Abstract
We obtain the optimal version of Cwikel-type estimates for the uniform operator norm which implies the optimality of M.Z. Solomyak’s results (Proc. Lond. Math. Soc. 71(1):53–75, 1995) within the classes of Orlicz/Lorentz spaces. Our methods are based on finding the distributional version of the Sobolev inequality.
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1 Introduction
This text is inspired by result of Solomyak [23, 24] establishing (for \(d\in 2\mathbb {N}\)) that the (symmetrized) Cwikel–Solomyak operator
belongs to the weak trace class ideal \(\mathcal {L}_{1,\infty }\) when \(f\in L\log L(\mathbb {T}^d).\) Here, \(\mathbb {T}^d\) is d-dimensional torus, \(M_f\) is an (unbounded) multiplication operator by f on \(L_2(\mathbb {T}^d)\) and \(\Delta _{\mathbb {T}^d}\) is the torical Laplacian.
The space \(L\log L\) is simultaneously characterised as the Orlicz space \(L_M\) with \(M(t)=t\log (e+t)\) and the Lorentz space \(\Lambda _{\phi }\) with \(\phi (t)=t\log \big (\frac{e}{t}\big )\) (see e.g. Theorem 5.1 in [5]). In this paper we show that the result of Solomyak is optimal in the classes of Orlicz and Lorentz spaces. The principal result of this paper(which implies this optimality result) asserts that the Marcinkiewicz space \(\texttt{M}_{\psi }\) with \(\psi (t)=\frac{1}{\log \big (\frac{e}{t}\big )}\) is the largest possible symmetric function space such that the symmetrized Cwikel–Solomyak operator is bounded. We combine our principal result with that of Solomyak to obtain new estimates in Schatten–Lorentz ideals in Corollary 25.
In the course of the proof of our principal result, we have made two notable contributions to the theory of Sobolev Inequality and Sobolev Embedding Theorem.
Firstly, we establish a distributional version of Sobolev inequality written in terms of Hardy–Littlewood submajorization (Theorem 3). We demonstrate that our version of this inequality is optimal in the sense that it cannot be improved in terms of the distribution function (Proposition 6).
Secondly, we show that our version of Sobolev inequality yields the optimal Sobolev embedding theorem (Proposition 7 and Corollary 14). These results enhance earlier results due to Hansson [12], Brezis and Wainger [7], and Cwikel and Pustylnik [9].
In the following section, we will establish any necessary notation and preliminaries. Technical details and the relation to prior work on Cwikel–Solomyak-type estimates are discussed in Sect. 3. Then in Sect. 4 we prove a distributional form of the Sobolev-type inequality, followed by an optimal Sobolev embedding theorem in Sect. 5. Finally, in Sect. 6 we use these results to prove our optimal Cwikel–Solomyak type estimates.
2 Preliminaries
2.1 Symmetric Function Spaces
For detailed information about decreasing rearrangements and symmetric spaces briefly discussed below, we refer the reader to the standard textbook [14] (see also [15, 16]). Let \((\Omega ,\nu )\) be a measure space. Let \(S(\Omega ,\nu )\) be the collection of all \(\nu \)-measurable functions on \(\Omega \) such that, for some \(n\in \mathbb {N},\) the function \(|f|\chi _{\{|f|>n\}}\) is supported on a set of finite measure. For every \(f\in S(\Omega ,\nu )\) one can define its decreasing rearrangement, \(\mu (f)\), see e.g. [16, Sect. 2.3] (and also [14, Sect. II.2], [15, pp. 116–117]). This is a positive decreasing function on \(\mathbb {R}_+\) equimeasurable with |f|. Note that the notation is somewhat unconventional. In the literature it is common to denote the decreasing rearrangement of |f| by \(f^{*}.\) Similarly, we denote \(f^{*}(s)\) by \(\mu (s,f).\)
Let \(E(\Omega ,\nu )\subset S(\Omega ,\nu )\) and let \(\Vert \cdot \Vert _E\) be Banach norm on \(E(\Omega ,\nu )\) such that
-
(1)
if \(f\in E(\Omega ,\nu )\) and \(g\in S(\Omega ,\nu )\) be such \(|g|\le |f|,\) then \(g\in E(\Omega ,\nu )\) and \(\Vert g\Vert _E\le \Vert f\Vert _E;\)
-
(2)
if \(f\in E(\Omega ,\nu )\) and \(g\in S(\Omega ,\nu )\) be such \(\mu (g)=\mu (f),\) then \(g\in E(\Omega ,\nu )\) and \(\Vert g\Vert _E=\Vert f\Vert _E.\)
We say that \((E(\Omega ,\nu ),\Vert \cdot \Vert _E)\) (or simply E) is a symmetric Banach function space (symmetric space, for brevity). The most well-known examples of symmetric spaces are the Lebesgue \(L_p\)-spaces, (\(L_p(\Omega ,\nu ), \Vert \cdot \Vert _p)\), for \(1\le p\le \infty .\) That such spaces are symmetric is easily verified.
