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

Hao, Zhiwei; Li, Mei; Tian, Hongli; Weisz, Ferenc

doi:10.3390/math14020267

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

¹

School of Mathematics and Statistics, Hunan University of Science and Technology, Xiangtan 411201, China

²

School of Mathematics and Statistics, Central South University, Changsha 410075, China

³

Department of Numerical Analysis, Eötvös L. University, Pázmány P. Sétány 1/C, H-1117 Budapest, Hungary

^*

Author to whom correspondence should be addressed.

Mathematics 2026, 14(2), 267; https://doi.org/10.3390/math14020267

Submission received: 17 December 2025 / Revised: 3 January 2026 / Accepted: 9 January 2026 / Published: 10 January 2026

(This article belongs to the Section C: Mathematical Analysis)

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Abstract

Let E be a rearrangement-invariant Banach function space. Let

P (Ω)

denote the collection of all measurable variable exponents

p (\cdot) : Ω \to (0, \infty)

such that

0 < ess \inf_{w \in Ω} p (w) \leq ess \sup_{w \in Ω} p (w) < \infty

. In this paper, with the help of a new atomic decomposition of the variable Hardy–Lorentz–Karamata space

H_{p (\cdot), q, b}^{M}

via

{(s, p (\cdot), E)}^{M}

-atoms, we characterize the dual space of

H_{p (\cdot), q, b}^{M}

for the two cases

0 < q \leq 1

and

1 < q \leq \infty

, respectively. Using this, some new John–Nirenberg theorems associated with variable exponents are also established.

Keywords: variable Lorentz–Karamata space; rearrangement-invariant Banach function space; the John–Nirenberg theorem

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MDPI and ACS Style

Hao, Z.; Li, M.; Tian, H.; Weisz, F. The John–Nirenberg Theorems for Martingales on Variable Lorentz–Karamata Spaces. Mathematics 2026, 14, 267. https://doi.org/10.3390/math14020267

AMA Style

Hao Z, Li M, Tian H, Weisz F. The John–Nirenberg Theorems for Martingales on Variable Lorentz–Karamata Spaces. Mathematics. 2026; 14(2):267. https://doi.org/10.3390/math14020267

Chicago/Turabian Style

Hao, Zhiwei, Mei Li, Hongli Tian, and Ferenc Weisz. 2026. "The John–Nirenberg Theorems for Martingales on Variable Lorentz–Karamata Spaces" Mathematics 14, no. 2: 267. https://doi.org/10.3390/math14020267

APA Style

Hao, Z., Li, M., Tian, H., & Weisz, F. (2026). The John–Nirenberg Theorems for Martingales on Variable Lorentz–Karamata Spaces. Mathematics, 14(2), 267. https://doi.org/10.3390/math14020267

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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

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