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Applications of interpolation with a function parameter to Lorentz, Sobolev and Besov spaces. (English) Zbl 0546.46061
Interpolation spaces and allied topics in analysis, Proc. Conf., Lund/Swed. 1983, Lect. Notes Math. 1070, 183-201 (1984).
[For the entire collection see Zbl 0534.00013.]
In this paper, we take an interest in real interpolation. The Lebesgue, Lorentz, Sobolev and Besov spaces are fundamentally connected to interpolation with the real parameter $$\Theta$$, $$0<\Theta <1$$, actually to interpolation with the function parameter $$t\to t^{\theta}$$. We show here that the generalized preceding spaces, that are those obtained by replacing the function $$t\to t^{\theta}$$ by a more general function f ”almost” sub-multiplicative on $$]0,+\infty [$$, have analogous interpolation properties with respect to interpolation with a function parameter.
In particular, we identify interpolation spaces between two Lorentz spaces $$\Lambda^ p(\phi)$$, two Sobolev spaces $$W^ m_{\Lambda^ p(\phi)}$$ (in each case we obtain a same type space) and two Sobolev spaces $$H_ p^{\phi}$$ (we obtain a Besov space $$B^{\psi}_{p,q})$$. We prove the A. P. Calderón theorem for the spaces $$\Lambda^ p(\phi)$$ and $$W^ m_{\Lambda^ p(\phi)}$$. Imbedding and trace theorems are also given for the spaces $$H_ p^{\phi}$$ and $$B^{\phi}_{p,q}$$. As in the classical case we show the connection between semi-groups and intepolation. This is applied to Besov spaces $$B^{\phi}_{p,q}$$ for which we also prove an approximation theorem.

##### MSC:
 46M35 Abstract interpolation of topological vector spaces 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems