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arXiv · 1808.01223

Weighted Alpert Wavelets

Robert Rahm·Eric T. Sawyer·Brett D. Wick
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Abstract

In this paper we construct a wavelet basis in weighted L^2 of Euclidean space possessing vanishing moments of a fixed order for a general locally finite positive Borel measure. The approach is based on a clever construction of Alpert in the case of Lebesgue measure that is appropriately modified to handle the general measures considered here. We then use this new wavelet basis to study a two-weight inequality for a general Calder\'on-Zygmund operator on the real line and show that under suitable natural conditions, including a weaker energy condition, the operator is bounded from one weighted L^2 space to another if certain stronger testing conditions hold on polynomials. An example is provided showing that this result is logically different than existing results in the literature.

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BibTeXRIS

Robert Rahm, Eric T. Sawyer, Brett D. Wick. 2018-08-03. Weighted Alpert Wavelets. https://arxiv.org/abs/1808.01223

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