Estimates for Schur Multipliers and Double Operator Integrals -- A Wavelet Approach

TL;DR

This paper employs wavelet techniques to simplify proofs of norm estimates for integral operators and Schur multiplier bounds, extending classical results and providing new insights into operator theory.

Contribution

It introduces a wavelet-based approach to prove norm estimates and Schur multiplier bounds, extending previous theorems to broader classes of symbols and kernels.

Findings

Simplified proof of norm estimates for integral operators with specific Besov space kernels.

Extended Schur multiplier bounds to symbols without compact domain.

Provided new insights into the approximation theory of integral operators.

Abstract

We discuss the work of Birman and Solomyak on the singular numbers of integral operators from the point of view of modern approximation theory, in particular with the use of wavelet techniques. We are able to provide a simple proof of norm estimates for integral operators with kernel in $B^{\frac{1}{p}-\frac{1}{2}}_{p,p}(\mathbb R,L_2(\mathbb R))$. This recovers, extends and sheds new light on a theorem of Birman and Solomyak. We also use these techniques to provide a simple proof of Schur multiplier bounds for double operator integrals, with bounded symbol in $B^{\frac{1}{p}-\frac{1}{2}}_{\frac{2p}{2-p},p}(\mathbb R,L_\infty(\mathbb R))$, which extends Birman and Solomyak's result to symbols without compact domain.