mathbb{T}^d$. Consider the covering $\{\Pi_x\}_{x\in \mathbb{T}^d}$ of $\mathbb{T}^d$. Let $c_d\in\mathbb{N}$ and the sets $(S_l)_{l=1}^{c_d}$ be as in Lemma 3.4. Consider an arbitrary finite subset $A_l\subset S_l$. Note that
$$
\begin{equation*}
\{\Pi_x\}_{x\in A_l}\cup\biggl\{\bigcap_{x\in A_l}(\Pi_x)^c\biggr\}
\end{equation*}
\notag
$$
is a partition of $\mathbb{T}^d$. By (3.1) and Lemma 3.2 we have In other words, $|A_l|\leqslant 4n$ for every finite subset of $S_l$. This implies that the set $S_l$ is finite and $|S_l|\leqslant 4n$.
Set $\Pi_k=\Pi_{l,x}$, where the index $k$ stands for a pair $(l,x)$, where $x\in S_l$. It follows from above that there are at most $4c_dn$ distinct indices $k$. This completes the proof. Theorem 3.1 is proved. § 4. Proof of Theorem 1.1The following fact is standard and only presented for the convenience of the reader and because of the lack of a proper reference. It asserts that the homogeneous seminorm behaves well with respect to scaling. Fact 4.1. Let $\Pi=(-\pi\epsilon,\pi\epsilon)^d$, $0<\epsilon\leqslant 1$. Then In particular, In the proof of the next lemma, which is an extension of Lemma 2 in [36] to the case of an arbitrary dimension, we exploit Fact 4.1 crucially. Lemma 4.1. Let $d\in\mathbb{N}$. Let $\Pi\subset\mathbb{T}^d$ be an open toric cube. Let $\Phi(t)=t\log(e+t)$, $t>0$, and $f\in L_{\Phi}(\mathbb{T}^d)$. Then for every $u\in W^{d/2,2}(\Pi)$ that is orthogonal (in $L_2(\Pi)$) to every polynomial of degree $<d/2$ we have Proof. Without loss of generality $\Pi=(-\pi\epsilon,\pi\epsilon)^d$. After scaling we have
$$
\begin{equation*}
\int_{\Pi}|f|\cdot |u|^2\,d\nu=\epsilon^d\int_{\mathbb{T}^d}|\sigma_{1/\epsilon}f|\cdot |\sigma_{1/\epsilon}u|^2\,d\nu.
\end{equation*}
\notag
$$
By Hölder’s inequality (see, for example, Theorem II.5.2 in [21])
$$
\begin{equation*}
\int_{\mathbb{T}^d}F|G|^2\,d\nu\leqslant c_{\mathrm{abs}}\|F\|_{L_{\Phi}(\mathbb{T}^d)}\||G|^2\|_{\exp(L_1)(\mathbb{T}^d)}= c_{\mathrm{abs}}\|F\|_{L_{\Phi}(\mathbb{T}^d)}\|G\|_{\exp(L_2)(\mathbb{T}^d)}^2
\end{equation*}
\notag
$$
for all $F\in L_{\Phi}(\mathbb{T}^d)$ and all $G\in\exp(L_2)(\mathbb{T}^d)$. Thus,
$$
\begin{equation*}
\int_{\Pi}|f|\cdot |u|^2\,d\nu\leqslant c_{\mathrm{abs}} \epsilon^d\|\sigma_{1/\epsilon}f\|_{L_{\Phi}(\mathbb{T}^d)}\| \sigma_{1/\epsilon}u\|_{\exp(L_2)(\mathbb{T}^d)}^2.
\end{equation*}
\notag
$$
Clearly, $\sigma_{1/\epsilon}u$ is orthogonal to every polynomial of degree $<d/2$ on $\mathbb{T}^d$. By Theorems 2.2 and 2.3 we have By the definition of $J_f^{\Phi}$, A combination of the last three equalities yields the assertion.
Lemma 4.1 is proved. The following fact is standard and is only presented here for the convenience of the reader and due to the lack of a proper reference. Fact 4.2. Let $\Pi\subset\mathbb{T}^d$ be an open toric cube and let $P\colon L_2(\Pi)\to L_2(\Pi)$ be the projection onto the subspace spanned by the polynomials of degree $<d/2$. Then the following hold. (i) For every $u\in L_2(\Pi)$ the function $u-Pu$ is orthogonal (in $L_2(\Pi)$) to every polynomial $v$ of degree $<d/2$. (ii) For every $u\in W^{d/2,2}(\Pi)$ we have $\|u-Pu\|_{W^{d/2,2}_{\mathrm{hom}}(\Pi)}=\|u\|_{W^{d/2,2}_{\mathrm{hom}}(\Pi)}$. Lemma 4.2. Let $(\Pi_k)_{k=1}^K$ be a sequence of open toric cubes in $\mathbb{T}^d$. Assume that each point in $\mathbb{T}^d$ belongs to at most $C$ cubes $\Pi_k$, $1\leqslant k\leqslant K$. Then Here the constant $c_{d,s}$ depends only on $d$ and $s$, but not on the sequence $(\Pi_k)_{k=1}^K$. Proof. Step 1. Suppose $s$ is an integer. Then
$$
\begin{equation*}
\sum_{k=1}^K\|u\|_{W^{s,2}(\Pi_k)}^2=\sum_{|\alpha|_1\leqslant s}\sum_{k=1}^K\|\nabla^{\alpha}u\|_{L_2(\Pi_k)}^2.
\end{equation*}
\notag
$$
By assumption, for all $\alpha\in\mathbb{Z}^d_+$ such that $|\alpha|_1\leqslant s$ we have Therefore, where the constant $c_{d,s}'$ depends only on $d$ and $s$.
Step 2. Suppose $s\in(0,1)$. We identify $\mathbb{T}^d$ and $(-\pi,\pi)^d$. For $1\leqslant k\leqslant K$ fix a point $t^k\in\mathbb{T}^d$ such that the cube $t^k+\Pi$ is Euclidean. By the definition of a Sobolev space on a Euclidean cube we have
$$
\begin{equation*}
\|u\|_{W^{s,2}(\Pi_k)}^2=\|u\|_{L_2(\Pi_k)}^2 +\int_{t^k+\Pi_k}\int_{t^k+\Pi_k}\frac{|u(x-t^k)-u(y-t^k)|^2}{|x-y|_2^{d+2s}}\,d\nu(x)\,d\nu(y).
\end{equation*}
\notag
$$
By the definition of the distance6 on a torus
$$
\begin{equation*}
|x-y|_2\geqslant \operatorname{dist}(x,y), \qquad x,y\in t^k+\Pi_k.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\|u\|_{W^{s,2}(\Pi_k)}^2\leqslant \|u\|_{L_2(\Pi_k)}^2+\|v_s\|_{L_2(\Pi_k\times\Pi_k)}^2,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
v_s(x,y)=\frac{u(x)-u(y)}{\operatorname{dist}(x,y)^{d/2+s}}, \qquad x,y\in\mathbb{T}^d.
\end{equation*}
\notag
$$
By assumption
$$
\begin{equation*}
\sum_{k=1}^K\|u\|_{L_2(\Pi_k)}^2\leqslant C\|u\|_{L_2(\mathbb{T}^d)}^2 \quad\text{and}\quad \sum_{k=1}^K\|v_s\|_{L_2(\Pi_k\times\Pi_k)}^2\leqslant C^2\|v_s\|_{L_2(\mathbb{T}^d\times\mathbb{T}^d)}^2.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\sum_{k=1}^K\|u\|_{W^{s,2}(\Pi_k)}^2\leqslant C\|u\|_{L_2(\mathbb{T}^d)}^2+C^2\|v_s\|_{L_2(\mathbb{T}^d\times\mathbb{T}^d)}^2.
\end{equation*}
\notag
$$
By Theorem 2.1 we have
$$
\begin{equation*}
\|v_s\|_{L_2(\mathbb{T}^d\times\mathbb{T}^d)}^2\leqslant c''_{d,s}\|u\|_{W^{s,2}(\mathbb{T}^d)}^2,
\end{equation*}
\notag
$$
where the constant $c_{d,s}''$ depends only on $d$ and $s$. Therefore,
$$
\begin{equation*}
\sum_{k=1}^K\|u\|_{W^{s,2}(\Pi_k)}^2\leqslant C\|u\|_{L_2(\mathbb{T}^d)}^2+c''_{d,s}C^2\|u\|_{W^{s,2}(\mathbb{T}^d)}^2.
\end{equation*}
\notag
$$
Step 3. Now let $s$ be an arbitrary noninteger. By the definition of a Sobolev space on a cube we have
$$
\begin{equation*}
\|u\|_{W^{s,2}(\Pi_k)}^2\leqslant\|u\|_{W^{\lfloor s\rfloor,2}(\Pi_k)}^2+\sum_{|\alpha|_1=\lfloor s\rfloor}\|\nabla^{\alpha}u\|_{W^{s-\lfloor s\rfloor,2}(\Pi_k)}^2.
\end{equation*}
\notag
$$
By Step 1 we have
$$
\begin{equation*}
\sum_{k=1}^K\|u\|_{W^{\lfloor s\rfloor,2}(\Pi_k)}^2\leqslant c'_{d,\lfloor s\rfloor}C\|u\|_{W^{\lfloor s\rfloor,2}(\mathbb{T}^d)}^2\leqslant c'_{d,\lfloor s\rfloor}C\|u\|_{W^{s,2}(\mathbb{T}^d)}^2.
