top
Let X be a homogeneous space and let E be a UMD Banach space with a normalized unconditional basis . Given an operator T from to L¹(X), we consider the vector-valued extension T̃ of T given by . 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 for 1 < p < ∞ and for a weight W in the Muckenhoupt class . 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.
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 -