Abstract
We characterize the Hilbert–Schmidt class membership of commutator with the Hilbert transform in the two weight setting. The characterization depends upon the symbol of the commutator being in a new weighted Besov space. This follows from a Schatten class Sp result for dyadic paraproducts, where \(1< p < \infty \). We discuss the difficulties in extending the dyadic result to the full range of Schatten classes for the Hilbert transform.
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M. Lacey is a 2020 Simons Fellow, his Research is supported in part by grant from the U.S. National Science Foundation, DMS–1949206. J. Li is supported by the Australian Research Council through the research grant DP220100285. B. D. Wick’s research is supported in part by U. S. National Science Foundation – DMS 1800057 and Australian Research Council – DP 190100970.
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Lacey, M., Li, J. & Wick, B.D. Schatten Classes and Commutator in the Two Weight Setting, I. Hilbert Transform. Potential Anal 60, 875–894 (2024). https://doi.org/10.1007/s11118-023-10072-x
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DOI: https://doi.org/10.1007/s11118-023-10072-x