\end{equation*}
\notag
$$
By Step 2,
$$
\begin{equation*}
\sum_{k=1}^K\|\nabla^{\alpha}u\|_{W^{s-\lfloor s\rfloor,2}(\Pi_k)}^2\leqslant C\|\nabla^{\alpha}u\|_{L_2(\mathbb{T}^d)}+c''_{d,s-\lfloor s\rfloor}C^2\|\nabla^{\alpha}u\|_{W^{s-\lfloor s\rfloor,2}(\mathbb{T}^d)}^2.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\begin{aligned} \, \sum_{k=1}^K\|u\|_{W^{s,2}(\Pi_k)}^2 &\leqslant c'_{d,\lfloor s\rfloor}C\|u\|_{W^{s,2}(\mathbb{T}^d)}^2+C\sum_{|\alpha|_1=\lfloor s\rfloor}\|\nabla^{\alpha}u\|_{L_2(\mathbb{T}^d)} \\ &\qquad +C^2\sum_{|\alpha|_1=\lfloor s\rfloor}\|\nabla^{\alpha}u\|_{W^{s-\lfloor s\rfloor,2}(\mathbb{T}^d)}^2\leqslant c_{d,s}C^2\|u\|_{W^{s,2}(\mathbb{T}^d)}^2. \end{aligned}
\end{equation*}
\notag
$$
Lemma 4.2 is proved. The following assertion was proved by Solomyak for even $d$ (see Theorem 1 in [36]). We prove it for an arbitrary dimension. Lemma 4.3. Let $d\in\mathbb{N}$. Let $\Phi(t)=t\log(e+t)$, $t>0$, and let $0\leqslant f\in L_{\Phi}(\mathbb{T}^d)$. Then for every $n\in\mathbb{N}$ there exists an operator $K_n\colon L_2(\mathbb{T}^d)\to L_2(\mathbb{T}^d)$ such that $\operatorname{rank}(K_n)\leqslant c_dn$ and The operators $K_n,K_n^{\ast}\colon L_2(\mathbb{T}^d)\to L_2(\mathbb{T}^d)$ extend to bounded operators $K_n,K_n^{\ast}$: $L_1(\mathbb{T}^d)\to L_{\infty}(\mathbb{T}^d)$. Proof. Let $(\Pi_k)_{1\leqslant k\leqslant m(n)}$ be the sequence of toric cubes constructed in Theorem 3.1. As always, $\chi_{\Pi_k}$ is the indicator function of $\Pi_k$, $1\leqslant k\leqslant m(n)$.
Let $P_k\colon L_2(\mathbb{T}^d)\to L_2(\mathbb{T}^d)$ be the projection such that
$$
\begin{equation*}
P_k=M_{\chi_{\Pi_k}}P_kM_{\chi_{\Pi_k}}, \qquad 1\leqslant k\leqslant m(n),
\end{equation*}
\notag
$$
where $P_k\colon L_2(\Pi_k)\to L_2(\Pi_k)$ is the projection onto the linear subspace of all polynomials of degree $<d/2$.
Set
$$
\begin{equation*}
\Delta_k=\Pi_k\setminus\bigcup_{l<k}\Pi_l, \qquad 1\leqslant k\leqslant m(n).
\end{equation*}
\notag
$$
By Theorem 3.1, (i), the sequence $(\Delta_k)_{k=1}^{m(n)}$ is a partition of $\mathbb{T}^d$. Set
It is immediate from the definition that $K_n,K_n^{\ast}\colon L_2(\mathbb{T}^d)\to L_2(\mathbb{T}^d)$ extend to bounded operators $K_n,K_n^{\ast}\colon L_1(\mathbb{T}^d)\to L_{\infty}(\mathbb{T}^d)$. Since $m(n)\leqslant c_dn$ by Theorem 3.1, it follows that $\operatorname{rank}(K_n)\leqslant c_dn$ (for another constant $c_d$). We have
$$
\begin{equation*}
\int_{\mathbb{T}^d}f|u-K_nu|^2\,d\nu=\sum_{k=1}^{m(n)} \int_{\Delta_k}f|u-K_nu|^2\,d\nu=\sum_{k=1}^{m(n)}\int_{\Delta_k}f|u-P_ku|^2\,d\nu.
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation}
\int_{\mathbb{T}^d}f|u-K_nu|^2\,d\nu\leqslant \sum_{k=1}^{m(n)}\int_{\Pi_k}f |u-P_ku|^2\,d\nu.
\end{equation}
\tag{4.1}
$$
By Fact 4.2, (i), the function $u-P_ku$ satisfies the assumptions of Lemma 4.1. By Lemma 4.1 and Fact 4.2, (ii),
$$
\begin{equation*}
\int_{\Pi_k}f|u-P_ku|^2\,d\nu\leqslant c_d J_f^{\Phi}(\Pi_k)\cdot \|u-P_ku\|_{W^{d/2,2}_{\mathrm{hom}}(\Pi_k)}^2 =c_d J_f^{\Phi}(\Pi_k)\cdot \|u\|_{W^{d/2,2}_{\mathrm{hom}}(\Pi_k)}^2.
\end{equation*}
\notag
$$
Combining the last estimate with Theorem 3.1, (iii), we obtain
$$
\begin{equation*}
\int_{\Pi_k}f|u-P_ku|^2\,d\nu\leqslant \frac{c_d}n\|f\|_{L_{\Phi}}\cdot \|u\|_{W^{d/2,2}_{\mathrm{hom}}(\Pi_k)}^2,
\end{equation*}
\notag
$$
and therefore, by (4.1),
$$
\begin{equation*}
\int_{\mathbb{T}^d}f|u-K_nu|^2\,d\nu\leqslant\frac{c_d}n\|f\|_{L_{\Phi}} \sum_{k=1}^{m(n)}\|u\|_{W^{d/2,2}_{\mathrm{hom}}(\Pi_k)}^2.
\end{equation*}
\notag
$$
Now the assertion follows from Lemma 4.2.
Lemma 4.3 is proved. The approximation given in Lemma 4.3 above yields a quasinorm estimate as in Theorem 1.1 in a standard fashion (see the schematic exposition on p. 58 in [37] and some earlier results: for example, Theorem 3.3 in [5]). We provide a complete proof for the convenience of the reader. Remark 4.1. In the proof below inner products are understood in the following sense: let $\xi,\eta\in L_1(\mathbb{T}^d)$ be such that $\xi\overline{\eta}\in L_1(\mathbb{T}^d)$; then set $\langle\xi,\eta\rangle=\int_{\mathbb{T}^d}\xi\overline{\eta}$. If the operators $K,K^{\ast}\colon L_2(\mathbb{T}^d)\to L_2(\mathbb{T}^d)$ extend to bounded operators $K,K^{\ast}\colon {L_1(\mathbb{T}^d)\to L_{\infty}(\mathbb{T}^d)}$, then $\langle K\xi,\eta\rangle=\langle\xi,K^{\ast}\eta\rangle$ whenever $\xi\overline{\eta}\in L_1(\mathbb{T}^d)$. Proof of Theorem 1.1. Without loss of generality assume that $f\geqslant0$.
Let $c_d$ be the constant in Lemma 4.3 (we assume it to be an integer). Take $m\in\mathbb{N}$ such that $m\geqslant 3c_d$. Let $n\in\mathbb{N}$ be such that $m\in[3c_dn,3c_d(n+1))$. Let the operator $K_n\colon L_2(\mathbb{T}^d)\to L_2(\mathbb{T}^d)$ be the one whose existence was established in Lemma 4.3. We have $\operatorname{rank}(K_n)\leqslant c_dn$ and
$$
\begin{equation*}
\int_{\mathbb{T}^d}f|u-K_nu|^2\,d\nu\leqslant\frac{c_d}n\|f\|_{L_{\Phi}} \|u\|_{W^{d/2,2}}^2, \qquad u\in W^{d/2,2}(\mathbb{T}^d).
\end{equation*}
\notag
$$
It is immediate that7
$$
\begin{equation*}
\begin{aligned} \, \int_{\mathbb{T}^d}f|u-K_nu|^2\,d\nu &=\langle f\cdot u,u\rangle-\langle f\cdot u,K_nu\rangle-\langle f\cdot K_nu,u\rangle+\langle f\cdot K_nu,K_nu\rangle \\ &=\langle M_fu,u\rangle-\langle K_n^{\ast}M_fu,u\rangle-\langle M_fK_nu,u\rangle+\langle K_n^{\ast}M_fK_nu,u\rangle \\ &=\langle T_nu,u\rangle, \end{aligned}
\end{equation*}
\notag
$$
where
Thus,
$$
\begin{equation*}
|\langle T_nu,u\rangle|\leqslant \frac{c_d}n\|f\|_{L_{\Phi}}\bigl\|(1-\Delta_{\mathbb{T}^d})^{d/4}u\bigr\|_2^2, \qquad u\in W^{d/2,2}(\mathbb{T}^d).
\end{equation*}
\notag
$$
By definition $(1-\Delta_{\mathbb{T}^d})^{\frac{d}{4}}$ is a bijection from $W^{d/2,2}(\mathbb{T}^d)$ to $L_2(\mathbb{T}^d)$. Therefore, we have
$$
\begin{equation*}
|\langle T_n(1-\Delta_{\mathbb{T}^d})^{-d/4}v,(1-\Delta_{\mathbb{T}^d})^{-d/4}v\rangle|\leqslant \frac{c_d}n\|f\|_{L_{\Phi}}\|v\|_2^2, \qquad v\in L_2(\mathbb{T}^d).
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
|\langle (1-\Delta_{\mathbb{T}^d})^{-d/4}T_n(1-\Delta_{\mathbb{T}^d})^{-d/4}v,v\rangle|\leqslant \frac{c_d}n\|f\|_{L_{\Phi}}\|v\|_2^2, \qquad v\in L_2(\mathbb{T}^d).
\end{equation*}
\notag
$$
Since $T_n$ is self-adjoint, from the definition of the operator norm we infer
$$
\begin{equation*}
\bigl\|(1-\Delta_{\mathbb{T}^d})^{-d/4}T_n(1-\Delta_{\mathbb{T}^d})^{-d/4}\bigr\|_{\infty}\leqslant \frac{c_d}n\|f\|_{L_{\Phi}}.
\end{equation*}
\notag
$$
Using the notation
$$
\begin{equation*}
L_n=(1-\Delta_{\mathbb{T}^d})^{-d/4}\cdot(K_n^{\ast}M_f+M_fK_n-K_n^{\ast}M_fK_n)\cdot (1-\Delta_{\mathbb{T}^d})^{-d/4},
\end{equation*}
\notag
$$
we rewrite the above inequality as
$$
\begin{equation*}
\bigl\|(1-\Delta_{\mathbb{T}^d})^{-d/4}M_f(1-\Delta_{\mathbb{T}^d})^{-d/4}-L_n\bigr\|_{\infty}\leqslant \frac{c_d}n\|f\|_{L_{\Phi}}.
