Weighted norm inequalities for vector-valued singular integrals on homogeneous spaces

Sergio Antonio Tozoni¹

¹ Instituto de Matemática Universidade Estadual de Campinas Caixa Postal 6065 13.081-970 Campinas – SP, Brazil

Studia Mathematica, Tome 161 (2004) no. 1, pp. 71-97

Résumé

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.

Détail

DOI : 10.4064/sm161-1-5

Keywords: homogeneous space umd banach space normalized unconditional basis geq given operator infty consider vector valued extension widetilde given widetilde sum sum j prove weighted integral inequality vector valued extension hardy littlewood maximal operator weighted fefferman stein inequality between vector valued extensions hardy littlewood sharp maximal operators context orlicz spaces sufficient conditions kernel singular integral operator have boundedness vector valued extension operator infty weight muckenhoupt class applications singular integral operators unit sphere finite product local fields mathbb given versions these results vector valued extensions operators functions defined homogeneous space values umd banach lattice given