Grand and Small Lebesgue Spaces and Their Analogs

Alberto Fiorenza; G. E. Karadzhov

doi:10.4171/zaa/1215

JournalszaaVol. 23, No. 4pp. 657–681

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

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Abstract

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}} ∥ f ∥_{L^{p)}} \approx \int_{0}^{1} (1 - ln t)^{- \frac{1}{p}} (\int_{0}^{t} [f^{*} (s)]^{p} d s)^{\frac{1}{p}} d t / t \approx 0 < t < 1 sup (1 - ln t)^{- \frac{1}{p}} (\int_{t}^{1} [f^{*} (s)]^{p} d s)^{\frac{1}{p}} (1 < p < \infty) (1 < p < \infty) .

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