Article Dans Une Revue Nonlinear Analysis: Theory, Methods and Applications Année : 2018

Characterization of interpolation between Grand, small or classical Lebesgue spaces

Résumé

In this paper, we show that the interpolation spaces between Grand, small or classical Lebesgue are so called Lorentz-Zygmund spaces or more generally GΓ-spaces. As a direct consequence of our results any Lorentz-Zygmund space L a,r (Log L) β , is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces provided that 1 < a < ∞, β = 0. The method consists in computing the so called K-functional of the interpolation space and in identifying the associated norm. Contents 1 Introduction. Main results 2 2 Notations. Preliminary Lemmas 7

3 Computation of some K-functionals and characterization of the interpolation spaces (L p),α , L q),α ) θ,r 11 4 Small Lebesgue space as interpolation of usual Lebesgue spaces 22 5 Interpolation between small, Grand Lebesgue spaces and the associated K-functional 24 6 The critical case p = q. The interpolation space (L p) , L (p ) and its Kfunctional 28

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hal-04869402 , version 1 (07-01-2025)

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Alberto Fiorenza, Maria Rosaria Formica, Amiran Gogatishvili, Tengiz Kopaliani, Jean Michel Rakotoson. Characterization of interpolation between Grand, small or classical Lebesgue spaces. Nonlinear Analysis: Theory, Methods and Applications, 2018, 177 (part B), pp.422-453. ⟨10.1016/j.na.2017.09.005⟩. ⟨hal-04869402⟩
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