Cwikel type estimate as a consequence of certain properties of the heat kernel

Abstract

Estimates for the singular values of the operator T fg := f (H)g(x) are investigated for suitable functions f (λ), λ ∈ R, g(x), x ∈ R d , and a selfadjoint operator H in L 2 (R d ). It is assumed that the kernel of the semigroup e −tH satisfies special conditions. Power-like estimates for the singular values of the operator T fg are obtained, in particular, in the case where T fg ∈ S 2 . Conditions for the operator T fg to belong to the trace class are established. Neither any smoothness conditions for the kernel of the operator T fg , nor any knowledge of the (partial) diagonalization of the operator H are required. The results admit further refinement under additional conditions imposed on the generalized eigenfunctions of the operator H. 835 Licensed to Stockholm University. Prepared on Sat Aug 1 15:53:46 EDT 2015 for download from IP 130.237.165.40. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 836 V. A. SLOUSHCHfor integral operators to belong to the classes S p,q , p < 2, which are contained in the Hilbert-Schmidt class, have been studied to a lesser extent. To my best knowledge, without additional assumptions about the smoothness of the kernel, such results have been available only for the operator f (i∇)g(x) (see [5, 6] and [7]) and also for the Dirac operator (see [8]). An overview of some of the results mentioned above can be found in [7] and [4]; the paper [7] covers some applications.In the present paper we deal with estimates for the singular values of operators of two similar types. First, consider an operator of the formIt is assumed that the semigroup e −tH satisfies the so-called Nash-Aronson estimate also known as the upper Gaussian bound (see Subsection 2 of the Introduction); in particular, the semigroup generated by a selfadjoint uniformly elliptic second order differential operator with uniformly bounded coefficients fulfills this requirement. We require neither any smoothness conditions for the coefficients of the operator H, nor any knowledge of its (partial) diagonalization. In Theorem 1.3 we obtain conditions for the operator T fg to be in the classes S p,q , p ∈ (0, 2), q ∈ (0, +∞]. We also give conditions (see Theorem 1.2) for the operator T fg to belong to the classes S p,q , p ∈ (2, +∞), q ∈ (0, +∞].Moreover, in the present paper we consider certain integral operators bordered by suitable scalar weights. Namely, let (K, dμ) be a separable measurable space with σ-finite measure, let t( · , · ) ∈ L ∞ (K × R d , dμ dx) be the kernel of a bounded integral operator T : L 2 (R d ) → L 2 (K, dμ) and let f : K → C, g : R d → C be suitable measurable functions. It is assumed that the operator T (partially) diagonalizes the selfadjoint operator H acting in L 2 (R d ) and satisfying the conditions described above. In Theorem 1.5 we give estimates for the singular values of the integral operator T fg = fT g with the kernel f (k)t(k, y)g(y). Conditions for the operator T fg to belong to the classes S p,q , p ∈ (0, 2), q ∈ (0, +∞], inclu...

Cwikel type estimates for the bordered Airy transform

“…Here S i = S i (κ, p, q, p 0 ), i = 1, 2, 3. By applying [6,Lemma 4.2] to the first integral in (3.8) and [6,Lemma 4.1] to the third integral, we get the required assertions of Theorem 2.1.…”

“…Such results with no additional requirements on the smoothness of the operators' kernel were obtained in the papers [3,4], and [2] for the operator f (i∇)g(x) and in the paper [5] for the operator f ( p H)g(x), where p H is the Dirac operator. In [6] the conditions of being in the classes S p,q , p < 2, were obtained for the operator f (H)g(x), provided that the self-adjoint and lower bounded operator H generates a semigroup satisfying the Nash-Aronson estimate (the upper Gaussian estimate). The results of [6] cannot be directly applied to the operator f Ag = Af (H)g because the Stark operator H is not bounded from below.…”

“…In [6] the conditions of being in the classes S p,q , p < 2, were obtained for the operator f (H)g(x), provided that the self-adjoint and lower bounded operator H generates a semigroup satisfying the Nash-Aronson estimate (the upper Gaussian estimate). The results of [6] cannot be directly applied to the operator f Ag = Af (H)g because the Stark operator H is not bounded from below. Nevertheless, after a certain modernization, the method developed in [6] applies to the operator f Ag.…”

“…The results of [6] cannot be directly applied to the operator f Ag = Af (H)g because the Stark operator H is not bounded from below. Nevertheless, after a certain modernization, the method developed in [6] applies to the operator f Ag. The present paper is devoted to the results obtained in this way.…”