If \(\Omega =\mathbb {R}_+,\) then the function
is called the fundamental function of E. This may also be extended to the case where \(\Omega \) is an interval or an arbitrary \(\sigma \)-finite measure space (see [14, Chap. II, Sect. 8]). The concrete examples of measure spaces \((\Omega , \nu )\) considered in this paper are d-dimensional tori \(\mathbb {T}^d\) (equipped with the respective Haar measures), \(\mathbb {R}_+\) and \(\mathbb {R}^d\) (equipped with Lebesgue measures), their measurable subsets and compact d-dimensional Riemannian manifolds (X, g).
In this paper, we focus on the following concrete examples of symmetric spaces: \(L_p\)-spaces, Orlicz spaces, Lorentz spaces and Marcinkiewicz spaces.
Given an Orlicz function M on \(\mathbb {R}_+\) (that is continuous convex increasing function M satisfying \(M(0)=0\) and \(M(t)\rightarrow \infty \) as \(t\rightarrow \infty \)) the Orlicz space \(L_M(\Omega ,\nu )\) is defined by setting
We equip it with a Banach norm
We refer the reader to [13,14,15] for further information about Orlicz spaces.
Given a concave increasing function \(\psi ,\) the Lorentz space \(\Lambda _{\psi }\) is defined by setting
The Marcinkiewicz space \(\texttt{M}_{\psi }\) is then defined by setting
Setting \(\psi (t)=t^{\frac{1}{2}}\), the spaces \(L_{2,1}\) and \(L_{2,\infty }\) are then the Lorentz and Marcinkiewicz spaces corresponding to \(\psi (t)\) (see e.g. [15]).
All the examples of symmetric spaces listed above are closed with respect to the Hardy–Littlewood submajorization; a pre-order (denoted \(\prec \prec \)) defined on equivalence classes of functions in \(L_1+L_{\infty }\) by \(y\prec \prec x\) if and only if
Let us fix, throughout the text, the two concave functions
and consider Orlicz functions
Throughout this text, the interplay between the Lorentz spaces \(\Lambda _{\psi }(0,1)\) and \(\Lambda _{\phi }(0,1)\), the Orlicz spaces \(L_M(0,1)\) and \(L_G(0,1)\), and the Marcinkiewicz spaces \(\texttt{M}_{\psi }(0,1)\) and \(\texttt{M}_{\phi }(0,1)\) plays a fundamental role. The space \(\texttt{M}_{\psi }\) (respectively, the space \(\Lambda _{\phi }\)) is the largest (respectively, the smallest) symmetric Banach function space with the fundamental function \(\phi .\)
First of all, we observe that the spaces \(\texttt{M}_{\phi }(0,1)\) and \(L_G(0,1)\) coincide (the corresponding norms are equivalent). Indeed, the fundamental functions of those spaces coincide, hence \(L_G(0,1)\subset \texttt{M}_{\phi }(0,1).\) On the other hand, \(\phi '\in L_G(0,1)\) and, therefore, \(\texttt{M}_{\phi }(0,1)\subset L_G(0,1).\) This fact was first observed in [4].
By invoking Köthe duality (see [14] and [15]), we infer that the spaces \(\Lambda _\phi (0,1)\) and \(L_M(0,1)\) also coincide and (the corresponding norms are equivalent).
In the statement of our main result, we use the 2-convexification of the symmetric space. Given a symmetric space \(E(\Omega ,\nu ),\) we set
More details on the 2-convexification of a function space may be found in the book [15].
