Optimal Cwikel–Solomyak Estimates

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.

optimality result) asserts that the Marcinkiewicz space M ψ with ψ(t) = 1 log e t 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 Sobolevtype 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.

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 ( , ν) be a measure space. Let S( , ν) be the collection of all ν-measurable functions on such that, for some n ∈ N, the function | f |χ {| f |>n} is supported on a set of finite measure. For every f ∈ S( , ν) one can define its decreasing rearrangement, μ( 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 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 μ(s, f ).
is called the fundamental function of E. This may also be extended to the case where is an interval or an arbitrary σ -finite measure space (see [14, Chap. II, Sect. 8]). The concrete examples of measure spaces ( , ν) considered in this paper are d-dimensional tori T d (equipped with the respective Haar measures), R + and R d (equipped with Lebesgue measures), their measurable subsets and compact ddimensional 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 R + (that is continuous convex increasing function 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 ψ, the Lorentz space ψ is defined by setting The Marcinkiewicz space M ψ is then defined by setting Setting ψ(t) = t 1 2 , the spaces L 2,1 and L 2,∞ are then the Lorentz and Marcinkiewicz spaces corresponding to ψ(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 ≺≺) defined on equivalence classes of functions in L 1 + L ∞ by y ≺≺ 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 ψ (0, 1) and φ (0, 1), the Orlicz spaces L M (0, 1) and L G (0, 1), and the Marcinkiewicz spaces M ψ (0, 1) and M φ (0, 1) plays a fundamental role. The space M ψ (respectively, the space φ ) is the largest (respectively, the smallest) symmetric Banach function space with the fundamental function φ.
In the statement of our main result, we use the 2-convexification of the symmetric space. Given a symmetric space E( , ν), 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 σ u , u > 0, which acts on S(R) by the formula We frequently use the mapping r d : R d → R + defined by the formula

Connection with Cwikel-Solomyak Type Estimates
Let d ∈ N and let f be a measurable function on d-dimensional torus T d and let M f be an (unbounded) multiplication operator by f on L 2 (T d ). Let T d be the torical Laplacian. In Proposition 19 we show that (symmetrized) Cwikel-Solomyak operator is bounded for f ∈ M ψ (T d ). We denote the * -algebra of all bounded linear operators on the Hilbert space Estimates of the symmetrized Cwikel-Solomyak operator in the weak-trace ideal L 1,∞ as well as in the ideal M 1,∞ are important in Non-commutative Geometry and in Mathematical Physics. For the background concerning weak trace ideals L p,∞ , 1 ≤ p < ∞ and Marcinkiewicz ideal M 1,∞ we refer to [16] (see also Sect. 6.3 below).
Estimates of the symmetrized Cwikel-Solomyak operator in L 1,∞ (on the torus) for even d appeared in the foundational papers [23,24]. The estimate there was given in terms of the Orlicz norm · L M (which is equivalent to · φ ). Recently, symmetrized Cwikel-Solomyak estimates in the ideal M 1,∞ were established (on the Euclidean space) in [17]. The estimate was given in terms of the Lorentz norm The surprising fact that φ (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 L ∞ for f ∈ M ψ and to L 1,∞ for f ∈ φ . 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.

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 ⊂ R d we refer the reader to Chap. 7 in [2].

Theorem 1 Let d be even and let be a bounded domain in R d (conditions apply).
There exists a constant c such that This result is optimal due to the following theorem (proved in [9]).

Theorem 2 Let d be even and let be a bounded domain in R d (conditions apply). If X be a Banach symmetric function space on such that
for some constant c , then (2) ψ ⊂ X . In the next sections, we prove the assertions which substantially strengthen theorems above and extend them to arbitrary dimensions d ≥ 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].

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.

Distributional Sobolev-Type Inequality
Let T : L 2 (0, 1) → L 2 (0, 1) be the operator defined by the formula The boundedness of T : L 2 (0, 1) → 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 R d may be informally understood as the boundedness of the operator We suggest to view this result as a statement concerning distribution functions of ele- Our distributional version of Sobolev inequality is as follows: Here, ≺≺ denotes the Hardy-Littlewood submajorization.
We need the following lemma stated in terms of convolutions.
Proof Lemma 1.5 in [20] states that Obviously, For t ∈ (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.

Proof of Theorem 3
We rewrite (1 − ) − 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 d 4 and is given in [3] (see formulae (2.7) and (2.10) there) as follows Let B d be the unit ball in R d . If d 4 is not integer, then it follows from formulae (9.6.2) and (9.6.10) in [1] If d 4 is integer, then formula (9.6.11) in [1] [1]. Thus, g ∈ (L 2,∞ ∩ L 1 )(R d ). The assertion follows from Lemma 4.

Optimality of the Distributional Sobolev-Type Inequality
We start with the following useful observation.
where ω d is the volume of the unit ball in R d .
Proof Indeed, for every interval (a, b), we have Hence, for every measurable set A ⊂ (0, ∞), we have In other words, the mapping γ d : R d → R preserves a measure. Thus 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
Proof As in the proof of Theorem 3 above, we rewrite (1 − ) − 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 B d be the unit ball in R d . Since g is strictly positive, it follows from the formulae (9.6.2) and (9.6.10) in [1] (when d = 0mod4) or from the formula (9.6.11) in [1] (when d = 0mod4) that Since x ≥ 0, it follows that When |t| < 1, we have Thus, Obviously, |t − s| ≤ |t| + |s| ≤ 2 max{|t|, |s|} 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.

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 ψ to (0, ∞) by setting

So-defined ψ is a concave increasing function on (0, ∞).
Below we consider the space ψ and its 2-convexification on R d .

Proposition 7 Let d ∈ N.
For every x ∈ L 2 (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]).

Note that m(A) ≥ s.
We have Thus, If m(C) ≥ 1 2 m(A), then In either case, we have Thus, The following lemma shows that the receptacle of the operator T is strictly smaller than the space suggested by the Moser-Trudinger inequality.

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 R d ) we have Applying the latter inequality to f = (1 − ) − d 4 x and using (2) and (3) we obtain 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.
The technical part of the proof of Theorem 10 is concentrated in the next lemma.
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.
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]).
Proof Fix t > 0 and denote We have (see e.g. inequality 2.24 in Chap. II of [14]) Proof of Theorem 10 Let z = μ(z) ∈ ψ (0, 1). Let y be as in Lemma 11 and let x be as in Lemma 12. It follows from Lemma 11 that y 2 ≤ 3 1 2 z (2) ψ . Obviously, y is positive and increasing. It follows from Lemma 12 that Thus, The next corollary shows that the result of Proposition 7 is optimal in the class of symmetric function spaces.

We consider operators
which act, respectively, on L 2 (R d ) and L 2 (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 ∈ E is the Marcinkiewicz space M ψ . For Euclidean space, this follows from Proposition 15 and Theorem 16 below. For torus, this follows from Proposition 19 and Theorem 20 below.

Estimates for Euclidean Space
Recall that ψ is extended to a concave increasing function on (0, ∞) in Sect. 5.

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
We have Proof Without loss of generality, h is real-valued and positive. We have ≤ sup Recall the post-critical Sobolev inequality (see e.g. Theorem 7.57 (c) in [2]): Therefore, Combining these estimates, we complete the proof.
It is of crucial importance that the estimate in the preceding lemma is given in terms of h 1 rather than h M ψ .
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.