Grand and Small Lebesgue Spaces and Their Analogs

  • Alberto Fiorenza

    Università degli Studi di Napoli Federico II, Italy
  • G. E. Karadzhov

    Bulgarian Acedemy of Sciences, Sofia, Bulgaria


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{alignat*}{2} \|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{alignat*} Similar results are proved for the generalized small and grand spaces.

Cite this article

Alberto Fiorenza, G. E. Karadzhov, Grand and Small Lebesgue Spaces and Their Analogs. Z. Anal. Anwend. 23 (2004), no. 4, pp. 657–681

DOI 10.4171/ZAA/1215