Extrapolation theory for the real interpolation method.
The first author has recently obtained improvements to a result of Yano on sublinear operators \(T\) mapping \(L\log L(\mu)\) into \(L^1(\nu)\) (when \(\mu\), \(\nu\) are finite measures) and one of Zygmund on linear operators mapping \(L^\infty(\mu)\) into \(L_{\exp}(\nu)\). In the present paper, the authors develop an abstract extrapolation theory for the real interpolation method that gives a better range space in the case of a Yano-type result and a better domain space in the case of a Zygmund-type result. The method also gives new endpoint estimates of the Sobolev embedding theorem for an arbitrary domain. The identification of the extrapolation spaces is readily made in the theory, and this means that it is easy to apply.
scientific article; zbMATH DE number 1943353