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Sharp weak-type inequalities for Fourier multipliers and second-order Riesz transforms

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Central European Journal of Mathematics

Abstract

We study sharp weak-type inequalities for a wide class of Fourier multipliers resulting from modulation of the jumps of Lévy processes. In particular, we obtain optimal estimates for second-order Riesz transforms, which lead to interesting a priori bounds for smooth functions on ℝd. The proofs rest on probabilistic methods: we deduce the above inequalities from the corresponding estimates for martingales. To obtain the lower bounds, we exploit the properties of laminates, important probability measures on the space of matrices of dimension 2×2, and some transference-type arguments.

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Authors and Affiliations

  1. Institute of Mathematics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097, Warsaw, Poland

    Adam Osękowski

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  1. Adam Osękowski
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Correspondence to Adam Osękowski.

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Osękowski, A. Sharp weak-type inequalities for Fourier multipliers and second-order Riesz transforms. centr.eur.j.math. 12, 1198–1213 (2014). https://doi.org/10.2478/s11533-014-0401-6

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  • DOI: https://doi.org/10.2478/s11533-014-0401-6

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