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): 

$$
\begin{align*} 
\|f\|_{L^{(p}} &\approx \int_0^1 (1-\ln t)^{-\frac{1}{p}}\left(\int_0^t [f^{\ast}(s)]^{p}ds\right)^{\frac{1}{p}} dt/t &\qquad &(1 < p < \infty) \\ 
\|f\|_{L^{p)}} &\approx \sup_{0 < t < 1} (1-\ln t)^{-\frac{1}{p}} \left(\int_{t}^1 [f^{\ast}(s)]^p ds\right)^{\frac{1}{p}} &\qquad &(1 < p < \infty). 
\end{align*} 
$$

Similar results are proved for the generalized small and grand spaces.

Grand and Small Lebesgue Spaces and Their Analogs

Alberto Fiorenza

G. E. Karadzhov

Abstract

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