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Optimal Banach function space generated with the cone of nonnegative increasing functions. (Russian. English summary) Zbl 1464.46030

Summary: The article deals with the effective constructions for the optimal Banach ideal and symmetric spaces (of functions \(f: [0,T]\to\mathbb{R}\)) containing a cone of nonnegative and monotone inceasing functions with respect to the natural functionally generated \(L_p\)-norm (\(1\leq p<\infty\)). The first of these spaces turns out to be the space of measurable functions \(f\) such that \(\|f\|_{L_\infty(\cdot,T)}\in L_p(0,T)\); this space can be endowed with the norm \(\|\,\|f\|_{L_\infty(\cdot,T)}\|f\|_{L_p(0,T)}\). \( \bigl\|\,\|f\|_{L_\infty(\cdot,T)} \,\bigr\|_{L_p(0,T)} \). The second coincides with the usual space \(L_p\).

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:

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