Applications of interpolation with a function parameter to Lorentz, Sobolev and besov spaces

Abstract

In this paper we show that interpolation with a function parameter is perfectly suited to identify interpolation spaces between two quasi-normed Lorentz spaces ∧^p(φ) and, in the case of Banach spaces, between two Sobolev spaces $W_{\Lambda ^p (\phi )}^m$. We deduce the A.P. Calderón theorem for these spaces. We prove the identity $(H_p^{\phi _0 } ,H_p^{\phi _1 } )_{f,q;K} = B_{p,q}^\psi$ where ψ is classically connected with φ₀, φ₁ and f, and in which the Sobolev space H ^φ_p and the Besov space B ^φ_p,q are constructed in the same way as in the classical case. Imbedding and trace theorems are given for these spaces, as well as equivalent norms on space H ^φ_p,q in connection with semi-groups and approximation theory.