We also need a definition of dilation operator \(\sigma _u,\) \(u>0,\) which acts on \(S(\mathbb {R})\) by the formula
We frequently use the mapping \(r_d:\mathbb {R}^d\rightarrow \mathbb {R}_+\) defined by the formula
3 Connection with Earlier Results
3.1 Connection with Cwikel–Solomyak Type Estimates
Let \(d\in \mathbb {N}\) and let f be a measurable function on d-dimensional torus \(\mathbb {T}^d\) and let \(M_f\) be an (unbounded) multiplication operator by f on \(L_2(\mathbb {T}^d).\) Let \(\Delta _{\mathbb {T}^d}\) be the torical Laplacian. In Proposition 19 we show that (symmetrized) Cwikel–Solomyak operator
is bounded for \(f\in \texttt{M}_{\psi }(\mathbb {T}^d).\) We denote the \(*\)-algebra of all bounded linear operators on the Hilbert space \(L_2(\mathbb {T}^d)\) by \(\mathcal {L}_{\infty }.\)
Estimates of the symmetrized Cwikel–Solomyak operator in the weak-trace ideal \(\mathcal {L}_{1,\infty }\) as well as in the ideal \(\mathcal {M}_{1,\infty }\) are important in Non-commutative Geometry and in Mathematical Physics. For the background concerning weak trace ideals \(\mathcal {L}_{p,\infty }\), \(1\le p<\infty \) and Marcinkiewicz ideal \(\mathcal {M}_{1,\infty }\) we refer to [16] (see also Sect. 6.3 below).
Estimates of the symmetrized Cwikel–Solomyak operator in \(\mathcal {L}_{1,\infty }\) (on the torus) for even d appeared in the foundational papers [23, 24]. The estimate there was given in terms of the Orlicz norm \(\Vert \cdot \Vert _{L_M}\) (which is equivalent to \(\Vert \cdot \Vert _{\Lambda _{\phi }}\)). Recently, symmetrized Cwikel–Solomyak estimates in the ideal \(\mathcal {M}_{1,\infty }\) were established (on the Euclidean space) in [17]. The estimate was given in terms of the Lorentz norm
The surprising fact that \(\Lambda _\phi (0,1)=L_M(0,1)\) demonstrates the convergence of those totally unrelated approaches.
The results in this paper complement the results cited in the preceding paragraph. Indeed, Cwikel–Solomyak operator belongs to \(\mathcal {L}_{\infty }\) for \(f\in \texttt{M}_{\psi }\) and to \(\mathcal {L}_{1,\infty }\) for \(f\in \Lambda _{\phi }.\) This opens an avenue for a given function f to determine (using interpolation methods) the least ideal to which the corresponding Cwikel–Solomyak operator belongs.
3.2 Connection with Sobolev-Type Estimates
The following Sobolev inequality is customarily credited to [12] and [7]. Actually, none of those papers contain a complete proof or even a clear-cut statement. The proof is available in [9]. For a notion of Sobolev space on \(\Omega \subset \mathbb {R}^d\) we refer the reader to Chap. 7 in [2].
Theorem 1
Let d be even and let \(\Omega \) be a bounded domain in \(\mathbb {R}^d\) (conditions apply). There exists a constant \(c_{\Omega }\) such that
This result is optimal due to the following theorem (proved in [9]).
Theorem 2
Let d be even and let \(\Omega \) be a bounded domain in \(\mathbb {R}^d\) (conditions apply). If X be a Banach symmetric function space on \(\Omega \) such that
for some constant \(c_{\Omega },\) then \(\Lambda _{\psi }^{(2)}\subset X.\)
In the next sections, we prove the assertions which substantially strengthen theorems above and extend them to arbitrary dimensions \(d\ge 1.\) In the course of the proof, we also recast Sobolev inequality using distribution functions. Those results are very much inspired by [9], however, our technique is a substantial improvement of that in [9]. We refer the reader to [18] and [19] for further development of the inequalities in [9].
4 Distributional Sobolev-Type Inequality
In this section, we introduce Hardy-type operator T (see e.g. [14]) and employ it as a technical tool for our distributional version of Sobolev inequality. In the second subsection, we show that our result is optimal.
4.1 Distributional Sobolev-Type Inequality
Let \(T:L_2(0,1)\rightarrow L_2(0,1)\) be the operator defined by the formula
The boundedness of \(T:L_2(0,1)\rightarrow L_2(0,1)\) is guaranteed by the fact that T is actually a Hilbert–Schmidt operator. Indeed, its integral kernel is given by the formula
which is, obviously, square-integrable.
