A generalized \(H^\infty\) functional calculus for operators on subspaces of \(L^p\) and application to maximal regularity
Let \((\Omega,\mu)\) be a measure space and let \(1<p<\infty\) be a number. Consider a closed operator \(A\) on \(L^p(\Omega)\) and assume that it admits a bounded \(H^\infty(\Sigma_\theta)\) functional calculus (in the sense of McIntosh) for some sector \(\Sigma_\theta= \{z\in \mathbb{C}^*:|\arg z|< \theta\}\). Let \(H\) be a Hilbert space and let \({\mathcal A}\) be the closure, which exists, of \(A\otimes I_H\) on \(L^p(\Omega; H)\). In recent joint work with F. Lancien, we showed that for any \(\nu>\theta\), the bounded holomorphic functional calculus of \(A\) naturally extends to a bounded \(H^\infty(\Sigma_\nu; B(H))\) functional calculus for \({\mathcal A}\). As a consequence, we could deduce abstract maximal regularity results on spaces of the form \(L^p(\Omega; H)\), for operators which are the sum of an operator acting on \(L^p(\Omega)\) and another one acting on \(H\). The purpose of this paper is to extend the above-mentioned results to the case \(p=1\) and to the situation where \(L^p(\Omega)\) is replaced by one of its closed subspaces. As a consequence, we get a new class of operators satisfying the \(L_p\)-maximal regularity property for the first-order Cauchy problem on intervals.
scientific article; zbMATH DE number 1193075