Grand and small Lebesgue spaces and their analogs
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.
scientific article; zbMATH DE number 2155001
scientific article