Abstract
In this paper we construct a wavelet basis in \(L^{2}({\mathbb {R}}^{n};\mu )\) possessing vanishing moments of a fixed order for a general locally finite positive Borel measure \(\mu \). The approach is based on a clever construction of Alpert in the case of the 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 Calderón–Zygmund operator on \({\mathbb {R}}\) and conjecture that under suitable natural conditions, including a weaker energy condition, the operator is bounded from \(L^{2}({\mathbb {R}};\sigma )\) to \(L^{2}({\mathbb {R}};\omega )\) if certain stronger testing conditions hold on polynomials. An example is provided showing that this conjecture is logically different from existing results in the literature.
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Acknowledgements
E. T. Sawyer research supported in part by a grant from the National Science and Engineering Research Council of Canada. B. D. Wick research supported in part by National Science Foundation DMS Grants # 1560955 and 1800057.
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Communicated by Stephan Dahlke.
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Rahm, R., Sawyer, E.T. & Wick, B.D. Weighted Alpert Wavelets. J Fourier Anal Appl 27, 1 (2021). https://doi.org/10.1007/s00041-020-09784-0
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DOI: https://doi.org/10.1007/s00041-020-09784-0