Sparse domination results for compactness on weighted spaces

Abstract

By means of appropriate sparse bounds, we deduce compactness on weighted $L^p(w)$ spaces, $1<p<\infty$, for all Calderón-Zygmund operators having compact extensions on $L^2(\mathbb{R}^n)$. Similar methods lead to new results on boundedness and compactness of Haar multipliers on weighted spaces. In particular, we prove weighted bounds for weights in a class strictly larger than the typical $A_p$ class.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof : Proof of Lemma \ref{['CZDecomposition']}
• proof : Proof of Lemma \ref{['HaarTruncationWeakType']}