 

Michael T. Lacey
Issues related to Rubio de Francia's LittlewoodPaley inequality


Published: 
January 18, 2007 
Keywords: 
LittlewoodPaley inequality, multipliers, square function, multipliers, BMO 
Subject: 
Primary: 42B25. Secondary: 42B30, 42B35 


Abstract
Let $\operatorname S_\omega f=\int_\omega \widehat f(\xi)e^{ix\xi}\; d\xi$ be the Fourier projection operator to an interval $\omega$ in the real line. Rubio de Francia's LittlewoodPaley inequality (Rubio de Francia, 1985) states that for any collection of disjoint intervals $\Omega$, we have \begin{equation*}\notag \NORM \Biggl[ \sum_{\omega\in\Omega} \abs{\operatorname S_\omega f}^2\Biggr]^{1/2} .p.\lesssim{}\norm f.p.,\qquad 2\le{}p<\infty. \end{equation*} We survey developments related to this inequality, including the higher dimensional case, and consequences for multipliers. 

Acknowledgements
Research supported in part by a National Science Foundation Grant. The author is a Guggenheim Fellow. 

Author information
School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332
lacey@math.gatech.edu
http://www.math.gatech.edu/~lacey 