\end{equation*}
\notag
$$
Since the rank of operator $K_n$ (and therefore of $K_n^{\ast}$) does not exceed $c_dn$, it follows that $\operatorname{rank}(L_n)\leqslant 3c_dn$. Hence
$$
\begin{equation*}
\inf_{\operatorname{rank}(S)\leqslant 3c_dn}\bigl\|(1-\Delta_{\mathbb{T}^d})^{-d/4} M_f(1-\Delta_{\mathbb{T}^d})^{-d/4}-S\bigr\|_{\infty}\leqslant \frac{c_d}n\|f\|_{L_{\Phi}}.
\end{equation*}
\notag
$$
That is,
$$
\begin{equation*}
\mu\bigl(3c_dn,(1-\Delta_{\mathbb{T}^d})^{-d/4}M_f(1-\Delta_{\mathbb{T}^d})^{-d/4})\leqslant \frac{c_d}n\|f\|_{L_{\Phi}}.
\end{equation*}
\notag
$$
Since $m+1\geqslant 3c_d(n+1)$, it follows that and
$$
\begin{equation}
\mu\bigl(m,(1-\Delta_{\mathbb{T}^d})^{-d/4}M_f(1-\Delta_{\mathbb{T}^d})^{-d/4}\bigr)\leqslant \frac{6c_d^2}{m+1}\|f\|_{L_{\Phi}}, \qquad m\geqslant 3c_d.
\end{equation}
\tag{4.2}
$$
Now, for $m\in\mathbb{Z}_+$ such that $m<3c_d$ we have
$$
\begin{equation*}
\begin{aligned} \, &\mu\bigl(m,(1-\Delta_{\mathbb{T}^d})^{-d/4}M_f(1-\Delta_{\mathbb{T}^d})^{-d/4}\bigr) \leqslant \mu\bigl(0,(1-\Delta_{\mathbb{T}^d})^{-d/4}M_f(1-\Delta_{\mathbb{T}^d})^{-d/4}\bigr) \\ &\qquad=\bigl\|(1-\Delta_{\mathbb{T}^d})^{-d/4}M_f(1-\Delta_{\mathbb{T}^d})^{-d/4}\bigr\|_{\infty} \leqslant c_d\|f\|_{L_{\Phi}}\leqslant\frac{3c_d^2}{m+1}\|f\|_{L_{\Phi}}. \end{aligned}
\end{equation*}
\notag
$$
Hence (4.2) also holds for $m<3c_d$. Thus,
$$
\begin{equation*}
\bigl\|(1-\Delta_{\mathbb{T}^d})^{-d/4}M_f(1-\Delta_{\mathbb{T}^d})^{-d/4}\bigr\|_{1,\infty}\leqslant 6c_d^2\|f\|_{L_{\Phi}}.
\end{equation*}
\notag
$$
Theorem 1.1 is proved. § 5. Symmetrized Solomyak-type estimate for $\mathcal{L}_{1,\infty}$ in $ {\mathbb{R}}^d$This section is devoted to the proof of Theorem 1.3. 5.1. The function $f$ has support on the unit cubeWhen $f$ has support on $(-1,1)^d$, we can extend $f$ to a function on $\mathbb{T}^d$ (for example, by identifying $\mathbb{T}^d$ with $(-\pi,\pi)^d$ and setting $f=0$ on $(-\pi,\pi)^2\setminus(-1,1)^d$). Lemma 5.1. Let $f$, $0\leqslant f\in L_{\infty}(\mathbb{R}^d)$, have support in $(-1,1)^d$. Then8 where $a\in l_{\infty}(\mathbb{Z}^d)$ does not depend on $f$ and satisfies This is, effectively, a combination of Lemmas 4.5 and 4.6 in [39]. These were established for the cube $(0,1)^d$, but taking $(-1,1)^d$ instead makes no difference. The following lemma yields the assertion of Theorem 1.3 in the special case when $f$ has support in the cube $(-1,1)^d$. Recall that $\Phi(t)=t\log(e+t)$, $t>0$. Proof. Without loss of generality let $f\geqslant0$. The operator
$$
\begin{equation*}
(1-\Delta_{\mathbb{R}^d})^{-d/4}M_f(1-\Delta_{\mathbb{R}^d})^{-d/4}
\end{equation*}
\notag
$$
is bounded. Using the standard identities
$$
\begin{equation*}
\mu(TT^{\ast})=\mu(T^{\ast}T) \quad\text{and}\quad \|TT^{\ast}\|_{1,\infty}=\|T^{\ast}T\|_{1,\infty}
\end{equation*}
\notag
$$
we conclude that
$$
\begin{equation*}
\bigl\|(1-\Delta_{\mathbb{R}^d})^{-d/4}M_f(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr\|_{1,\infty} =\bigl\|M_{f^{1/2}}(1-\Delta_{\mathbb{R}^d})^{-d/2}M_{f^{1/2}}\bigr\|_{1,\infty}.
\end{equation*}
\notag
$$
By Lemma 5.1,
$$
\begin{equation*}
\begin{aligned} \, &\bigl\|M_{f^{1/2}}(1-\Delta_{\mathbb{R}^d})^{-d/2}M_{f^{1/2}}\bigr\|_{1,\infty} =\|M_{f^{1/2}}a(\nabla_{\mathbb{T}^d})M_{f^{1/2}}\|_{1,\infty} \\ &\quad\leqslant c_d\|M_{f^{1/2}}(1-\Delta_{\mathbb{T}^d})^{-d/2}M_{f^{1/2}}\|_{1,\infty} =c_d\bigl\|(1-\Delta_{\mathbb{T}^d})^{-d/4}M_f(1-\Delta_{\mathbb{T}^d})^{-d/4}\bigr\|_{1,\infty}. \end{aligned}
\end{equation*}
\notag
$$
The assertion follows now from Theorem 1.1.
The lemma is proved. 5.2. The function $f$ has support outside the unit ballIn what follows we equip the unit ball $\mathbb{B}^d$ in $\mathbb{R}^d$ with the Lebesgue measure. Lemma 5.3. The operator is unitary on $L_2(\mathbb{R}^d)$. Consequently, the operator $V\colon L_1(\mathbb{R}^d)\to L_1(\mathbb{R}^d)$ given by the formula is an isometry. Proof. Let $s_k=\frac{t_k}{|t|^2}$. Then
$$
\begin{equation*}
\frac{\partial s_k}{\partial t_l}=-\frac{2t_kt_l}{|t|^4}, \qquad k\neq l,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\frac{\partial s_k}{\partial t_k}=\frac{|t|^2-2t_k^2}{|t|^4}.
\end{equation*}
\notag
$$
Hence we can write the Jacobian as
$$
\begin{equation*}
J=|t|^{-2}\cdot\biggl(1-2\biggl(\frac{t_k}{|t|}\cdot\frac{t_l}{|t|}\biggr)_{1\leqslant k,l\leqslant d}\biggr).
\end{equation*}
\notag
$$
The matrix is obviously a rank-one projection in Hilbert space $\mathbb{C}^d$. In other words, it is unitarily equivalent to the matrix unit $E_{11}$ (that is, to the matrix with $(1,1)$-entry one and other entries zeros). Hence
It follows that
$$
\begin{equation*}
\int_{\mathbb{R}^d}\eta(s)\,d\nu(s)=\int_{\mathbb{R}^d}\eta\biggl(\frac{t}{|t|^2}\biggr)\cdot |\operatorname{det}(J)(t)|\,d\nu(t)=\int_{\mathbb{R}^d}\eta\biggl(\frac{t}{|t|^2}\biggr)\cdot |t|^{-2d}\,d\nu(t).
\end{equation*}
\notag
$$
Setting $\eta=|\xi|^2$ we can write
$$
\begin{equation*}
\int_{\mathbb{R}^d}|\xi|^2(s)\,d\nu(s)=\int_{\mathbb{R}^d}|\xi|^2\biggl(\frac{t}{|t|^2}\biggr) \cdot |t|^{-2d}\,d\nu(t).
\end{equation*}
\notag
$$
In other words,
$$
\begin{equation*}
\|\xi\|_{L_2(\mathbb{R}^d)}^2=\|U\xi\|_{L_2(\mathbb{R}^d)}^2.
\end{equation*}
\notag
$$
The lemma is proved. It is important to note that $U=U^{-1}$. The following lemma can either be established via a (lengthy) direct calculation or derived from general geometric results (see, for example, Ch. III, § 7, in [19]). The symbol $\partial_k$ denotes the partial derivative with respect to the $k$th coordinate. Corollary 5.1. For every $n\in\mathbb{N}$, Here the polynomials $p_{\gamma}$ for $|\gamma|_1=2n$ are of order $4n$ (in fact, they are scalar multiples of $h_{4n}$), while the polynomials $p_{\gamma}$ for $|\gamma|_1<2n$ have lower orders. Proof. By Lemma 5.4, is a differential operator of order $2$ with polynomial coefficients of degree $4$ or less. Hence $U(1-\Delta_{\mathbb{R}^d})^nU^{-1}$ is a differential operator of order $2n$ with polynomial coefficients of degree $4n$ or less. The degrees of the polynomials $p_{\gamma}$ can be evaluated using the Leibniz rule.
The corollary is proved. Indeed,
$$
\begin{equation*}
\mu(TSS^{\ast}T^{\ast})=\mu^2(TS)\leqslant\|S\|_{\infty}^2\mu^2(T)=\|S\|_{\infty}^2\mu(TT^{\ast}).
\end{equation*}
\notag
$$
Let $C^n(\mathbb{R}^d)$ be the collection of all $n$-fold continuously differentiable complex-valued functions such that the function itself and all of its derivatives up to order $n$ are bounded. We have
$$
\begin{equation*}
\partial^{\gamma}M_g=\sum_{\substack{\gamma_1+\gamma_2=\gamma\\ \gamma_1,\gamma_2\geqslant0}}c_{\gamma_1,\gamma_2} M_{\partial^{\gamma_1}g}\,\partial^{\gamma_2}.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\|\partial^{\gamma}M_g(1-\Delta_{\mathbb{R}^d})^{-n}\|_{\infty} \leqslant\sum_{\substack{\gamma_1+\gamma_2=\gamma\\ \gamma_1,\gamma_2\geqslant0}}|c_{\gamma_1,\gamma_2}|\, \|M_{\partial^{\gamma_1}g}\|_{\infty}\|\partial^{\gamma_2}(1-\Delta_{\mathbb{R}^d})^{-n}\|_{\infty}.
