Littlewood-Paley and multiplier theorems on weighted 𝐿^{𝑝} spaces

Abstract

The Littlewood-Paley operator γ ( f ) \gamma (f) , for functions f defined on R n {{\textbf {R}}^n} , is shown to be a bounded operator on certain weighted L p {L^p} spaces. The weights satisfy an A p {A_p} condition over the class of all n-dimensional rectangles with sides parallel to the coordinate axes. The necessity of this class of weights demonstrates the 1-dimensional nature of the operator. Results for multipliers are derived, including weighted versions of the Marcinkiewicz Multiplier Theorem and Hörmander’s Multiplier Theorem.