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

Authors:	S Ya Novikov

Abstract:	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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