\end{equation*}
\notag
$$
The operator $\partial^{\gamma_2}(1-\Delta_{\mathbb{R}^d})^{-n}$ on the right-hand side is bounded in virtue of functional calculus. By assumption we have
$$
\begin{equation*}
\|M_{\partial^{\gamma_1}g}\|_{\infty}\leqslant\|g\|_{C^{2n}(\mathbb{R}^d)},
\end{equation*}
\notag
$$
and the assertion follows. The following lemma (for $z=d/4$) is the crucial technical tool in the proof of Theorem 1.3. Its proof relies on Hadamard’s three-lines theorem. Lemma 5.5. For every real-valued $\phi\in C^{\infty}_c(\mathbb{R}^d)$ the operator is well defined and bounded on $L_2(\mathbb{R}^d)$. Here $h_z(t)=|t|^z$, $t\in\mathbb{R}^d$. Proof. First note that the operator $M_{h_{4z}\phi}U^{-1}(1-\Delta_{\mathbb{R}^d})^{-z}$ is bounded on $L_2(\mathbb{R}^d)$ (as a composition of bounded operators). If $\xi\in L_2(\mathbb{R}^d)$, then $M_{h_{4z}\phi}U^{-1}(1-\Delta_{\mathbb{R}^d})^{-z}\xi$ is also an element of $L_2(\mathbb{R}^d)$, and therefore it is a tempered distribution. Hence is also a tempered distribution. We aim to show that the latter tempered distribution is actually an element of $L_2(\mathbb{R}^d)$.
Let $\eta\in\mathcal{S}(\mathbb{R}^d)$ (that is, $\eta$ is a Schwartz function). Consider the function
$$
\begin{equation*}
F\colon z\to\langle T_z\xi,\eta\rangle=\bigl\langle M_{h_{4z}\phi}U^{-1}(1-\Delta_{\mathbb{R}^d})^{-z}\xi, (1-\Delta_{\mathbb{R}^d})^{\overline{z}}\eta\bigr\rangle, \qquad \operatorname{Re}(z)\geqslant0.
\end{equation*}
\notag
$$
The function
$$
\begin{equation*}
z\to M_{h_{4z}\phi}U^{-1}(1-\Delta_{\mathbb{R}^d})^{-z}\xi, \qquad \operatorname{Re}(z)\geqslant0,
\end{equation*}
\notag
$$
is $L_2(\mathbb{R}^d)$-valued analytic (and continuous on the boundary). The function
$$
\begin{equation*}
z\to (1-\Delta_{\mathbb{R}^d})^{\overline{z}}\eta, \qquad\operatorname{Re}(z)\geqslant0,
\end{equation*}
\notag
$$
is $L_2(\mathbb{R}^d)$-valued anti-analytic (and continuous on the boundary). Thus, $F$ is analytic and continuous on the boundary.
We have
$$
\begin{equation*}
\begin{aligned} \, |F(i\lambda)| &\leqslant\|M_{h_{4i\lambda}\phi}U^{-1} (1-\Delta_{\mathbb{R}^d})^{-i\lambda}\xi\|_{L_2(\mathbb{R}^d)}\| (1-\Delta_{\mathbb{R}^d})^{-i\lambda}\eta\|_{L_2(\mathbb{R}^d)} \\ &\leqslant \|\phi\|_{L_{\infty}(\mathbb{R}^d)}\|\xi\|_{L_2(\mathbb{R}^d)}\|\eta\|_{L_2(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
Also,
$$
\begin{equation*}
\begin{aligned} \, |F(n+i\lambda)| &\leqslant\|T_{n+i\lambda}\xi\|_{L_2(\mathbb{R}^d)}\|\eta\|_{L_2(\mathbb{R}^d)} \\ &\leqslant \|U(1-\Delta_{\mathbb{R}^d})^nM_{h_{4n+4i\lambda}\phi}U^{-1} (1-\Delta_{\mathbb{R}^d})^{-n}\|_{\infty}\|\xi\|_{L_2(\mathbb{R}^d)} \|\eta\|_{L_2(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
For brevity set $\alpha(t)=t/|t|^2$, $t\in\mathbb{R}^d$. By Corollary 5.1 we have
$$
\begin{equation*}
\begin{aligned} \, &U(1-\Delta_{\mathbb{R}^d})^nM_{h_{4n+4i\lambda}\phi}U^{-1}(1-\Delta_{\mathbb{R}^d})^{-n} \\ &\qquad =U(1-\Delta_{\mathbb{R}^d})^nU^{-1}\cdot UM_{h_{4n+4i\lambda}\phi}U^{-1}\cdot (1-\Delta_{\mathbb{R}^d})^{-n} \\ &\qquad =\sum_{|\gamma|_1\leqslant 2n}\partial^{\gamma}M_{p_{\gamma}}\cdot M_{h_{-4n-4i\lambda}\cdot (\phi\mathbin{\circ}\alpha)}\cdot (1-\Delta_{\mathbb{R}^d})^{-n}, \end{aligned}
\end{equation*}
\notag
$$
where the last equality follows from
$$
\begin{equation*}
UM_{h_z\phi}U^{-1}=M_{h_{-z}\cdot (\phi\mathbin{\circ}\alpha)}, \qquad z\in\mathbb{C}.
\end{equation*}
\notag
$$
Note that $\phi\circ\alpha$ vanishes in a neighbourhood of $0$. Fix $\epsilon>0$ such that $\phi\circ\alpha=0$ on $\epsilon\mathbb{B}^d$. An elementary calculation shows that
$$
\begin{equation*}
p_{\gamma}\cdot h_{-4n-4i\lambda}\in C^{2n}(\mathbb{R}^d\setminus\epsilon\mathbb{B}^d)
\end{equation*}
\notag
$$
and, moreover,
$$
\begin{equation*}
\|p_{\gamma}\cdot h_{-4n-4i\lambda}\|_{C^{2n}(\mathbb{R}^d\setminus\epsilon\mathbb{B}^d)}\leqslant c_{n,\gamma}(1+|\lambda|)^{2n}.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
p_{\gamma}\cdot h_{-4n-4i\lambda}\cdot (\phi\mathbin{\circ}\alpha)\in C^{2n}(\mathbb{R}^d)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\|p_{\gamma}\cdot h_{-4n-4i\lambda}\cdot (\phi\mathbin{\circ}\alpha)\|_{C^{2n}(\mathbb{R}^d)}\leqslant c_{n,\gamma,\phi}(1+|\lambda|)^{2n}.
\end{equation*}
\notag
$$
By the triangle inequality and Fact 5.2 we have
$$
\begin{equation*}
\begin{aligned} \, &\|U(1-\Delta_{\mathbb{R}^d})^nM_{h_{4n+4i\lambda}\phi}U^{-1} (1-\Delta_{\mathbb{R}^d})^{-n}\|_{\infty} \\ &\qquad \leqslant \sum_{|\gamma|_1\leqslant 2n}c_{n,\gamma}c_{n,\gamma,\phi}(1+|\lambda|)^{2n} =c_{n,\phi}(1+|\lambda|)^{2n}. \end{aligned}
\end{equation*}
\notag
$$
We conclude that
$$
\begin{equation*}
|F(n+i\lambda)|\leqslant c_{n,\phi}(1+|\lambda|)^{2n}\|\xi\|_{L_2(\mathbb{R}^d)}\|\eta\|_{L_2(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
Next, we claim that $F$ is bounded in the strip $\{0\leqslant \operatorname{Re}(z)\leqslant n\}$. Indeed,
$$
\begin{equation*}
\begin{aligned} \, |F(z)| &\leqslant\|h_{4z}\cdot\phi\|_{L_{\infty}(\mathbb{R}^d)} \|(1-\Delta_{\mathbb{R}^d})^{-z}\xi\|_{L_2(\mathbb{R}^d)}\| (1-\Delta_{\mathbb{R}^d})^{\overline{z}}\eta\|_{L_2(\mathbb{R}^d)} \\ &\leqslant c_{n,\phi}'\|\xi\|_{L_2(\mathbb{R}^d)}\|(1-\Delta_{\mathbb{R}^d})^n\eta\|_{L_2(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
Let
$$
\begin{equation*}
G(z)=e^{z^2}F(z), \qquad \operatorname{Re}(z)\geqslant0.
\end{equation*}
\notag
$$
It follows that
$$
\begin{equation*}
|G(i\lambda)|\leqslant \|\phi\|_{L_{\infty}(\mathbb{R}^d)}\|\xi\|_{L_2(\mathbb{R}^d)}\|\eta\|_{L_2(\mathbb{R}^d)} \quad\!\!\text{and}\!\!\quad |G(n+i\lambda)|\leqslant c''_{n,\phi}\|\xi\|_{L_2(\mathbb{R}^d)}\|\eta\|_{L_2(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
In addition to that, the function $G$ is bounded in the strip $\{0\leqslant \operatorname{Re}(z)\leqslant n\}$ as $F$ is bounded there. Now we are in a position to apply Hadamard’s three-lines theorem, which yields
$$
\begin{equation*}
|G(z)|\leqslant \max\{\|\phi\|_{L_{\infty}(\mathbb{R}^d)},c'_{n,\phi}\}\cdot \|\xi\|_{L_2(\mathbb{R}^d)}\|\eta\|_{L_2(\mathbb{R}^d)}, \qquad 0\leqslant \operatorname{Re}(z)\leqslant n.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
|F(z)|\leqslant |e^{-z^2}|\cdot \max\{\|\phi\|_{L_{\infty}(\mathbb{R}^d)},c'_{n,\phi}\}\cdot \|\xi\|_{L_2(\mathbb{R}^d)}\|\eta\|_{L_2(\mathbb{R}^d)}, \qquad 0\leqslant \operatorname{Re}(z)\leqslant n.
\end{equation*}
\notag
$$
In other words, the functional
$$
\begin{equation*}
\eta\to \langle T_z\xi,\eta\rangle, \qquad \eta\in\mathcal{S}(\mathbb{R}^d),
\end{equation*}
\notag
$$
extends to a bounded functional on $L_2(\mathbb{R}^d)$ (and the norm of this functional is controlled by $c_z\|\xi\|_{L_2(\mathbb{R}^d)}$). By Riesz’s lemma $T_z\xi\in L_2(\mathbb{R}^d)$ and
$$
\begin{equation*}
\|T_z\xi\|_{L_2(\mathbb{R}^d)}\leqslant c_z\|\xi\|_{L_2(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
Since $\xi\in L_2(\mathbb{R}^d)$ is arbitrary, it follows that $T_z$ is well defined and bounded on $L_2(\mathbb{R}^d)$.
