Relative weak compactness in infinite-dimensional Fefferman-Meyer duality
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929548690456576 |
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| author | Melnikov, Vasily |
| author_facet | Melnikov, Vasily |
| contents | Let $E$ be a Banach space such that $E'$ has the Radon-Nikodým property. The aim of this work is to connect relative weak compactness in the $E$-valued martingale Hardy space $H^{1}(μ,E)$ to a convex compactness criterion in a weaker topology, such as the topology of uniform convergence on compacts in measure. These results represent a dynamic version of the deep result of Diestel, Ruess, and Schachermayer on relative weak compactness in $L^{1}(μ,E)$. In the reflexive case, we obtain a Kadec-Pełczyński dichotomy for $H^{1}(μ,E)$-bounded sequences, which decomposes a subsequence into a relatively weakly compact part, a pointwise weakly convexly convergent part, and a null part converging to zero uniformly on compacts in measure. As a corollary, we investigate a parameterized version of the vector-valued Komlós theorem without the assumption of $H^{1}(μ,E)$-boundedness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13416 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relative weak compactness in infinite-dimensional Fefferman-Meyer duality Melnikov, Vasily Functional Analysis Probability 46A50, 46E40, 60G44 Let $E$ be a Banach space such that $E'$ has the Radon-Nikodým property. The aim of this work is to connect relative weak compactness in the $E$-valued martingale Hardy space $H^{1}(μ,E)$ to a convex compactness criterion in a weaker topology, such as the topology of uniform convergence on compacts in measure. These results represent a dynamic version of the deep result of Diestel, Ruess, and Schachermayer on relative weak compactness in $L^{1}(μ,E)$. In the reflexive case, we obtain a Kadec-Pełczyński dichotomy for $H^{1}(μ,E)$-bounded sequences, which decomposes a subsequence into a relatively weakly compact part, a pointwise weakly convexly convergent part, and a null part converging to zero uniformly on compacts in measure. As a corollary, we investigate a parameterized version of the vector-valued Komlós theorem without the assumption of $H^{1}(μ,E)$-boundedness. |
| title | Relative weak compactness in infinite-dimensional Fefferman-Meyer duality |
| topic | Functional Analysis Probability 46A50, 46E40, 60G44 |
| url | https://arxiv.org/abs/2404.13416 |