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Singular Integrals with Bilinear Phases

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Abstract

We prove the boundedness from L p (T 2) to itself, 1 < p < ∞, of highly oscillatory singular integrals Sf(x, y) presenting singularities of the kind of the double Hilbert transform on a non–rectangular domain of integration, roughly speaking, defined by |y'| > |x'|, and presenting phases λ(Ax+By) with 0 ≤ A, B ≤ 1 and λ ≥ 0. The norms of these oscillatory singular integrals are proved to be independent of all parameters A, B and λ involved. Our method extends to a more general family of phases. These results are relevant to problems of almost everywhere convergence of double Fourier and Walsh series.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma "Tor Vergata", Roma 00133, Italy

    Elena Prestini

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  1. Elena Prestini
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Correspondence to Elena Prestini.

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Supported by Progetto cofimansiato HIUR "Amatisi Armomica"

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Prestini, E. Singular Integrals with Bilinear Phases. Acta Math Sinica 22, 251–260 (2006). https://doi.org/10.1007/s10114-005-0562-0

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  • DOI: https://doi.org/10.1007/s10114-005-0562-0

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