Weighted fully measurable grand Lebesgue spaces and the maximal theorem

Abstract

Anatriello and Fiorenza (J Math Anal Appl 422:783–797, 2015) introduced the fully measurable grand Lebesgue spaces on the interval (0 , 1) ⊂ R, which contain some known Banach spaces of functions, among which there are the classical and the grand Lebesgue spaces, and the EXPα spaces (α> 0). In this paper we introduce the weighted fully measurable grand Lebesgue spaces and we prove the boundedness of the Hardy–Littlewood maximal function. Namely, let (Formula presented.), where w is a weight, 0 < δ(·) ≤ 1 ≤ p(·) < ∞, we show that if p+= ‖ p‖ ∞< + ∞, the inequality ǁMfǁp[·], δ[·], w ≤ cǁfǁp[·], δ[·], w holds with some constant c independent of f if and only if the weight w belongs to the Muckenhoupt class Ap+

Identification of Fully Measurable Grand Lebesgue Spaces

“…Second, the attempts to look for the associate spaces of the fully measurable Lebesgue spaces, made in [4,5], both solve completely the characterization of the duality. Our result may be relevant also for the weighted variant of the fully measurable Lebesgue spaces which recently attracted much interest ( [6][7][8][9]). …”

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

“…It turned out that in the theory of PDEs, the grand Lebesgue spaces are appropriate to the existence and uniqueness solution and, also, the regularity problem for various nonlinear differential equation. The boundedness problems for fundamental integral operators of harmonic analysis were intensively studied in other works [18][19][20][21][22][23][24][25][26][27][28][29][30][31] (see also the monograph, 32 chapter 14, and references therein). In this paper, the analogous problem is treated in Banach-valued weighted grand Lebesgue spaces.…”

Sawyer's duality principle for grand Lebesgue spaces

“…In , those authors constructed weighted grand Lebesgue spaces, denoted by , and studied the boundedness of the maximal operator in the framework of these spaces. Subsequently, the boundedness of various other integral operators in spaces were studied by different authors, e.g., see .…”