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

Sergio Antonio Tozoni. "Weighted norm inequalities for vector-valued singular integrals on homogeneous spaces." Studia Mathematica 161.1 (2004): 71-97. <http://eudml.org/doc/284718>.

@article{SergioAntonioTozoni2004,
abstract = {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.},
author = {Sergio Antonio Tozoni},
journal = {Studia Mathematica},
keywords = {singular integral; maximal function; homogeneous space; UMD Banach space; -weights},
language = {eng},
number = {1},
pages = {71-97},
title = {Weighted norm inequalities for vector-valued singular integrals on homogeneous spaces},
url = {http://eudml.org/doc/284718},
volume = {161},
year = {2004},
}

TY - JOUR
AU - Sergio Antonio Tozoni
TI - Weighted norm inequalities for vector-valued singular integrals on homogeneous spaces
JO - Studia Mathematica
PY - 2004
VL - 161
IS - 1
SP - 71
EP - 97
AB - 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.
LA - eng
KW - singular integral; maximal function; homogeneous space; UMD Banach space; -weights
UR - http://eudml.org/doc/284718
ER -