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

Abstract

Abstract. Let X be a homogeneous space and let E be a UMD Banach space with a normalized unconditional basis (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, W dµ; 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 S n and on a finite product of local fields 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.

Fefferman–Stein Inequalities for the Hardy–Littlewood Maximal Function on the Infinite Rooted k-ary Tree

“…At this point we would also like to note that Fefferman and Stein noted in [11] that w ∈ A 1 is a necessary condition for this inequality to hold. Since those works, the theory of weights and more in particular Fefferman-Stein inequalities and related variants have been studied in a variety of contexts [23,36,2,27] and for singular integrals [8,24,28,10,5,16] and their commutators [25,26,15]. See also [17,3,7].…”

On integral operators in weighted grand Lebesgue spaces of Banach‐valued functions

“…In this section, we will follow some definitions from Tozoni. 16 We are going to discuss Banach-valued extensions of operators for functions defined on a homogeneous type space X and with values in a UMD Banach lattice.…”

“…The theory of Banach-valued function spaces and integral operators in the context of UMD property was intensively studied in other studies. [6][7][8][9][10][11][12][13][14][15][16] For scalar functions grand Lebesgue spaces, L p) were introduced by Iwaniec and Sbordone in 1992. 17 The theory of Iwaniec-Sbordone space is nowadays one of intensively developing directions in modern analysis.…”