## Journal of the Mathematical Society of Japan

### Weak-type inequalities for Fourier multipliers with applications to the Beurling-Ahlfors transform

#### Abstract

The paper contains the study of weak-type constants of Fourier multipliers resulting from modulation of the jumps of Léevy processes. We exhibit a large class of functions $m: {\mathbb R}^d\to {\mathbb C}$, for which the corresponding multipliers $T_m$ satisfy the estimates

$\|T_m f\|_{L^{p,\infty}({\mathbb R}^d)} \leq \bigg[ \frac{1}{2}\Gamma \bigg( \frac{2p-1}{p-1} \bigg) \bigg]^{(p-1)/p} \|f\|_{L^p({\mathbb R}^d)}$

for 1 < $p$ < 2, and

$\|T_m f\|_{L^{p,\infty}({\mathbb R}^d)} \leq \bigg[ \frac{p^{p-1}}{2} \bigg]^{1/p} \|f\|_{L^p({\mathbb R}^d)}$

for $2 \leq p$ < $\infty$. The proof rests on a novel duality method and a new sharp inequality for differentially subordinated martingales. We also provide lower bounds for the weak-type constants by constructing appropriate examples for the Beurling-Ahlfors operator on ${\mathbb C}$.

#### Article information

Source
J. Math. Soc. Japan, Volume 66, Number 3 (2014), 745-764.

Dates
First available in Project Euclid: 24 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1406206971

Digital Object Identifier
doi:10.2969/jmsj/06630745

Mathematical Reviews number (MathSciNet)
MR3238316

Zentralblatt MATH identifier
1306.42020

#### Citation

OSȨKOWSKI, Adam. Weak-type inequalities for Fourier multipliers with applications to the Beurling-Ahlfors transform. J. Math. Soc. Japan 66 (2014), no. 3, 745--764. doi:10.2969/jmsj/06630745. https://projecteuclid.org/euclid.jmsj/1406206971

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