Mathematical Research Letters

Volume 21 (2014)

Number 4

Fourier multipliers on weighted $L^p$ spaces

Pages: 807 – 830

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n4.a11

Author

Sebastian Król (Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Torń, Poland; and Institut für Analysis, Fachrichtung Mathematik, Dresden, Germany)

Abstract

The paper provides a complement to the classical results on Fourier multipliers on $L^p$ spaces. In particular, we prove that if $q \in (1, 2)$ and a function $m : \mathbb{R} \to \mathbb{C}$ is of bounded $q$-variation uniformly on the dyadic intervals in $\mathbb{R}$, i.e., $m \in V_q (\mathcal{D})$, then $m$ is a Fourier multiplier on $L^p (\mathbb{R}, w \: dx)$ for every $p \geq q$ and every weight $w$ satisfying Muckenhoupt’s $A_{p/q}$-condition. We also obtain a higher-dimensional counterpart of this result as well as of a result by $\mathrm{E}$. Berkson and T.A. Gillespie including the case of the $V_q (\mathcal{D})$ spaces with $q \gt 2$. New weighted estimates for modified Littlewood-Paley functions are also provided.

Keywords

weighted Fourier multipliers, weighted inequalities, Littlewood-Paley square functions, Muckenhoupt weights

2010 Mathematics Subject Classification

Primary 42B25. Secondary 42B15.

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