Lemma 5.5 is proved. The assertion of Lemma 5.6 is of crucial importance in the proof of Theorem 1.3. Lemma 5.6. Suppose $f\in L_{\infty}(\mathbb{R}^d)$ has support on the set $\mathbb{R}^d\setminus\mathbb{B}^d$. Then Proof. Recall the notation $\alpha(t)=t/|t|^2$, $t\in\mathbb{R}^d$. We have
Fix a real-valued function $\phi\in C^{\infty}_c(\mathbb{R}^d)$ such that $\phi=1$ on $\mathbb{B}^d$. Since $f\circ\alpha$ has support on $\mathbb{B}^d$, it follows that
$$
\begin{equation*}
f\mathbin{\circ}\alpha = (f\mathbin{\circ}\alpha)\cdot\phi=Uf\cdot h_d\phi.
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
\begin{aligned} \, &M_{f\mathbin{\circ}\alpha} \cdot U^{-1}(1-\Delta_{\mathbb{R}^d})^{-d/2}U\cdot M_{f\mathbin{\circ}\alpha} \\ &\qquad=M_{Uf}\cdot M_{h_d\phi} U^{-1}(1-\Delta_{\mathbb{R}^d})^{-d/2}UM_{h_d\phi}\cdot M_{Uf}=TSS^{\ast}T^{\ast}, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
T=M_{Uf}(1-\Delta_{\mathbb{R}^d})^{-d/4} \quad\text{and}\quad S=(1-\Delta_{\mathbb{R}^d})^{d/4}M_{h_d\phi}U^{-1}(1-\Delta_{\mathbb{R}^d})^{-d/4}.
\end{equation*}
\notag
$$
Combining Lemma 5.5 and Fact 5.1 we complete the proof.
The lemma is proved. 5.3. Proof of Theorem 1.3The proof of the following proposition is deferred to § 7. Now we are ready to prove the main result in this section. Proof of Theorem 1.3. Without loss of generality let $f\geqslant0$. First assume that $f\in L_{\infty}(\mathbb{R}^d)$.
It is obvious that
$$
\begin{equation*}
\begin{aligned} \, (1-\Delta_{\mathbb{R}^d})^{-d/4}M_f(1-\Delta_{\mathbb{R}^d})^{-d/4} &=(1-\Delta_{\mathbb{R}^d})^{-d/4}M_{f\chi_{\mathbb{B}^d}}(1-\Delta_{\mathbb{R}^d})^{-d/4} \\ &\qquad+(1-\Delta_{\mathbb{R}^d})^{-d/4}M_{f\chi_{\mathbb{R}^d \setminus\mathbb{B}^d}}(1-\Delta_{\mathbb{R}^d})^{-d/4}. \end{aligned}
\end{equation*}
\notag
$$
By the quasi-triangle inequality
$$
\begin{equation*}
\begin{aligned} \, &\bigl\|(1-\Delta_{\mathbb{R}^d})^{-d/4}M_f(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr\|_{1,\infty} \leqslant2\bigl\|(1\,{-}\,\Delta_{\mathbb{R}^d})^{-d/4}M_{f\chi_{\mathbb{B}^d}} (1\,{-}\,\Delta_{\mathbb{R}^d})^{-d/4}\bigr\|_{1,\infty} \\ &\qquad\qquad +2\bigl\|(1-\Delta_{\mathbb{R}^d})^{-d/4}M_{f\chi_{\mathbb{R}^d \setminus\mathbb{B}^d}}(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr\|_{1,\infty}. \end{aligned}
\end{equation*}
\notag
$$
By Lemma 5.6 as applied to $f^{1/2}\chi_{\mathbb{R}^d\setminus\mathbb{B}^d}$, we have
$$
\begin{equation*}
\begin{aligned} \, &\bigl\|(1-\Delta_{\mathbb{R}^d})^{-d/4} M_{f\chi_{\mathbb{R}^d\setminus\mathbb{B}^d}}(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr\|_{1,\infty} \\ &\qquad=\bigl\|M_{f^{1/2}\chi_{\mathbb{R}^d\setminus\mathbb{B}^d}} (1-\Delta_{\mathbb{R}^d})^{-d/2}M_{f^{1/2} \chi_{\mathbb{R}^d\setminus\mathbb{B}^d}}\bigr\|_{1,\infty} \\ &\!\!\quad \stackrel{\text{Lemma }5.6}{\leqslant} c_{\mathrm{abs}}\bigl\|M_{U(f^{1/2}\chi_{\mathbb{R}^d\setminus\mathbb{B}^d})} (1-\Delta_{\mathbb{R}^d})^{-d/2}M_{U(f^{1/2}\chi_{\mathbb{R}^d \setminus\mathbb{B}^d})}\bigr\|_{1,\infty} \\ &\qquad\leqslant c_{\mathrm{abs}}\bigl\|(1-\Delta_{\mathbb{R}^d})^{-d/4}M_{(U(f^{1/2} \chi_{\mathbb{R}^d\setminus\mathbb{B}^d}))^2}(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr\|_{1,\infty} \\ &\qquad \leqslant c_{\mathrm{abs}}\bigl\|(1-\Delta_{\mathbb{R}^d})^{-d/4}M_{(Vf) \chi_{\mathbb{B}^d}}(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr\|_{1,\infty}. \end{aligned}
\end{equation*}
\notag
$$
By Lemma 5.2,
$$
\begin{equation*}
\bigl\|(1-\Delta_{\mathbb{R}^d})^{-d/4}M_f(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr\|_{1,\infty} \leqslant c_d\bigl(\|f\chi_{\mathbb{B}^d}\|_{L_{\Phi}(\mathbb{R}^d)} +\|(Vf)\chi_{\mathbb{B}^d}\|_{L_{\Phi}(\mathbb{R}^d)}\bigr).
\end{equation*}
\notag
$$
Now the assertion (for bounded $f$) follows from Proposition 5.1.
Next, let $f\in L_{\Phi}(\mathbb{R}^d)$ be arbitrary. Set
$$
\begin{equation*}
f_n=f\chi_{\{|f|\leqslant n\}},\qquad n\in\mathbb{N}.
\end{equation*}
\notag
$$
We have already established the inequality for bounded function (in particular, the inequality holds for $f_n$). For every $n\in\mathbb{N}$ we have
$$
\begin{equation*}
\begin{aligned} \, &\bigl\|(1-\Delta_{\mathbb{R}^d})^{-d/4}M_{f_n}(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr\|_{1,\infty} \\ &\qquad \leqslant c_d\biggl(\|f\|_{L_{\Phi}(\mathbb{R}^d)}+\int_{\mathbb{R}^d}|f(s)|\log(1+|s|)\,d\nu(s)\biggr). \end{aligned}
\end{equation*}
\notag
$$
On the other hand it follows from Theorem 2.3 in [23] (the Lorentz space $\Lambda_1(\mathbb{R}^d)$ in [23] is known to coincide with the space $L_{\Phi}(\mathbb{R}^d)$) that
$$
\begin{equation*}
\begin{aligned} \, & \bigl\|(1-\Delta_{\mathbb{R}^d})^{-d/4}M_{f_n}(1-\Delta_{\mathbb{R}^d})^{-d/4}- (1-\Delta_{\mathbb{R}^d})^{-d/4}M_f(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr\|_{\infty} \\ &\qquad\leqslant c_d\|f-f_n\|_{L_{\Phi}(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
It is easy to see that
$$
\begin{equation*}
\|f-f_n\|_{L_{\Phi}(\mathbb{R}^d)}\to0, \qquad n\to\infty.
\end{equation*}
\notag
$$
It follows from the Fatou property of $\mathcal{L}_{1,\infty}$ that
$$
\begin{equation*}
\bigl\|(1-\Delta_{\mathbb{R}^d})^{-d/4}M_f(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr\|_{1,\infty}\leqslant c_d\biggl(\|f\|_{L_{\Phi}(\mathbb{R}^d)}+\int_{\mathbb{R}^d}|f(s)|\log(1+|s|)\,d\nu(s)\biggr).
\end{equation*}
\notag
$$
Theorem 1.3 is proved. § 6. Solomyak estimate for $\mathcal{L}_{1,\infty}$ does not hold in $ {\mathbb{R}}^d$This section is devoted to the proof of Theorem 1.2. 6.1. Simple facts used in the proofIn the following lemma the notation $\bigoplus_{k\in\mathbb{Z}^d}T_k$ is a shorthand for the element $\sum_{k\in\mathbb{Z}^d}T_k\otimes e_k$ of the von Neumann algebra $B(H)\mathbin{\overline{\otimes}}l_{\infty}(\mathbb{Z}^d)$. Here $e_k$ is the unit vector whose only nonzero component is at the $k$th position. Similarly, $A^{\oplus n}$ is a shorthand for the element $\sum_{k=0}^{n-1}A\otimes e_k$ in the von Neumann algebra $B(H)\mathbin{\overline{\otimes}}l_{\infty}(\mathbb{Z})$. Hardy-Littlewood submajorization is defined by the formula
$$
\begin{equation*}
S\prec\prec T \quad\text{if and only if}\quad \int_0^t\mu(s,S)\,d\nu(s)\leqslant\int_0^t\mu(s,T)\,d\nu(s), \qquad t>0,
\end{equation*}
\notag
$$
where we use the identification of the sequence of singular values with the corresponding step function. The following facts are well known. We include their proofs for the convenience of the reader. Fact 6.2. If $T\in\mathcal{L}_{2,\infty}$ and $S\prec\prec T$, then $S\in\mathcal{L}_{2,\infty}$ and For every $t>0$ we have
$$
\begin{equation*}
\begin{aligned} \, t\mu(t,S) &\leqslant\int_0^t\mu(s,S)\,d\nu(s)\leqslant\int_0^t\mu(s,T)\,d\nu(s) \\ &\leqslant\|T\|_{2,\infty}\int_0^ts^{-1/2}\,d\nu(s)=2t^{1/2}\|T\|_{2,\infty}. \end{aligned}
\end{equation*}
\notag
$$
Dividing by $t^{1/2}$ and taking the supremum over $t>0$ we complete the proof. For every $t>0$ we have
$$
\begin{equation*}
\mu(t,A+B)\leqslant\mu\biggl(\frac{t}{2},A\biggr)+\mu\biggl(\frac{t}{2},B\biggr).