The gist of (the critical case of the) Sobolev inequality on \(\mathbb {R}^d\) may be informally understood as the boundedness of the operator \((1-\Delta )^{-\frac{d}{4}}\) from \(L_2(\mathbb {R}^d)\) into a symmetric Banach function space on \(\mathbb {R}^d\) which is sufficiently close to \(L_{\infty }(\mathbb {R}^d).\) We suggest to view this result as a statement concerning distribution functions of elements \((1-\Delta )^{-\frac{d}{4}}x\) and \(T(\mu (x)\chi _{(0,1)})\) where \(x\in L_2(\mathbb {R}^d).\)
Our distributional version of Sobolev inequality is as follows:
Theorem 3
Let \(d\in \mathbb {N}.\) For every \(x\in L_2(\mathbb {R}^d),\) we have
Here, \(\prec \prec \) denotes the Hardy–Littlewood submajorization.
We need the following lemma stated in terms of convolutions.
Lemma 4
Let \(x\in L_2(\mathbb {R}^d)\) and \(g\in (L_{2,\infty }\cap L_1)(\mathbb {R}^d).\) We have
Proof
Lemma 1.5 in [20] states that
Obviously,
For \(t\in (0,1),\) we have
We, therefore, have
where
Computing the derivative, we obtain
Next,
Thus,
It is now immediate that
Combining this equation with (1), we obtain
This is exactly the required assertion. \(\square \)
Proof of Theorem 3
We rewrite \((1-\Delta )^{-\frac{d}{4}}\) as a convolution operator. Namely,
where g is the Fourier transform of the function
Precise expression for the function g involves Macdonald function \(K_{\frac{d}{4}}\) and is given in [3] (see formulae (2.7) and (2.10) there) as follows
Let \(\mathbb {B}^d\) be the unit ball in \(\mathbb {R}^d.\) If \(\frac{d}{4}\) is not integer, then it follows from formulae (9.6.2) and (9.6.10) in [1] that \(g\chi _{\mathbb {B}^d}\in L_{2,\infty }(\mathbb {B}^d)\subset L_1(\mathbb {B}^d).\) If \(\frac{d}{4}\) is integer, then formula (9.6.11) in [1] yields that \(g\chi _{\mathbb {B}^d}\in L_{2,\infty }(\mathbb {B}^d)\subset L_1(\mathbb {B}^d).\) Also, \(g\chi _{\mathbb {R}^d\backslash \mathbb {B}^d}\in (L_1\cap L_{\infty })(\mathbb {R}^d\backslash \mathbb {B}^d)\) by formula (9.7.2) in [1]. Thus, \(g\in (L_{2,\infty }\cap L_1)(\mathbb {R}^d).\) The assertion follows from Lemma 4. \(\square \)
4.2 Optimality of the Distributional Sobolev-Type Inequality
We start with the following useful observation.
Lemma 5
For \(x\in S(0,\infty ),\) we have
where \(\omega _d\) is the volume of the unit ball in \(\mathbb {R}^d.\)
Proof
Indeed, for every interval (a, b), we have
Hence, for every measurable set \(A\subset (0,\infty ),\) we have
Set \(\gamma _d(t)=\omega _d|t|^d,\) \(t\in \mathbb {R}^d.\) In other words, the mapping \(\gamma _d:\mathbb {R}^d\rightarrow \mathbb {R}\) preserves a measure. Thus
\(\square \)
The following proposition shows (with the help of Lemma 5) that the result of Theorem 3 is optimal.
Proposition 6
There exists a strictly positive constant \(c_d'\) depending only on d such that
for every \(0\le x\in L_2(0,\infty )\) supported in the interval (0, 1).
Proof
As in the proof of Theorem 3 above, we rewrite \((1-\Delta )^{-\frac{d}{4}}\) as a convolution operator. Namely,
where g is given by the formula
By formula (9.6.23) in [1], this is a strictly positive function.
Let \(\mathbb {B}^d\) be the unit ball in \(\mathbb {R}^d.\) Since g is strictly positive, it follows from the formulae (9.6.2) and (9.6.10) in [1] (when \(d\ne 0\textrm{mod}4\)) or from the formula (9.6.11) in [1] (when \(d=0\textrm{mod}4\)) that
Since \(x\ge 0,\) it follows that
When \(|t|<1,\) we have
Thus,
Obviously,
and, therefore,
It follows that
Passing to spherical coordinates, we obtain
Making the substitution \(r^d=u,\) we write
Combining the last two equations, we complete the proof. \(\square \)
5 Sobolev Embedding Theorem in Arbitrary Dimension
In this section, we extend Theorems 1 (see Proposition 7) and 2 (see Corollary 14) to Euclidean spaces of arbitrary dimension. We provide new proofs of both theorems in their original setting (for both even and odd d).
