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 \(B^{1/p-1/2}_{p,p}(\mathbb R,L_2(\mathbb R))\). 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 \(B^{1/p-1/2}_{2p/(2-p),p}(\mathbb R,L_\infty(\mathbb R))\), which extends Birman and Solomyak’s result to symbols without compact domain.

中文翻译：

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Schur 乘子和双算子积分的估计——一种小波方法

摘要

我们从现代近似理论的角度讨论了 Birman 和 Solomyak 在积分算子奇异数方面的工作，特别是使用小波技术。我们能够为内核中的积分运算符提供范数估计的简单证明\(B^{1/p-1/2}_{p,p}(\mathbb R,L_2(\mathbb R))\) . 这恢复、扩展并阐明了伯曼和索洛米亚克定理。我们还使用这些技术为在\(B^{1/p-1/2}_{2p/(2-p),p}(\mathbb R,L_\infty(\mathbb R)))\)，将 Birman 和 Solomyak 的结果扩展到没有紧凑域的符号。