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
\begin{aligned} \, \|A+B\|_{2,\infty} &\leqslant \sup_{t>0}t^{1/2}\biggl(\mu\biggl(\frac{t}{2},A\biggr) +\mu\biggl(\frac{t}{2},B\biggr)\biggr) \\ &=2^{1/2}\sup_{t>0}t^{1/2}(\mu(t,A)+\mu(t,B))\leqslant 2^{1/2}\|A\|_{2,\infty}+2^{1/2}\|B\|_{2,\infty}. \end{aligned}
\end{equation*}
\notag
$$
Indeed,
$$
\begin{equation*}
\mu(A^{\oplus n})=\sigma_n\mu(A)\geqslant\sigma_n(\|A\|_{\infty}\chi_{(0,1)})=\|A\|_{\infty}\chi_{(0,n)}.
\end{equation*}
\notag
$$
The first equality here is one of relations (5) in [40]. In the next lemma we estimate the product of the operator $(1-\Delta_{\mathbb{R}^d})^{d/4+1/2}$ and the commutator $\bigl[M_{\phi},(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr]$. Lemma 6.1. If $\phi\in C^{\infty}_c(\mathbb{R}^d)$, then the operator (defined originally as a mapping from $\mathcal{S}(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$) extends to a bounded operator on $L_2(\mathbb{R}^d)$. Proof. The operator $(1-\Delta_{\mathbb{R}^d})^{-d/4}$ is a pseudo-differential operator of order $-d/2$. The operator $M_{\phi}$ is a pseudo-differential operator of order $0$. By Theorem 2.5.1 in [32]
$$
\begin{equation*}
\bigl[M_{\phi},(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr]
\end{equation*}
\notag
$$
is a pseudo-differential operator of order $-d/2-1$. Consequently,
$$
\begin{equation*}
(1-\Delta_{\mathbb{R}^d})^{d/4+1/2}\bigl[M_{\phi}, (1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr]
\end{equation*}
\notag
$$
is a pseudo-differential operator of order $0$. By Theorem 2.4.2 in [32] it is bounded.
The lemma is proved. 6.2. Proof of Theorem 1.2The following proposition is the key to the proof of Theorem 1.2. It provides a concrete example of a function for which the estimate in Theorem 1.2 holds. Proposition 6.1. If then there exists a constant $c_d'$ (depending only on $d$, but not on $n$) such that Proof. Let $K=[-1/2,1/2]^d$, and let $p_k=M_{\chi_{k+K}}$, $k\in\mathbb{Z}^d$. Using Fact 6.1 we obtain
$$
\begin{equation*}
\bigoplus_{k\in\mathbb{Z}^d}M_{\chi_{k+K}}M_{f_n}(1-\Delta_{\mathbb{R}^d})^{-d/4} M_{\chi_{k+K}}\prec\prec M_{f_n}(1-\Delta_{\mathbb{R}^d})^{-d/4}.
\end{equation*}
\notag
$$
For $n\geqslant 2$ we have
$$
\begin{equation*}
M_{\chi_{k+K}}M_{f_n}=M_{\chi_{k+\frac1{n}\mathbb{B}^d}}.
\end{equation*}
\notag
$$
For $n\geqslant2$, from Fact 6.2 for we infer
Let $\phi\in C^{\infty}_c(\mathbb{R}^d)$ have support in $K$ and satisfy $\phi=1$ on $K/2$ and $\|\phi\|_{\infty}=1$. Let $\phi_k(t)=\phi(t-k)$, $t\in\mathbb{R}^d$. Then
$$
\begin{equation}
2\bigl\|M_{f_n}(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr\|_{2,\infty} \geqslant\biggl\|\bigoplus_{k\in\{0,\dots,n-1\}^d}M_{\chi_{k+\frac1{n}\mathbb{B}^d}} (1-\Delta_{\mathbb{R}^d})^{-d/4}M_{\phi_k}\biggr\|_{2,\infty}.
\end{equation}
\tag{6.1}
$$
For $n\geqslant4$ we have
$$
\begin{equation*}
M_{\chi_{k+\frac1{n}\mathbb{B}^d}}=M_{\chi_{k+\frac1{n}\mathbb{B}^d}}M_{\phi_k}.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\begin{aligned} \, &\bigoplus_{k\in\{0,\dots,n-1\}^d}M_{\chi_{k+\frac1{n}\mathbb{B}^d}} (1-\Delta_{\mathbb{R}^d})^{-d/4}=\bigoplus_{k\in\{0,\dots,n-1\}^d} M_{\chi_{k+\frac1{n}\mathbb{B}^d}}(1-\Delta_{\mathbb{R}^d})^{-d/4}M_{\phi_k} \\ &\qquad\qquad+ \bigoplus_{k\in\{0,\dots,n-1\}^d}M_{\chi_{k+\frac1{n}\mathbb{B}^d}} \bigl[M_{\phi_k},(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr]. \end{aligned}
\end{equation*}
\notag
$$
It follows from the quasi-triangle inequality (see Fact 6.3) that
$$
\begin{equation*}
\|A+B\|_{2,\infty}\leqslant 2^{1/2}\|A\|_{2,\infty}+2^{1/2}\|B\|_2.
\end{equation*}
\notag
$$
Consequently,
$$
\begin{equation*}
\begin{aligned} \, &\biggl\|\bigoplus_{k\in\{0,\dots,n-1\}^d} M_{\chi_{k+\frac1{n}\mathbb{B}^d}}(1-\Delta_{\mathbb{R}^d})^{-d/4}\biggr\|_{2,\infty} \\ &\qquad \leqslant 2^{1/2}\biggl\|\bigoplus_{k\in\{0,\dots,n-1\}^d}M_{\chi_{k+\frac1{n}\mathbb{B}^d}} (1-\Delta_{\mathbb{R}^d})^{-d/4}M_{\phi_k}\biggr\|_{2,\infty} \\ &\qquad\qquad +2^{1/2}\biggl\|\bigoplus_{k\in\{0,\dots,n-1\}^d} M_{\chi_{k+\frac1{n}\mathbb{B}^d}}\bigl[M_{\phi_k},(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr]\biggr\|_2. \end{aligned}
\end{equation*}
\notag
$$
Using (6.1) we obtain
$$
\begin{equation}
\begin{aligned} \, &\biggl\|\bigoplus_{k\in\{0,\dots,n-1\}^d}M_{\chi_{k+\frac1{n}\mathbb{B}^d}} (1-\Delta_{\mathbb{R}^d})^{-d/4}\biggr\|_{2,\infty}\leqslant 2^{3/2}\bigl\|M_{f_n}(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr\|_{2,\infty} \nonumber \\ &\qquad\qquad +\biggl(2\sum_{k\in\{0,\dots,n-1\}^d}\bigl\|M_{\chi_{k+\frac1{n}\mathbb{B}^d}} [M_{\phi_k},(1-\Delta_{\mathbb{R}^d})^{-d/4}]\bigr\|_2^2\biggr)^{1/2}. \end{aligned}
\end{equation}
\tag{6.2}
$$
Now we estimate the second summand on the right-hand side of (6.2):
$$
\begin{equation*}
\begin{aligned} \, &\sum_{k\in\{0,\dots,n-1\}^d}\bigl\|M_{\chi_{k+\frac1{n}\mathbb{B}^d}} [M_{\phi_k},(1-\Delta_{\mathbb{R}^d})^{-d/4}]\bigr\|_2^2 \\ &\qquad\leqslant \sum_{k\in\{0,\dots,n-1\}^d}\bigl\|M_{\chi_{k+\frac1{n}\mathbb{B}^d}} (1-\Delta_{\mathbb{R}^d})^{-d/4-1/2}\bigr\|_2^2 \\ &\qquad\qquad\times \bigl\|(1-\Delta_{\mathbb{R}^d})^{d/4+1/2}[M_{\phi_k}, (1-\Delta_{\mathbb{R}^d})^{-d/4}]\bigr\|_{\infty}^2 \\ &\qquad= \biggl(\sum_{k\in\{0,\dots,n-1\}^d}\bigl\|M_{\chi_{k+\frac1{n}\mathbb{B}^d}} (1-\Delta_{\mathbb{R}^d})^{-d/4-1/2}\bigr\|_2^2\biggr) \\ &\qquad\qquad\times \bigl\|(1-\Delta_{\mathbb{R}^d})^{d/4+1/2}[M_{\phi}, (1-\Delta_{\mathbb{R}^d})^{-d/4}]\bigr\|_{\infty}^2 \\ &\qquad= \bigl\|M_{f_n}(1-\Delta_{\mathbb{R}^d})^{-d/4-1/2} \bigr\|_2^2\cdot\bigl\|(1-\Delta_{\mathbb{R}^d})^{d/4+1/2} [M_{\phi},(1-\Delta_{\mathbb{R}^d})^{-d/4}]\bigr\|_{\infty}^2 \\ &\ \, \stackrel{\text{Lemma }6.1}{=}\frac12(c_d')^2. \end{aligned}
\end{equation*}
\notag
$$
To estimate the left-hand side of (6.2) (from below) it remains to note that the operators
$$
\begin{equation*}
\bigl\{M_{\chi_{k+\frac1{n}\mathbb{B}^d}} (1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr\}_{k\in\{0,\dots,n-1\}^d}
\end{equation*}
\notag
$$
are pairwise unitarily equivalent (via a shift operator) and, thus,
$$
\begin{equation*}
\biggl\|\bigoplus_{k\in\{0,\dots,n-1\}^d}M_{\chi_{k+\frac1{n}\mathbb{B}^d}} (1-\Delta_{\mathbb{R}^d})^{-d/4}\biggr\|_{2,\infty} \stackrel{\text{Fact }6.4}{\geqslant} n^{d/2}\bigl\|M_{\chi_{\frac1{n}\mathbb{B}^d}}(1-\Delta_{\mathbb{R}^d})^{-d/4} \bigr\|_{\infty}.
