Open Access
2014 On restricted weak-type constants of Fourier multipliers
Adam Oseękowski
Publ. Mat. 58(2): 415-443 (2014).


We exhibit a large class of symbols $m\colon \mathbb{R}^d\to \mathbb{C}$ for which the corresponding Fourier multipliers $T_m$ satisfy the following restricted weak-type estimates: if $A\subset \mathbb{R}^d$ has finite Lebesgue measure, then

$$||T_m\chi_A||_{p,\infty}\leq \frac{p}{2}e^{(2-p)/p}||\chi_A||_p,\quad p\geq 2.$$

In particular, this leads to novel sharp estimates for the real and imaginary part of the Beurling-Ahlfors operator on $\mathbb{C}$. The proof rests on probabilistic methods: we exploit a stochastic representation of the multipliers in terms of Lévy processes and appropriate sharp inequalities for differentially subordinated martingales.


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Adam Oseękowski. "On restricted weak-type constants of Fourier multipliers." Publ. Mat. 58 (2) 415 - 443, 2014.


Published: 2014
First available in Project Euclid: 21 July 2014

zbMATH: 1304.30068
MathSciNet: MR3264505

Primary: 42B15 , 60G44
Secondary: 42B20

Keywords: Beurling-Ahlfors transform , Differential subordination , Fourier multiplier , martingale , singular integral

Rights: Copyright © 2014 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.58 • No. 2 • 2014
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