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Grand and small Lebesgue spaces and their analogs. (English) Zbl 1071.46023

Summary: We give the following, equivalent, explicit expressions for the norms of the small and grand Lebesgue spaces, which depend only on the non-decreasing rearrangement (we assume here that the underlying measure space has measure 1): \[ \|f\|_{L^(p}\approx\int^1_0(1-\ln t)^{-\frac 1p}\left(\int^t_0 \bigl[f^*(s) \bigr]^pds\right)^{\frac 1p}dt/t \quad (1<p<\infty) \]
\[ \|f\|_{L^p)}\approx \sup_{0<t<1}(1-\ln t)^{-\frac 1p}\left(\int^1_t\bigl[f^*(s)\bigr]^pds \right)^{\frac 1p}\quad (1< p<\infty). \] Similar results are proven for the generalized small and grand spaces.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46B70 Interpolation between normed linear spaces
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