\end{equation*}
\notag
$$
Proposition 6.1 is proved. The following important result was proved in [41], Theorem 16. Here
$$
\begin{equation*}
\psi(t)= \begin{cases} \dfrac1{\log(e/t)},& t\in(0,1), \\ t,& t\geqslant 1, \end{cases}
\end{equation*}
\notag
$$
and $\mathcal{M}_{\psi}$ is the corresponding Marcinkiewicz space (see [21]). Proof of Theorem 1.2. Let $n\geqslant 4$, and let $f_n$ be as in Proposition 6.1. Then
$$
\begin{equation*}
n^{d/2}\|M_{\chi_{\frac1{n}\mathbb{B}^d}}(1-\Delta_{\mathbb{R}^d})^{-d/4}\|_{\infty}\leqslant 2^{3/2}\bigl\|M_{f_n}(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr\|_{2,\infty}+c_d', \qquad n\in\mathbb{N}.
\end{equation*}
\notag
$$
By Proposition 6.2 we have
$$
\begin{equation*}
\begin{aligned} \, &\|M_{\chi_{\frac1{n}\mathbb{B}^d}}(1-\Delta_{\mathbb{R}^d})^{-d/4}\|_{\infty} =\|(1-\Delta_{\mathbb{R}^d})^{-d/4}M_{\chi_{\frac1{n}\mathbb{B}^d}} (1-\Delta_{\mathbb{R}^d})^{-d/4}\|_{\infty}^{1/2} \\ &\qquad\geqslant c_d^{1/2}\|\chi_{(0,n^{-d})}\|_{\mathcal{M}_{\psi}}^{1/2}\geqslant d^{1/2}c_d^{1/2} n^{-d/2}\log^{1/2}(n), \qquad n\in\mathbb{N}. \end{aligned}
\end{equation*}
\notag
$$
A combination of these inequalities yields Consequently,
We have
$$
\begin{equation*}
\begin{aligned} \, \mu(f_n) &=\mu\biggl(\bigoplus_{k\in\{0,\dots,n-1\}^d}\chi_{k+\frac1{n}\mathbb{B}^d}\biggr) =\mu\biggl(\bigoplus_{k\in\{0,\dots,n-1\}^d}\chi_{\frac1{n}\mathbb{B}^d}\biggr) \\ &=\mu\bigl(\chi_{\frac1{n}\mathbb{B}^d}^{\oplus n^d}\bigr) =\sigma_{n^d}\mu(\chi_{\frac1{n}\mathbb{B}^d})=\mu(\chi_{\mathbb{B}^d}). \end{aligned}
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
\mu(f_n)=\chi_{(0,\operatorname{Vol}(\mathbb{B}^d))} \quad\text{and}\quad\|f_n\|_E=\|\chi_{(0,\operatorname{Vol}(\mathbb{B}^d))}\|_E, \qquad n\in\mathbb{N},
\end{equation*}
\notag
$$
for every symmetric quasi-Banach function space $E$.
Let $C_E$ be the concavity modulus of $E$ (that is, the constant in the quasi-triangle inequality). Choose a sequence $\{n_k\}_{k\geqslant1}$ such that
$$
\begin{equation}
\bigl(d^{1/2}c_d^{1/2} \log^{1/2}(n_k)-c_d'\bigr)_+^2\geqslant k^3C_E^k, \qquad k\geqslant 1.
\end{equation}
\tag{6.4}
$$
Set We claim that $f\in E$. Indeed, by the quasi-triangle inequality,
$$
\begin{equation*}
\|f\|_E\leqslant\sum_{k\geqslant1}C_E^k\|k^{-2}C_E^{-k}f_{n_k}\|_E =\sum_{k\geqslant1}k^{-2}\|f_{n_k}\|_E=\frac{\pi^2}{6}\cdot \|\chi_{(0,\operatorname{Vol}(\mathbb{B}^d))}\|_E.
\end{equation*}
\notag
$$
Since each $f_{n_k}$ is positive, it follows that
$$
\begin{equation*}
(1-\Delta_{\mathbb{R}^d})^{-d/4}M_f(1-\Delta_{\mathbb{R}^d})^{-d/4} \geqslant k^{-2}C_E^{-k} (1-\Delta_{\mathbb{R}^d})^{-d/4}M_{f_{n_k}}(1-\Delta_{\mathbb{R}^d})^{-d/4}\geqslant0.
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
\begin{aligned} \, &\bigl\|(1-\Delta_{\mathbb{R}^d})^{-d/4}M_f(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr\|_{1,\infty} \\ &\qquad\geqslant k^{-2}C_E^{-k}\bigl\|(1-\Delta_{\mathbb{R}^d})^{-d/4}M_{f_{n_k}} (1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr\|_{1,\infty}. \end{aligned}
\end{equation*}
\notag
$$
By (6.3) we have
$$
\begin{equation*}
\begin{aligned} \, &\bigl\|(1-\Delta_{\mathbb{R}^d})^{-d/4}M_f(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr\|_{1,\infty} \\ &\qquad\geqslant k^{-2}C_E^{-k}\cdot \bigl(d^{1/2}c_d^{1/2} \log^{1/2}(n_k)-c_d'\bigr)_+^2 \stackrel{(6.4)}{\geqslant}k^{-2}C_E^{-k}\cdot k^3C_E^k=k, \qquad k\geqslant1. \end{aligned}
\end{equation*}
\notag
$$
This inequality shows that
$$
\begin{equation*}
(1-\Delta_{\mathbb{R}^d})^{-d/4}M_f(1-\Delta_{\mathbb{R}^d})^{-d/4}\notin\mathcal{L}_{1,\infty}.
\end{equation*}
\notag
$$
Theorem 1.2 is proved. § 7. Proof of Proposition 5.1In this section we simplify the expressions used in the proof of Theorem 1.3. Our argument extends the one in Theorem 3.1 of [33]. Proof. Without loss of generality let $f\geqslant0$. Suppose that
$$
\begin{equation*}
\int_{\mathbb{R}^d\setminus\mathbb{B}^d}f(s)\log(1+|s|)\,d\nu(s)\leqslant 1\quad\text{and} \quad \|f\|_{L_{\Phi}(\mathbb{R}^d\setminus\mathbb{B}^d)}\leqslant 1.
\end{equation*}
\notag
$$
By definition (2.1) of the Orlicz norm, the latter inequality is equivalent to
$$
\begin{equation*}
\int_{\mathbb{R}^d\setminus\mathbb{B}^d}\Phi(f(s))\,d\nu(s)\leqslant 1.
\end{equation*}
\notag
$$
Since $\Phi(t)\geqslant t$, $t>0$, it follows that
$$
\begin{equation*}
\int_{\mathbb{R}^d\setminus\mathbb{B}^d}f(s)\,d\nu(s)\leqslant 1.
\end{equation*}
\notag
$$
Hence (we use the concrete form of the function $\Phi$ as well as the formula for the Jacobian computed in the proof of Lemma 5.3)
$$
\begin{equation*}
\begin{aligned} \, &\int_{\mathbb{B}^d}\Phi((Vf)(u))\,d\nu(u) =\int_{\mathbb{B}^d}\Phi\biggl(|u|^{-2d}f\biggl(\frac{u}{|u|^2}\biggr)\biggr)\,d\nu(u) \\ &\qquad=\int_{\mathbb{R}^d\setminus\mathbb{B}^d}\Phi(|s|^{2d}f(s))|s|^{-2d}\,d\nu(s) =\int_{\mathbb{R}^d\setminus\mathbb{B}^d}f(s)\cdot \log(e+|s|^{2d}f(s))\,d\nu(s). \end{aligned}
\end{equation*}
\notag
$$
We have Hence
$$
\begin{equation*}
\begin{aligned} \, \log(e+|s|^{2d}f(s)) &\leqslant 1+\log(1+|s|^{2d})+\log(1+f(s)) \\ &\leqslant 1+2d\log(1+|s|)+\log(e+f(s)). \end{aligned}
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
\begin{aligned} \, \int_{\mathbb{B}^d}\Phi((Vf)(t))\,d\nu(t) &\leqslant \int_{\mathbb{R}^d\setminus\mathbb{B}^d}f(s)\,d\nu(s) +2d\int_{\mathbb{R}^d\setminus\mathbb{B}^d}f(s)\log(1+|s|)\,d\nu(s) \\ &\qquad+\int_{\mathbb{R}^d\setminus\mathbb{B}^d}\Phi(f(s))\,d\nu(s)\leqslant 1+2d+1=2d+2. \end{aligned}
\end{equation*}
\notag
$$
Lemma 7.1 is proved. Proof. For brevity set $g=Vf$ and note that $f=Vg$. Without loss of generality let $f\geqslant0$. The assertion is homogeneous. Therefore, it is sufficient to prove it in the case when the right-hand side is equal to $1$. In other words, we assume that It follows that
$$
\begin{equation*}
\begin{aligned} \, &\int_{\mathbb{R}^d\setminus\mathbb{B}^d}\Phi(f(u))\,d\nu(u) =\int_{\mathbb{R}^d\setminus\mathbb{B}^d}M \biggl(|u|^{-2d}g\biggl(\frac{u}{|u|^2}\biggr)\biggr)\,d\nu(u) \\ &\qquad =\int_{\mathbb{B}^d}\Phi(|s|^{2d}g(s))|s|^{-2d}\,d\nu(s) =\int_{\mathbb{B}^d}g(s)\cdot \log(e+|s|^{2d}g(s))\,d\nu(s) \\ &\qquad\leqslant \int_{\mathbb{B}^d}g(s)\cdot \log(e+g(s))\,d\nu(s)\leqslant 1. \end{aligned}
\end{equation*}
\notag
$$
The lemma is proved. Proof. For brevity set $g=(Vf)\chi_{\mathbb{B}^d}$ and note that $f=Vg$. Thus,
$$
\begin{equation*}
\begin{aligned} \, &\int_{\mathbb{R}^d\setminus\mathbb{B}^d}|f(s)|\log(1+|s|)\,d\nu(s) =\int_{\mathbb{R}^d\setminus\mathbb{B}^d}|s|^{-2d}\, \biggl|g\biggl(\frac{s}{|s|^2}\biggr)\biggr|\log(1+|s|)\,d\nu(s) \\ &\qquad=\int_{\mathbb{B}^d}|g(u)|\log\biggl(1+\frac1{|u|}\biggr)\,d\nu(u) =\int_{\mathbb{B}^d}|g(u)|\,|h(u)|\,d\nu(u), \end{aligned}
\end{equation*}
\notag
$$
where $h(u)=\log(1+1/|u|)$, $u\in\mathbb{B}^d$. It follows that
$$
\begin{equation*}
\int_{\mathbb{R}^d\setminus\mathbb{B}^d}|f(s)|\log(1+|s|)\,d\nu(s) \leqslant\int_0^{\operatorname{Vol}(\mathbb{B}^d)}\mu(t,g)\mu(t,h)\,d\nu(t).
