Grand and Small Lebesgue Spaces and Their Analogs
Grand and Small Lebesgue Spaces and Their Analogs
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): \begin{align*} \|f\|_{L^{(p}} &\approx \int_0^1 (1-\ln t)^{-\frac{1}{p}}\left(\int_0^t [f^{\ast}(s)]^{p}ds\right)^{\frac{1}{p}} dt/t &\qquad &(1 < p < \infty) \\ …