Complex interpolation of quasi-Banach spaces with an A-convex containing space
Let \((X_ 0,X_ 1)\) be a compatible pair of quasi-Banach spaces, and the pair \((X_ 0,x_ 1)\) has an \(A\)-convex containing space \({\mathcal U}\) (i.e. \(X_ 0+X_ 1\subset{\mathcal U}\) with continuous inclusion). The space \({\mathcal H}({\mathcal U})\) consists of all functions \(f\in A(S,{\mathcal U})\), the class of \({\mathcal U}\)-valued bounded analytic functions on the strip \(S=\{z\): \(0<\text{re} z<1\}\) that are continuous up to the boundary of \(S\), such that \(f(j+it)\in X_ j\) and \(f(j+it)\) defines a bounded continuous function \(f_ j\) from \(\mathbb{R}\) to \(X_ j\), \(j=0,1\). Let \(\| f\|_{{\mathcal H}({\mathcal U})}=\max_{j=0,1} \sup_{t\in\mathbb{R}}\| f_ j(t)\|_ j\), where \(\|\cdot\|_ j\) denotes the quasi-norm of \(X_ j\), \(j=0,1\), and \({\mathcal F}({\mathcal U})\) be the \(\|\cdot\|_{{\mathcal H}({\mathcal U})}\)-closure of \(\{\sum_{n=1}^ N \phi_ n x_ n\): \(x_ n\in X_ 0\cap X_ 1\), \(\phi_ n\in A(S,\mathbb{C})\), \(N\in\mathbb{N}\}\), where \(\mathbb{C}\) is the complex field. For \(0\leq\theta\leq 1\), \((X_ 0,X_ 1)_{[\theta],{\mathcal U}}\) denote the space of all \(f(\theta)\), \(f\in{\mathcal F}({\mathcal U})\), with the complete quasi-norm \(\| x\|_{[\theta],{\mathcal U}}=\inf\{\| f\|_{{\mathcal H}({\mathcal U})}\): \(f\in {\mathcal F}({\mathcal U}\), \(f(\theta)=x\}\). The authors show some properties of the spaces \((X_ 0,X_ 1)_{[\theta],{\mathcal U}}\) and discuss a number of examples.
scientific article; zbMATH DE number 60547
scientific article