Cwikel estimates revisited

Abstract

In this paper, we extend and strengthen classical estimates for singular values of integral operators originally due to Cwikel, Birman and Solomyak. Suppose that A1 and A2 are two semifinite von Neumann algebras with representations π1:scriptA1→Bfalse(scriptHfalse) and π2:scriptA2→Bfalse(scriptHfalse) such that false∥π1false(xfalse)π2false(yfalse)∥scriptL2false(scriptHfalse)⩽const ∥x∥scriptL2false(A1false)∥y∥scriptL2false(scriptAfalse).Our first result asserts that the inequality false∥π1false(xfalse)π2false(yfalse)∥E(H)⩽CE∥x⊗y∥E(scriptA1⊗scriptA2)holds in any interpolation space E for (L2,L∞). In the special case, when scriptA1=scriptA2=L∞false(Rdfalse) and π1, π2 are given by multipliers and Fourier multipliers, respectively, our result yields a (strengthened version) of well‐known Cwikel estimates [Cwikel, Ann. of Math. (2) 106 (1977) 93–100]. We demonstrate further the applicability of our result by considering noncommutative Euclidean space (Moyal plane) and magnetic Laplacian. Our second direction relates to Birman–Solomyak estimates for interpolation space E for the (quasi)‐Banach couple (ℓp,ℓ2), p<2. In this setting, our technique yields substantial strengthening of results from [Birman and Solomyak, Russian Math. Surveys 32 (1977) 15–89] and Chapter VI of Simon [Trace ideals and their applications (American Mathematical Society, Providence, RI, 2005)]. Finally, we provide Cwikel–Birman–Solomyak estimates for the crucial case of weak Schatten p‐ideals, 1⩽p<2, in the setting of noncommutative Euclidean space.

Quantum Differentiability on Quantum Tori

“…Hence, (3.2) holds and thus by [33,Corollory 3.5] it follows that (3.1) holds. Now, we take E = L p in (3.1) for p ∈ (2, ∞).…”

“…After proving (3.1), we explain how it entails the results in the statement of the theorem. In fact (3.1) can be obtained by a direct application of [33,Corollary 3.5]. Here we have two von Neumann algebras L ∞ (T d θ ) and ℓ ∞ (Z d ) represented on the same Hilbert space L 2 (T d θ ) by left multiplication and Fourier multiplication respectively.…”

“…Here we have two von Neumann algebras L ∞ (T d θ ) and ℓ ∞ (Z d ) represented on the same Hilbert space L 2 (T d θ ) by left multiplication and Fourier multiplication respectively. In this setting, we can use [33,Corollary 3.5] which states that if we have an estimate of the form:…”

Quantum Differentiability on Noncommutative Euclidean Spaces

“…Noncommutative Euclidean spaces admit several equivalent definitions; here we follow the approach in [39], where the authors define L ∞ (R d θ ) as twisted group von Neumann algebra and then define function spaces on R d θ as being operator spaces associated to that algebra. We refer the reader to [39] for more details on this approach, and give a brief introduction here.…”

“…Quantum Euclidean spaces have also been studied as an interesting noncommutative setting for classical and harmonic analysis, and for this we refer the reader to recent work such as [24,39,46,67].…”