In what follows, we extend \(\psi \) to \((0,\infty )\) by setting
So-defined \(\psi \) is a concave increasing function on \((0,\infty ).\)
Below we consider the space \(\Lambda _{\psi }\) and its 2-convexification on \(\mathbb {R}^d.\)
Proposition 7
Let \(d\in \mathbb {N}.\) For every \(x\in L_2(\mathbb {R}^d),\) we have
The proof of Proposition 7 requires some preparation.
With some effort, the crucial lemma below can be inferred from the proof of Theorem 5.1 in [10]. We provide a direct proof. For the definition of real interpolation method employed below we refer the reader to [6] (see also [15]).
Lemma 8
We have
Proof
Let \(t\in (0,1)\) and set \(s=\psi ^{-1}(t).\)
Let \(x\in (\texttt{M}_{\phi }+L_{\infty })(0,1)\) and let \(x=f+g,\) where \(f\in \texttt{M}_{\phi }(0,1)\) and where \(g\in L_{\infty }(0,1).\) Set
Note that \(m(A)\ge s.\)
We have
Thus,
So, \(u\in B^c\cap C^c\) implies that \(u\in A^c.\) That is, \(B^c\cap C^c\subset A^c\) or, equivalently, \(A\subset B\cup C.\)
If \(m(B)\ge \frac{1}{2} m(A),\) then
If \(m(C)\ge \frac{1}{2} m(A),\) then
In either case, we have
Thus,
\(\square \)
The following lemma shows that the receptacle of the operator T is strictly smaller than the space suggested by the Moser–Trudinger inequality.
Lemma 9
We have \(T:L_2(0,1)\rightarrow \Lambda _{\psi }^{(2)}(0,1).\)
Proof
Let \(x\in L_{2,1}(0,1).\) It is immediate that
Thus,
Let \(x\in L_{2,\infty }(0,1).\) It is immediate that
Obviously,
That is,
Since the function \(t\rightarrow \log \big (\frac{e^2}{t}\big ),\) \(t\in (0,1),\) falls into \(\texttt{M}_{\phi }(0,1),\) it follows that
By real interpolation, we have
is a bounded mapping. By Lemma 8, we have
Thus, \(T:L_2(0,1)\rightarrow \Lambda _{\psi }^{(2)}(0,1)\) is a bounded mapping. \(\square \)
Proof of Proposition 7
By Theorem 3 and Lemma 9, we have
On the other hand, we trivially have
For every measurable function f (e.g. on \(\mathbb {R}^d\)) we have
Applying the latter inequality to \(f=(1-\Delta )^{-\frac{d}{4}}x\) and using (2) and (3) we obtain
\(\square \)
The next assertion should be compared with Theorem 4 in [9] (it is proved in the companion paper [10] and constitutes the key part of the proof of Theorem 5.7 in that paper). Note that our T is different from that in [9]. This difference is the reason why our proof is so much simpler.
Theorem 10
For every \(z=\mu (z)\in \Lambda _{\psi }^{(2)}(0,1),\) there exists \(x=\mu (x)\in L_2(0,1)\) such that
The technical part of the proof of Theorem 10 is concentrated in the next lemma.
Lemma 11
Let \(z\in \Lambda _{\psi }^{(2)}(0,1)\) and let
We have
Proof
Recall that the Cesaro operator \(C:L_2(0,1)\rightarrow L_2(0,1)\) is defined by the formula
We claim that (here C is the classical Cesaro operator)
The functional on the left hand side is normal, subadditive and positively homogeneous. By Lemma II.5.2 in [14], it suffices to prove the inequality for the indicator functions.
Let \(w=\chi _A\) and let \(m(A)=u.\) Obviously, \(\mu (w)=\chi _{(0,u)}\) and
Thus,
The function \(s\rightarrow s^{-1}\psi ^2(s),\) \(s\in (0,1),\) decreases on the interval \((0,e^{-1})\) and increases on the interval \((e^{-1},1).\) Thus,
For \(u\in (e^{-1},1),\) we have
For \(u\in (0,e^{-1}),\) we have
This yields the claim.
Let \(w=\mu ^2(z)\in \Lambda _{\psi }.\) Since \(\mu (w)\le C\mu (w),\) it follows from the preceding paragraph that
Noting that
we complete the proof. \(\square \)
The following lemma affords a very substantial simplification and streamlining of the arguments employed in Theorem 5.7 in [10] (see also Theorem 4 in [9]).
