Atomic Decompositions and John-Nirenberg Theorem of Grand Martingale Hardy Spaces with Variable Exponents

Libo Li,Zhiwei Hao,Tianqing An

doi:10.1155/2022/9021391

Libo Li, Zhiwei Hao + Show 1 more

Open Access

https://doi.org/10.1155/2022/9021391

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Abstract

Let θ ≥ 0 and p · be a variable exponent, and we introduce a new class of function spaces L p · , θ in a probabilistic setting which unifies and generalizes the variable Lebesgue spaces with θ = 0 and grand Lebesgue spaces with p · ≡ p and θ = 1 . Based on the new spaces, we introduce a kind of Hardy-type spaces, grand martingale Hardy spaces with variable exponents, via the martingale operators. The atomic decompositions and John-Nirenberg theorem shall be discussed in these new Hardy spaces.

Highlights

The martingale theory is widely studied in the field of mathematical physics, stochastic analysis, and probability
Let 1 < p < ∞, and the grand Lebesgue space LpÞðEÞ introduced by Iwaniec and Sbordone [15] is defined as the Banach function space of the measurable functions f on finite E such that k f kLpÞ
We find that the framework of grand Lebesgue spaces with variable exponents has not yet been studied in martingale theory

Summary

Introduction

The martingale theory is widely studied in the field of mathematical physics, stochastic analysis, and probability. Weisz [1] presented the atomic decomposition theorem for martingale Hardy spaces. We find that the framework of grand Lebesgue spaces with variable exponents has not yet been studied in martingale theory. We first present the atomic characterization of grand Hardy martingale spaces with variable exponents. Ð2, pð·Þ, θÞ-atoms, ð3, pð·Þ, θÞ-atoms), τk are Journal of Function Spaces stopping times satisfying ða1Þ and ða2Þ in Definition 1, and μk are nonnegative numbers and k〠∈Zμk χfτk

Preliminaries

Atomic Characterization

The Generalized John-Nirenberg Theorem