Embeddings of B-valued Lorentz-Karamata spaces associated with martingale transforms

Kaituo Liu; Xiqin Jiang; Kaituo Liu; Xiqin Jiang

doi:10.3934/era.2026389

Electronic Research Archive

2026, Volume 34, Issue 12: 9250-9266. doi: 10.3934/era.2026389

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Embeddings of B-valued Lorentz-Karamata spaces associated with martingale transforms

Kaituo Liu ^,,
Xiqin Jiang

Department of Mathematics and Statistics, Hubei University of Automotive Technology, Shiyan 442002, China

Received: 22 July 2026 Revised: 20 October 2026 Accepted: 20 October 2026 Published: 28 October 2026

In this article, the embeddings of Banach space valued (briefly by $ {\bf{B}} $-valued) Lorentz-Karamata martingale spaces are established. Different from the previous approach, we employ the technique of martingale transforms defined by some operator sequences, which can often supply a clear and succinct method in the research of martingale theory and Harmonic analysis. It's worth mentioning that the results obtained here are connected closely with the geometrical properties of the Banach spaces in which the martingales take values, such as $ p $-uniform smoothness and $ q $-uniform convexity.
- martingale transform,
- Lorentz-Karamata space,
- embedding,
- $ {\bf{B}} $-valued martingale,
- uniform convexity,
- uniform smoothness
Citation: Kaituo Liu, Xiqin Jiang. Embeddings of B-valued Lorentz-Karamata spaces associated with martingale transforms[J]. Electronic Research Archive, 2026, 34(12): 9250-9266. doi: 10.3934/era.2026389

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Abstract

In this article, the embeddings of Banach space valued (briefly by $ {\bf{B}} $-valued) Lorentz-Karamata martingale spaces are established. Different from the previous approach, we employ the technique of martingale transforms defined by some operator sequences, which can often supply a clear and succinct method in the research of martingale theory and Harmonic analysis. It's worth mentioning that the results obtained here are connected closely with the geometrical properties of the Banach spaces in which the martingales take values, such as $ p $-uniform smoothness and $ q $-uniform convexity.

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