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Zeitschrift für Analysis und ihre Anwendungen

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Volume 23, Issue 4, 2004, pp. 657–681
DOI: 10.4171/ZAA/1215

Published online: 2004-12-31

Grand and Small Lebesgue Spaces and Their Analogs

Alberto Fiorenza[1] and G. E. Karadzhov[2]

(1) Università degli Studi di Napoli Federico II, Italy
(2) 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.

Keywords: Extrapolation, interpolation, Lorentz-Karamata spaces

Fiorenza Alberto, Karadzhov G. E.: Grand and Small Lebesgue Spaces and Their Analogs. Z. Anal. Anwend. 23 (2004), 657-681. doi: 10.4171/ZAA/1215