Limiting case for interpolation spaces generated by holomorphic semigroups
type: Item
instance of: scholarly article
title: Limiting case for interpolation spaces generated by holomorphic semigroups
zbMATH Open document ID: 0941.47035
DOI: 10.1007/PL00005986
author: Gabriella Di Blasio
published in: Semigroup Forum
publication date: 2000-08-01T00:00:00Z
review text: Let \(A\) be the generator of an analytic semigroup \(\{T(t)\); \(t\geq 0\}\) of bounded linear operators on a Banach space \((X, \|\cdot\|)\). Denote by \(X_{\theta, p}\) \((0 < \theta< 1)\) the subspace of all elements \(x\in X\) satisfying \[ \|x\|_{\theta,p} := \left(\int^{\infty}_0 (t^{1-\theta} \|AT(t) x\|)^p t^{-1} dt\right)^{1/p}< \infty, \quad p < \infty \] and \[ \|x\|_{\theta,\infty} := \sup_t t^{1-\theta} \|AT(t)x\|. \] The author investigates the spaces generated by the norm \(\|x\|_{\theta, p}\) in the limiting case \(\theta = 0\). In particular, it is proved that these spaces are spaces of maximal regularity for the semigroups \(T_p(t)\); for different operators \(A\) and \(B\) sufficient conditions for the equality \(X^{A}_{0, p} = X^{B}_{0, p}\) are given. This paper continues the investigation begun in [ibid. 47, No. 1, 105-114 (1993; Zbl 0816.47044)].
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46B70
zbMATH DE Number: 1268033
zbMATH Keywords: analytic semigroup
zbMATH Keywords: spaces of maximal regularity
MaRDI profile type: Publication
full work available at URL: pl00005986
OpenAlex ID: W1966228003
description: scientific article; zbMATH DE number 1268033