What question did this study set out to answer?

The research aims to investigate the dual space of Hardy–Lorentz–Karamata spaces with variable exponents.

January 14, 2026Open Access

The John–Nirenberg Theorems for Martingales on Variable Lorentz–Karamata Spaces

ZHZhiwei HaoHunan University of Science and Technology LMLi MeiHunan University of Science and Technology HTHongli TianCentral South University

Key Points

The research aims to investigate the dual space of Hardy–Lorentz–Karamata spaces with variable exponents.
Utilized atomic decomposition involving M-atoms in the variable Hardy–Lorentz–Karamata space.
Characterized cases for dual spaces under different constraints on variable exponents.
Established new John–Nirenberg theorems related to variable exponents in the context of dual spaces.

Abstract

Let E be a rearrangement-invariant Banach function space. Let P(Ω) denote the collection of all measurable variable exponents p(·):Ω→(0,∞) such that 0<essinfw∈Ωp(w)≤esssupw∈Ωp(w)<∞. In this paper, with the help of a new atomic decomposition of the variable Hardy–Lorentz–Karamata space Hp(·),q,bM via (s,p(·),E)M-atoms, we characterize the dual space of Hp(·),q,bM for the two cases 0<q≤1 and 1<q≤∞, respectively. Using this, some new John–Nirenberg theorems associated with variable exponents are also established.

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