The K-functional for the quasi-normed couples \((A_ 0,(A_ 0,A_ 1)^ K_ E)\) and \(((A_ 0,A_ 1)^ K_ E,A_ 1)\)
Let \(\bar A=(A_ 0,A_ 1)\) be a couple of quasi-normed spaces, i.e., \(A_ 0\) and \(A_ 1\) are two quasi-normed spaces continuously embedded in some Hausdorff topological vector space \({\mathcal A}\). For every \(a\in A_ 0+A_ 1\) we define the K-functional \[ K(t,a;\bar A)=\inf \{\| a_ 0\|_{A_ 0}+t\| a_ 1\|_{A_ 1}:a=a_ 0+a_ 1\}, \] where \(t\in {\mathbb{R}}_+=(0,\infty)\). Let \(\bar A=(A_ 0,A_ 1)\) be a couple of quasi-normed spaces. For any quasi-function spaces of measurable functions E on \(({\mathbb{R}}_+,dt/t)\) containing min(1,t) the K- space \(\bar A{}^ K_ E\) consists of all \(a\in A_ 0+A_ 1\) such that K(\(\cdot,a;\bar A)\in E\). Put \(\| a\|_{\bar A^ K}=\| K(\cdot,a;\bar A)\|_ E\). If \(E=L^ p_{t^{- \theta}}({\mathbb{R}}_+,dt/t)\), \(0<\theta <1\), \(0<p\leq \infty\) the space \(\bar A{}^ K_ E\) coincides with the familiar space \(\bar A{}_{\theta,p}\) of Lions-Peetre. In this paper we give estimations for the following couples of quasi- normed spaces \((A_ 0,(A_ 0,A_ 1)^ K_ E)\), \(((A_ 0,A_ 1)^ K_ E,A_ 1)\). These results are applied to describe some interpolation spaces (in particular Lions-Peetre spaces). Moreover under some assumptions on a symmetric space X on \({\mathbb{R}}^ n\) we obtain estimates of the K-functional for the Banach couples (X,BMO) and \((H^ 1,X)\).
scientific article; zbMATH DE number 4004956