1In 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.
We compute the K-functional related to some couple of spaces as small or classical Lebesgue space or Lorentz-Marcinkiewicz spaces completing the results of [12]. This computation allows to determine the interpolation space in the sense of Peetre for such couple. It happens that the result is always a GΓ-space, since this last space covers many spaces. The motivations of such study are various, among them we wish to obtain a regularity estimate for the so called very weak solution of a linear equation in a domain Ω with data in the space of the integrable function with respect to the distance function to the boundary of Ω.
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