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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