\end{equation*}
\notag
$$
Clearly,
$$
\begin{equation*}
\mu(t,h)=\log\biggl(1+\biggl(\frac{t}{\operatorname{Vol}(\mathbb{B}^d)}\biggr)^{-1/d}\biggr), \qquad 0<t<\operatorname{Vol}(\mathbb{B}^d),
\end{equation*}
\notag
$$
hence
$$
\begin{equation*}
\mu(t,h)\leqslant c_d\biggl(1+\log_+\biggl(\frac1t\biggr)\biggr), \qquad t>0.
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
\int_{\mathbb{R}^d\setminus\mathbb{B}^d}|f(s)|\log(1+|s|)\, d\nu(s)\leqslant c_d\int_0^{\infty}\mu(t,g)\biggl(1+\log_+\biggl(\frac1t\biggr)\biggr)\,d\nu(t).
\end{equation*}
\notag
$$
The right-hand side is equal to the norm $\|g\|_{\Lambda_1}$, where $\Lambda_1$ is the Lorentz space featuring in [23]. Since the Orlicz space $L_{\Phi}$ coincides with $\Lambda_1$, the assertion follows.
Lemma 7.3 is proved. Proof of Proposition 5.1. By Lemma 7.1 we have
$$
\begin{equation*}
\begin{aligned} \, \|Vf\|_{L_{\Phi}(\mathbb{B}^d)} &\leqslant (2d+2)\|f\|_{L_{\Phi}(\mathbb{R}^d\setminus\mathbb{B}^d)} +(2d+2)\int_{\mathbb{R}^d\setminus\mathbb{B}^d}|f(s)|\log(1+|s|)\,ds \\ &\leqslant (2d+2)\|f\|_{L_{\Phi}(\mathbb{R}^d)}+(2d+2)\int_{\mathbb{R}^d}|f(s)|\log(1+|s|)\,d\nu(s). \end{aligned}
\end{equation*}
\notag
$$
It is immediate that Thus,
On the other hand it follows from the triangle inequality and Lemma 7.2 that
$$
\begin{equation*}
\|f\|_{L_{\Phi}(\mathbb{R}^d)}\leqslant \|f\|_{L_{\Phi}(\mathbb{B}^d)}+\|f\|_{L_{\Phi}(\mathbb{R}^d\setminus\mathbb{B}^d)}\leqslant \|f\|_{L_{\Phi}(\mathbb{B}^d)}+\|Vf\|_{L_{\Phi}(\mathbb{B}^d)}.
\end{equation*}
\notag
$$
By Lemma 7.3 we have
$$
\begin{equation*}
\int_{\mathbb{R}^d\setminus\mathbb{B}^d}|f(s)|\log(1+|s|)\,d\nu(s)\leqslant c_d\|Vf\|_{L_{\Phi}(\mathbb{B}^d)}.
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
\begin{aligned} \, &\|f\|_{L_{\Phi}(\mathbb{R}^d)}+\int_{\mathbb{R}^d}|f(s)|\log(1+|s|)\,d\nu(s) \\ &\qquad \leqslant (1+c_d)\bigl(\|f\chi_{\mathbb{B}^d}\|_{L_{\Phi}(\mathbb{R}^d)} +\|(Vf)\chi_{\mathbb{B}^d}\|_{L_{\Phi}(\mathbb{R}^d)}\bigr). \end{aligned}
\end{equation*}
\notag
$$
Combining this with the reverse inequality established in the preceding paragraph, we complete the argument.
Proposition 5.1 is proved. § 8. Proof of Theorem 1.3 for $d=2$This section contains a short proof of Theorem 1.3 for $d=2$. The proof was communicated to us by Prof. Frank and is presented here with his kind permission. For a (possibly unbounded) self-adjoint operator $S$ we denote the number of eigenvalues of $S$ in the interval $I$ by $N(I,S)$. This is set to be $+\infty$ if the spectrum of $S$ on $I$ is not discrete. The proof is based on the main result in [33], which can be read as follows. Strictly speaking, in [33] the right-hand side is written as
$$
\begin{equation*}
1+\|f\|_{L_{\mathcal{B}}(\mathbb{R}^2)}+\int_{\mathbb{R}^2}|f(s)|\log(1+|s|)\,d\nu(s),
\end{equation*}
\notag
$$
where This quantity is equivalent to the one on the right-hand side of the above statement since the Orlicz functions $M$ and $\mathcal{B}$ are equivalent for large values of $t$. Spectral estimates for Schrödinger operators and Solomyak-type estimates are related via the Birman-Schwinger principle. An abstract version of the Birman-Schwinger principle which is suitable for our purposes can be found, for instance, in Proposition 7.2 in [35] (or Proposition 2.3 in [29], or Lemma 1.4 in [10]). Theorem 8.2. Let $T$ be a positive, boundedly invertible operator. Let $V$ be a positive bounded operator. Suppose that $V^{1/2}T^{-1/2}$ is compact. Then Now we are ready to prove the main result in this section. Proof of Theorem 1.3 for $d=2$. We may assume without loss of generality that $f\geqslant0$ is bounded and has compact support. The approximation argument required to prove the assertion in full generality repeats mutatis mutandi the one in the proof of Theorem 1.3.
Let $t>0$. By Theorem 2.3 in [23] we have
$$
\begin{equation*}
\bigl\|(1-\Delta_{\mathbb{R}^2})^{-1/2}M_f (1-\Delta_{\mathbb{R}^2})^{-1/2}\bigr\|_{\infty}\leqslant c_1\|f\|_{L_{\Phi}(\mathbb{R}^2)}.
\end{equation*}
\notag
$$
A somewhat weaker bound, which is, however, sufficient for the proof of Theorem 1.3, can also be deduced directly from [36] and [33]. Therefore,
$$
\begin{equation*}
N\bigl((t,\infty),(1-\Delta_{\mathbb{R}^2})^{-1/2}M_f(1-\Delta_{\mathbb{R}^2})^{-1/2}\bigr)=0,
\end{equation*}
\notag
$$
whenever
Now suppose that By the Birman-Schwinger principle and Theorem 8.1,
$$
\begin{equation*}
\begin{aligned} \, &N\bigl((t,\infty),(1-\Delta_{\mathbb{R}^2})^{-1/2}M_f(1-\Delta_{\mathbb{R}^2})^{-1/2}\bigr) \\ &\qquad=N\bigl((1,\infty),(1-\Delta_{\mathbb{R}^2})^{-1/2}M_{t^{-1}f}(1-\Delta_{\mathbb{R}^2})^{-1/2}\bigr) \\ &\qquad= N\bigl((-\infty,0),1-\Delta_{\mathbb{R}^d}-M_{t^{-1}f}\bigr) =N\bigl((-\infty,-1),-\Delta_{\mathbb{R}^d}-M_{t^{-1}f}\bigr) \\ &\qquad\leqslant N\bigl((-\infty,0),-\Delta_{\mathbb{R}^d}-M_{t^{-1}f}\bigr) \\ &\qquad\leqslant 1+\frac{c_2}{t}\biggl(\|f\|_{L_{\Phi}(\mathbb{R}^2)} +\int_{\mathbb{R}^2}|f(s)|\log(1+|s|)\,d\nu(s)\biggr). \end{aligned}
\end{equation*}
\notag
$$
By the assumption on $t$ we have
$$
\begin{equation*}
1\leqslant \frac{c_1}{t}\|f\|_{L_{\Phi}(\mathbb{R}^2)}.
\end{equation*}
\notag
$$
It follows that
$$
\begin{equation*}
\begin{aligned} \, &N\bigl((t,\infty),(1-\Delta_{\mathbb{R}^2})^{-1/2}M_f(1-\Delta_{\mathbb{R}^2})^{-1/2}\bigr) \\ &\qquad\leqslant \frac{c_1+c_2}{t}\biggl(\|f\|_{L_{\Phi}(\mathbb{R}^2)} +\int_{\mathbb{R}^2}|f(s)|\log(1+|s|)\,d\nu(s)\biggr). \end{aligned}
\end{equation*}
\notag
$$
Combining the estimates in the preceding paragraphs we obtain
$$
\begin{equation*}
\begin{aligned} \, &N\bigl((t,\infty),(1-\Delta_{\mathbb{R}^2})^{-1/2}M_f(1-\Delta_{\mathbb{R}^2})^{-1/2}\bigr) \\ &\qquad \leqslant \frac{c_1+c_2}{t}\biggl(\|f\|_{L_{\Phi}(\mathbb{R}^2)} +\int_{\mathbb{R}^2}|f(s)|\log(1+|s|)\,d\nu(s)\biggr), \qquad t>0. \end{aligned}
\end{equation*}
\notag
$$
Setting $t=\mu(n,A)-\epsilon$, note that
$$
\begin{equation*}
\begin{aligned} \, &\sup_{t>0}tN((t,\infty),A)\geqslant \sup_{n\geqslant0}\sup_{\epsilon>0}\, (\mu(n,A)-\epsilon) N\bigl((\mu(n,A)-\epsilon,\infty),A\bigr) \\ &\qquad \geqslant\sup_{n\geqslant0}\mu(n,A)N\bigl((\mu(n,A)-0,\infty),A\bigr) \geqslant\sup_{n\geqslant0}\, (n+1)\mu(n,A)=\|A\|_{1,\infty}. \end{aligned}
\end{equation*}
\notag
$$
Now we can conclude that
$$
\begin{equation*}
\begin{aligned} \, &\bigl\|(1-\Delta_{\mathbb{R}^2})^{-1/2}M_f(1-\Delta_{\mathbb{R}^2})^{-1/2}\bigr\|_{1,\infty} \\ &\qquad \leqslant (c_1+c_2)\biggl(\|f\|_{L_{\Phi}(\mathbb{R}^2)} +\int_{\mathbb{R}^2}|f(s)|\log(1+|s|)\,d\nu(s)\biggr). \end{aligned}
\end{equation*}
\notag
$$
Theorem 1.3 is proved.
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