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Sequence Spaces l p,q in Probabilistic Characterizations of Weak Type Operators

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We study operators \(T:X \mapsto L_ \circ ([0,1],{\mathcal{M}},m)\) (not necessarily linear) defined on a quasi-Bahach space X and taking values in the space of real-valued Lebesgue-measurable functions. Factorization theorems for linear and superlinear operators with values in the space \(L_ \circ \) are proved with the help of the Lorentz sequence spaces \(l_{p,q} \). Sequences of functions belonging to fixed bounded sets in the spaces \(L_{p,\infty } \) are characterized for \(0 < p < \infty \) and \(0 < q \leqslant p\). The possibility of distinguishing weak type operators (bounded in the space \(L_{p,\infty } \)) from operators factorizable through \(L_{p,\infty } \) is obtained in terms of sequences of independent random variables. A criterion under which an operator is symmetrically bounded in order in \(L_{p,r} ,{\text{ }}0 < r \leqslant \infty \), is established. Some refinements of the above-mentioned results are obtained for translation shift-invariant sets and operators. Bibliography: 30 titles.

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  1. S. Ya. Novikov
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Novikov, S.Y. Sequence Spaces l p,q in Probabilistic Characterizations of Weak Type Operators. Journal of Mathematical Sciences 120, 1733–1751 (2004). https://doi.org/10.1023/B:JOTH.0000018872.60131.b4

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  • DOI: https://doi.org/10.1023/B:JOTH.0000018872.60131.b4

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