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.