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Singularities of embedding operators between symmetric function spaces on \([0,1]\). (English. Russian original) Zbl 0914.46029

Math. Notes 62, No. 4, 457-468 (1997); translation from Mat. Zametki 62, No. 4, 549-563 (1997).
Summary: The properties of the identity embedding operator \(I(X_1, X_2)\) \((X_1\subset X_2)\) between symmetric function spaces on \([0,1]\) such as weak compactness, strict singularity (in two versions), and the property of being absolutely summing are examined. Banach and quasi-Banach spaces are considered. A complete description of the linear hull closed with respect to measure of a sequence \((g^{(r)}_n)\) of independent symmetric equidistributed random variables with \[ d(g^{(r)}_n; t)= \text{meas}(\omega:| g^{(r)}_n(\omega)|> t)= {1\over t^r}\quad\text{for }t\geq 1\quad\text{and }0< r<\infty \] is obtained, and the boundaries for this space on the scale of symmetric spaces are found.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Full Text: DOI

References:

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