Publishing house / Journals and Serials / Studia Mathematica / All issues

Abstract

Let $X$ be a homogeneous space and let $E$ be a UMD Banach space with a normalized unconditional basis $(e_j)_{j\geq 1}$. Given an operator $T$ from $L^{\infty }_{\rm c}(X)$ to $L^1(X)$, we consider the vector-valued extension ${\widetilde T}$ of $T$ given by ${\widetilde T}(\sum _jf_je_j)=\sum _jT(f_j)e_j$. We prove a weighted integral inequality for the vector-valued extension of the Hardy–Littlewood maximal operator and a weighted Fefferman–Stein inequality between the vector-valued extensions of the Hardy–Littlewood and the sharp maximal operators, in the context of Orlicz spaces. We give sufficient conditions on the kernel of a singular integral operator to have the boundedness of the vector-valued extension of this operator on $L^p(X,Wd\mu ;E)$ for $1< p< \infty $ and for a weight $W$ in the Muckenhoupt class $A_p(X)$. Applications to singular integral operators on the unit sphere $S^n$ and on a finite product of local fields ${ \mathbb K}^n$ are given. The versions of all these results for vector-valued extensions of operators on functions defined on a homogeneous space $X$ and with values in a UMD Banach lattice are also given.