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Wavelet representation of singular integral operators

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Abstract

This article develops a novel approach to the representation of singular integral operators of Calderón–Zygmund type in terms of continuous model operators, in both the classical and the bi-parametric setting. The representation is realized as a finite sum of averages of wavelet projections of either cancellative or noncancellative type, which are themselves Calderón–Zygmund operators. Both properties are out of reach for the established dyadic-probabilistic technique. Unlike their dyadic counterparts, our representation reflects the additional kernel smoothness of the operator being analyzed. Our representation formulas lead naturally to a new family of T(1) theorems on weighted Sobolev spaces whose smoothness index is naturally related to kernel smoothness. In the one parameter case, we obtain the Sobolev space analogue of the \(A_2\) theorem; that is, sharp dependence of the Sobolev norm of T on the weight characteristic is obtained in the full range of exponents. In the bi-parametric setting, where local average sparse domination is not generally available, we obtain quantitative \(A_p\) estimates which are best known, and sharp in the range \(\max \{p,p'\}\ge 3\) for the fully cancellative case.

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Notes

  1. The exponent of the \(A_2\) constant obtained in [4,  Corollary 3.2] is 10, in contrast with power 3 obtained in (6.20). Therein, it is claimed that tracking the constants in the argument of [27] yields power 8.

  2. For instance, when proving Theorems A, B below, any \(D\ge 8(\max \{k_1,k_2\}+ d_1 + d_2)\) will suffice.

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Acknowledgements

The authors are deeply thankful to Alexander Barron, Henri Martikainen and Yumeng Ou for illuminating conversations on bi-parameter T(1) theorems and weighted norm inequalities. The authors gratefully acknowledge Walton Green for his insightful reading and suggestions which led to significant improvements to the clarity of the statements and exposition.

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni Università degli Studi di Napoli “Federico II” Via Cintia, Monte S. Angelo I-80126 Napoli, Italy

    Francesco Di Plinio

  2. Department of Mathematics, Washington University in Saint Louis, 1 Brookings Drive, Saint Louis, MO, 63130, USA

    Brett D. Wick & Tyler Williams

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  1. Francesco Di Plinio
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Correspondence to Brett D. Wick.

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F. Di Plinio was partially supported by the National Science Foundation under the grants NSF-DMS-2000510, NSF-DMS-2054863. B. D. Wick’s research partially supported in part by NSF grant NSF-DMS-1800057 as well as ARC DP190100970. Data sharing not applicable to this article as no datasets were generated or analyzed.

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Di Plinio, F., Wick, B.D. & Williams, T. Wavelet representation of singular integral operators. Math. Ann. 386, 1829–1889 (2023). https://doi.org/10.1007/s00208-022-02443-3

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