Lemma 12
Let \(y\in L_2\big ((0,1),\frac{dt}{t}\big )\) and let \(x(t)=t^{-\frac{1}{2}}y(t),\) \(0<t<1.\) If y is positive and increasing, then
Proof
Suppose \(t\in (0,e^{-1}).\) We have
Suppose \(t\in (e^{-1},1).\) We have
\(\square \)
Lemma 13
For every \(0\le x\in L_2(0,1),\) we have \(Tx\le T\mu (x).\)
Proof
Fix \(t>0\) and denote
We have (see e.g. inequality 2.24 in Chap. II of [14])
Since \(f_t=\mu (f_t),\) it follows that
\(\square \)
Proof of Theorem 10
Let \(z=\mu (z)\in \Lambda _{\psi }^{(2)}(0,1).\) Let y be as in Lemma 11 and let x be as in Lemma 12. It follows from Lemma 11 that \(\Vert y\Vert _2\le 3^{\frac{1}{2}}\Vert z\Vert _{\Lambda _{\psi }^{(2)}}.\) Obviously, y is positive and increasing. It follows from Lemma 12 that
Thus,
\(\square \)
The next corollary shows that the result of Proposition 7 is optimal in the class of symmetric function spaces.
Corollary 14
Let \(E(\mathbb {R}^d)\) be a symmetric Banach function space on \(\mathbb {R}^d.\) If
then \(\Lambda _{\psi }^{(2)}(\mathbb {R}^d)\subset (E+L_{\infty })(\mathbb {R}^d).\)
Proof
Clearly, \(E(\mathbb {R}^d)\subset (E+L_{\infty })(\mathbb {R}^d).\) Thus,
Hence, we may assume without loss of generality that \(E+L_{\infty }=E\) so that \(L_{\infty }\subset E.\)
Take \(z=\mu (z)\in \Lambda _{\psi }^{(2)}(0,1)\) and, using Theorem 10, find \(x=\mu (x)\in L_2(0,1)\) such that \(\mu (z)\le Tx.\) Extend x to \((0,\infty )\) by setting \(x=0\) outside (0, 1). By Proposition 6, we have
Since \(x\in L_2(0,\infty ),\) it follows that \(x\circ r_d\in L_2(\mathbb {R}^d).\) By assumption, we have \((1-\Delta )^{-\frac{d}{4}}(x\circ r_d)\in E(\mathbb {R}^d).\) Thus, \((Tx)\circ r_d\in E(\mathbb {R}^d)\) and \(Tx\in E(0,1).\) Since \(\mu (z)\le Tx,\) it follows that \(z\in E(0,1).\) Thus, \(\Lambda _{\psi }^{(2)}(0,1)\subset E(0,1).\) \(\square \)
6 Cwikel–Solomyak Estimate in \(\mathcal {L}_{\infty }\)
We consider operators
which act, respectively, on \(L_2(\mathbb {R}^d)\) and \(L_2(\mathbb {T}^d)\) and evaluate their uniform norms. We show that the maximal (symmetric Banach function) space E such that the operators above are bounded for every \(f\in E\) is the Marcinkiewicz space \(\texttt{M}_{\psi }.\) For Euclidean space, this follows from Proposition 15 and Theorem 16 below. For torus, this follows from Proposition 19 and Theorem 20 below.
6.1 Estimates for Euclidean Space
Recall that \(\psi \) is extended to a concave increasing function on \((0,\infty )\) in Sect. 5.
Proposition 15
Let \(d\in \mathbb {N}.\) We have
Proof
Without loss of generality, f is real-valued and positive. Recall that, for positive operator \(A:L_2(\mathbb {R}^d)\rightarrow L_2(\mathbb {R}^d),\) we have
Thus,
By Hölder inequality, we have
Thus,
The assertion follows now from Proposition 7. \(\square \)
The proof of the converse inequality will be based on Proposition 6 and Theorem 10. Recall the notation \(r_d\) from Lemma 5.
