Sbornik: Mathematics, 2022, Volume 213, Issue 9, Pages 1250–1289 DOI: https://doi.org/10.4213/sm9732e (Mi sm9732)

DOI: https://doi.org/10.4213/sm9732e

Russian version:
Matematicheskii Sbornik, 2022, Volume 213, Number 9, Pages 97–137
DOI: https://doi.org/10.4213/sm9732

Bibliographic databases:

§ 1. Introduction

Estimates 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 states^1[x]1In quantum physics bound states are eigenfunctions of a Schrödinger operator that correspond to eigenvalues outside the essential spectrum. 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

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:

In various applications (both to mathematical physics and noncommutative geometry) the following estimates are of primary interest. We fix the function $g$ to be

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

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.

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 $\mathcal{L}_{1,\infty}$.

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 space^2[x]2Strictly speaking, the condition in [23] is stated as $f\in\Lambda_1(\mathbb{R}^d)$, where $\Lambda_1$ is a certain Lorentz space. However, $\Lambda_1(\mathbb{R}^d)$ and the Orlicz space $L_{\Phi}(\mathbb{R}^d)$ are known to coincide (see a similar result in Lemma 4.6.2 in [3], for instance). $L_{\Phi}(\mathbb{R}^d)$.

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 proof

Our 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

In § 4 we restate Theorem 1.1 as

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

Acknowledgements

The 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. Preliminaries

Throughout 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 spaces

Let $(\Omega,\omega)$ 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

Let $E(\Omega,\omega)\subset S(\Omega,\omega)$, and let $\|\cdot\|_E$ be a quasi-Banach norm on $E(\Omega,\omega)$ such that

• 1) if $f\in E(\Omega,\omega)$ and $g\in S(\Omega,\omega)$ satisfy $|g|\leqslant|f|$, then $g\in E(\Omega,\omega)$ and ${\|g\|_E\leqslant\|f\|_E}$;
• 2) if $f\in E(\Omega,\omega)$ and $g\in S(\Omega,\omega)$ satisfy $\mu(g)=\mu(f)$, then $g\in E(\Omega,\omega)$ and $\|g\|_E=\|f\|_E$.

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

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

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

2.2. Trace ideals

The 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

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

One ideal of particular interest is $\mathcal{L}_{1,\infty}$.

2.3. Sobolev spaces on cubes

Let $m\in\mathbb{Z}_+$. For every open cube $\Pi\subset\mathbb{R}^d$ we define the Sobolev space $W^{m,2}(\Pi)$ as follows:

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:

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]):

We also need the notion of Sobolev spaces on the torus:

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

2.5. Sobolev spaces on toric cubes

We recall that by a toric cube we always mean a Cartesian product of (open) arcs^3[x]3An arc is a segment of the circle $\mathbb{T}$. 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 often^4[x]4Often, but not always! For example, let $d=2$. Under this identification, the union of the following Euclidean squares is a toric cube: $(-\pi,-\pi/{2})\times(-\pi,-\pi/2)$, $(-\pi,-\pi/2)\times(\pi/2,\pi)$, $(\pi/2,\pi)\times(-\pi,-\pi/2)$ and $(\pi/2,\pi)\times(\pi/2,\pi)$. 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

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 spaces

In what follows we need the notion of the homogeneous Sobolev seminorm: for $s=m\in\mathbb{Z}_+$, it is defined by

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.

§ 3. Solomyak-type theorem on coverings

Formally, 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

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.

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 $(x_i)_{i\in\mathbb{I}}$.

The next lemma delivers the subadditivity of the functional $J_f^{\Phi}$ and is an easy consequence of Lemma 3.1.

We equip the $\sigma$-algebra of Lebesgue measurable sets in $\mathbb{T}^d$ with the usual metric

The following assertion improves Lemma 4 in [36] slightly and adjusts it to the case of $\mathbb{T}^d$.

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.

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.

§ 4. Proof of Theorem 1.1

The 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.

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.

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.

The following assertion was proved by Solomyak for even $d$ (see Theorem 1 in [36]). We prove it for an arbitrary dimension.

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.

§ 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 cube

When $f$ 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 $(-\pi,\pi)^2\setminus(-1,1)^d$).

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 $\Phi(t)=t\log(e+t)$, $t>0$.

5.2. The function $f$ has support outside the unit ball

In what follows we equip the unit ball $\mathbb{B}^d$ in $\mathbb{R}^d$ with the Lebesgue measure.

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.

Indeed,

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

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.

The assertion of Lemma 5.6 is of crucial importance in the proof of Theorem 1.3.

5.3. Proof of Theorem 1.3

The proof of the following proposition is deferred to § 7.

Now we are ready to prove the main result in this section.

§ 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 proof

In 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

The following facts are well known. We include their proofs for the convenience of the reader.

For every $t>0$ we have

For every $t>0$ we have

Indeed,

In the next lemma we estimate the product of the operator $(1-\Delta_{\mathbb{R}^d})^{d/4+1/2}$ and the commutator $\bigl[M_{\phi},(1-\Delta_{\mathbb{R}^d})^{-d/4}\bigr]$.

6.2. Proof of Theorem 1.2

The 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.

The following important result was proved in [41], Theorem 16. Here

§ 7. Proof of Proposition 5.1

In 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].

§ 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

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]).

Now we are ready to prove the main result in this section.

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Citation in format AMSBIB

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\by F.~A.~Sukochev, D.~V.~Zanin
\paper Solomyak-type eigenvalue estimates for the Birman-Schwinger operator
\jour Mat. Sb.
\yr 2022
\vol 213
\issue 9
\pages 97--137
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\transl
\jour Sb. Math.
\yr 2022
\vol 213
\issue 9
\pages 1250--1289
\crossref{https://doi.org/10.4213/sm9732e}
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