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

Sergio Antonio Tozoni

Studia Mathematica, Tome 162 (2004), p. 71-97 / Harvested from The Polish Digital Mathematics Library

Résumé

Let X be a homogeneous space and let E be a UMD Banach space with a normalized unconditional basis $(e_{j})_{j≥1}$. Given an operator T from $L^{∞}_{c}(X)$ to L¹(X), we consider the vector-valued extension T̃ of T given by $T̃(∑_{j} f_{j}e_{j}) = ∑_{j} T(f_{j})e_{j}$. 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 $L^{p}(X,Wdμ;E)$ for 1 < p < ∞ and for a weight W in the Muckenhoupt class $A_{p}(X)$. Applications to singular integral operators on the unit sphere Sⁿ and on a finite product of local fields 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

Publié le : 2004-01-01

Zbl 1044.42013

EUDML-ID : urn:eudml:doc:284718