Theorem 16
Let \(d\in \mathbb {N}.\) Let \(f=\mu (f)\in \texttt{M}_{\psi }(0,\infty ).\) We have
Proof
We have
Let us now restrict the supremum to the radial functions \(\xi .\) That is, let \(\xi =x\circ r_d,\) where \(x\in L_2(0,\infty )\) and \(\Vert x\Vert _2\le \omega _d^{-\frac{1}{2}}\) (see Lemma 5). We have
Let us further assume that \(x=\mu (x)\) is supported on the interval (0, 1). By Proposition 6 we have
By Theorem 10, we have
By Theorem II.5.2 in [14] we have
Since \(f=\mu (f)\) and since \(\psi \) is linear on \((1,\infty ),\) it follows that
Combining the last three equations, we complete the proof. \(\square \)
6.2 Estimates for the Torus
The following lemma is taken from [25] (see Lemmas 4.5 and 4.6 there).
Lemma 17
Let h be a measurable function on \([-1,1]^d.\) We have
where
Lemma 18
Let \(h\in L_1(\mathbb {T}^d).\) We have
Proof
Without loss of generality, h is real-valued and positive. We have
Recall the post-critical Sobolev inequality (see e.g. Theorem 7.57 (c) in [2]):
Therefore,
Combining these estimates, we complete the proof. \(\square \)
It is of crucial importance that the estimate in the preceding lemma is given in terms of \(\Vert h\Vert _1\) rather than \(\Vert h\Vert _{\texttt{M}_{\psi }}.\)
The following proposition is a version of Proposition 15 for \(\mathbb {T}^d.\) A direct computation yields \(\psi '\notin L_M(0,1),\) \(M(t)=t\log (e+t),\) \(t>0.\) By Theorem II.5.7 in [14], Marcinkiewicz space \(\texttt{M}_{\psi }(0,1)\) contains (and, therefore, strictly contains) the Orlicz space \(L_M(0,1).\) The result below delivers the sharp estimate of the symmetrized Cwikel–Solomyak operator
The sharpness of the estimate will be demonstrated below in Theorem 20.
Proposition 19
Let \(d\in \mathbb {N}.\) Let \(f\in \texttt{M}_{\psi }(\mathbb {T}^d).\) We have
Proof
Without loss of generality, f is real-valued, positive and supported on \([-1,1]^d.\)
In this proof, we frequently use the following property: let \(H_0\subset H\) be a Hilbert subspace and let \(A:H\rightarrow H_0.\) If A vanishes on the orthogonal complement of \(H_0,\) then \(\Vert A\Vert _{\infty }=\Vert A|_{H_0}\Vert _{\infty }.\)
Firstly, note that
By Lemma 17, we have
By triangle inequality, we have
The first summand is estimated in Proposition 15. The second summand is estimated in Lemma 18. Combining these estimates, we complete the proof. \(\square \)
The following theorem is a version of Theorem 16 for \(\mathbb {T}^d.\) Here, f is a measurable function on (0, 1) and \(f\circ r_d\) is a measurable function on \(\mathbb {B}^d\) extended to \([-\pi ,\pi ]^d\) by setting \(f=0\) outside of \(\mathbb {B}^d\) and identified with a measurable function on \(\mathbb {T}^d.\)
Theorem 20
Let \(d\in \mathbb {N}.\) Let \(f=\mu (f)\in \texttt{M}_{\psi }(0,1).\) We have
Proof
Firstly, note that
By Lemma 17, we have
By triangle inequality, we have
Recall that
Thus,
By Theorem 16 and Lemma 18, we have
In other words, we have
On the other hand, we have
Thus,
Combining (4) and (5), we obtain
This completes the proof. \(\square \)
6.3 Optimality of Cwikel–Solomyak Estimates Within the Classes of Orlicz and Lorentz Spaces
The following material is standard; for more details we refer the reader to [16, 21]. Let H be a complex separable infinite dimensional Hilbert space, and 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 as:
Let \(p \in (0,\infty ).\) The weak Schatten class \(\mathcal {L}_{p,\infty }\) is the set of all operators \(T\in K(H)\) such that \(\mu (T)\) is in the weak \(L_p\)-space \(l_{p,\infty }\), with quasi-norm:
Obviously, \(\mathcal {L}_{p,\infty }\) is an ideal in B(H).
Our next corollary follows from Theorem 20 and provides one of the two key ingredients in the proof of one of our main results.
Corollary 21
Let \(E(\mathbb {T}^d)\) be a symmetric Banach function space on \(\mathbb {T}^d.\) Suppose that
It follows that \(E\subset \texttt{M}_{\psi }.\)
Proof
Take \(h=\mu (h)\in E(0,1)\) and let \(f=h\circ r_d\in E(\mathbb {T}^d)\) (by Lemma 5). We have that
By Theorem 20, it follows that
This completes the proof. \(\square \)
The next lemma demonstrates efficiency of general theory of symmetric function spaces in the study of Cwikel–Solomyak estimates.
