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Non-symmetric differentially subordinate martingales and sharp weak-type bounds for Fourier multipliers

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Abstract

Let p> 2 be a given exponent. In this paper, we prove, with the best constant, the weak-type (p,p) inequality

$$\Vert T_{m}f\Vert_{L^{p,\infty}(\mathbb{R}^{d})}\leqslant C_{p}\Vert f\Vert_{L^{p}(\mathbb{R}^{d})}$$

for a large class of non-symmetric Fourier multipliers Tm obtained via modulation of jumps of certain Lévy processes. In particular, the estimate holds for appropriate linear combinations of second-order Riesz transforms and skew versions of the Beurling-Ahlfors operator on the complex plane. The proof rests on a novel probabilistic bound for Hilbert-space-valued martingales satisfying a certain non-symmetric subordination principle. Further applications to harmonic functions and Riesz systems on Euclidean domains are indicated.

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Acknowledgements

Yong Jiao was supported by National Natural Science Foundation of China (Grant Nos. 12125109 and 11961131003). The authors thank the referees for their careful reading of the paper and several helpful suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Hunan Key Laboratory of Analytical Mathematics and Applications, Central South University, Changsha, 410075, China

    Meryem Akboudj & Yong Jiao

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

    Adam Osękowski

Authors
  1. Meryem Akboudj
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  2. Yong Jiao
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  3. Adam Osękowski
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Correspondence to Adam Osękowski.

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Akboudj, M., Jiao, Y. & Osękowski, A. Non-symmetric differentially subordinate martingales and sharp weak-type bounds for Fourier multipliers. Sci. China Math. (2024). https://doi.org/10.1007/s11425-023-2147-6

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