Cwikel estimates revisited

By G. Levitina, Fedor Sukochev, Dmitriy Zanin. Proceedings of the London Mathematical Society, 2017

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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:A1→B(H) and π2:A2→B(H) such that ∥π1(x)π2(y)∥L2(H)⩽const ∥x∥L2(A1)∥y∥L2(A).Our first result asserts that the inequality ∥π1(x)π2(y)∥E(H)⩽CE∥x⊗y∥E(A1⊗A2)holds in any interpolation space E for (L2,L∞) . In the special case, when A1=A2=L∞(Rd) 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.

Subfields: Applied Mathematics, Mathematical Physics

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