Lemma 22
If \(L_N(0,1)\) is an Orlicz space such that \(L_N(0,1)\subset \texttt{M}_{\psi }(0,1),\) then \(L_N(0,1)\subset L_M(0,1),\) where \(M(t)=t\log (e+t),\) \(t>0.\)
Proof
We have
Thus,
Setting \(u=t^{-1},\) we write
Setting \(v=N^{-1}(u),\) we write
Equivalently,
By convexity,
Thus,
Thus,
\(\square \)
Lemma 23
If \(\Lambda _{\theta }(0,1)\) is a Lorentz space such that \(\Lambda _{\theta }(0,1)\subset \texttt{M}_{\psi }(0,1),\) then \(\Lambda _{\theta }(0,1)\subset \Lambda _{\phi }(0,1).\)
Proof
By assumption, there exists a constant \(c_{\theta }\) such that
Setting \(x=\chi _{(0,t)},\) we obtain
Thus, \(\Lambda _{\theta }(0,1)\subset \Lambda _{\phi }(0,1).\) \(\square \)
The following corollary is one of our main results. It demonstrates that Cwikel inequality proved by Solomyak (for even dimensions d) cannot be improved within the classes of Orlicz and Lorentz spaces. Observe that a version of Solomyak inequality for an arbitrary dimension d (in more general setting of Riemannian manifolds) is established in [26].
It is interesting to compare the result of the following corollary with Theorem 9.4 in [22] which is proved under an artificial condition on the Orlicz function N. In contrast, the following result holds for an arbitrary Orlicz function.
Corollary 24
If \(L_N\) is an Orlicz space such that
then \(L_N(0,1)\subset L_M(0,1).\)
If \(\Lambda _{\theta }\) is a Lorentz space such that
then \(\Lambda _{\theta }(0,1)\subset L_M(0,1).\)
Proof
By Corollary 21, \(L_N\subset \texttt{M}_{\psi }\) (respectively, \(\Lambda _{\theta }(0,1)\subset \texttt{M}_{\psi }(0,1)\)). By Lemma 22 (respectively, by Lemma 23), \(L_N(0,1)\subset L_M(0,1)\) (respectively, \(\Lambda _{\theta }(0,1)\subset \Lambda _{\psi }(0,1)\)). \(\square \)
6.4 Cwikel–Solomyak Estimates in Schatten–Lorentz Ideals
Proposition 19 yields the sharp uniform norm estimate for the Cwikel–Solomyak operator, whereas Solomyak [23, 24] proved the \(\Vert \cdot \Vert _{1,\infty }\)-quasi-norm estimate for it. The next natural step is to apply real interpolation (see e.g. [6, 15]) to obtain Lorentz (p, q)-quasi-norm estimates for the Cwikel operator. Recall that the Lorentz (p, q)-quasi-norm is defined by the formula
For discussion of function spaces \(L_{p,q}(\Omega , \nu )\) and their noncommutative counterparts, we refer the reader to [15, pp. 142–144, pp. 228–229] and [11, 16, 21] respectively.
Corollary 25
Let \(1<p<\infty \) and \(1\le q\le \infty .\) We have
Proof
Let \(A:\texttt{M}_{\psi }(\mathbb {T}^d)\rightarrow \mathcal {L}_{\infty }\) be a bounded operator defined by the setting
We have that \(A:\Lambda _{\psi }(\mathbb {T}^d)\rightarrow \mathcal {L}_{1,\infty }\) is bounded. By real interpolation, we have
Combining Proposition 2.g.20 in [15] and Theorem 3.2 in [11], we obtain
This completes the proof. \(\square \)
Let us note that the spaces \([\Lambda _{\phi },\texttt{M}_{\psi }]_{1-\frac{1}{p},q}(\mathbb {T}^d)\) are different for different pairs (p, q)—see e.g. Theorem 4.6.24 in [8].
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Communicated by Mieczyslaw Mastylo.
Dedicated to the memory of M.Z. Solomyak.
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Sukochev, F., Zanin, D. Optimal Cwikel–Solomyak Estimates. J Fourier Anal Appl 29, 21 (2023). https://doi.org/10.1007/s00041-023-10003-9
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DOI: https://doi.org/10.1007/s00041-023-10003-9