\(L_ p\)-regularity of the Cauchy problem and the geometry of Banach spaces
type: Item
instance of: scholarly article
title: \(L_ p\)-regularity of the Cauchy problem and the geometry of Banach spaces
zbMATH Open document ID: 0845.47028
published in: Illinois Journal of Mathematics
publication date: 1996-09-16T00:00:00Z
review text: We investigate the different notions of regularity of the vector-valued Cauchy problem: \(u'+ Au= f\), \(u(0)= 0\), that is: if \(L[0,T ]\) is one of the lattices \(L_p [0,T ]\), \(1\leq p\leq +\infty\) or \(C[0,T ]\), then \(\text{CP}_A\) is \(L\)-regular if there exists a constant \(C\) such that, for all \(f\in L([0,T ]; X): |Au |_{L(X)}= |A({d\over dt}+ A)^{-1}f|_{L(X)}\leq C|f|_{L(X)}\). We prove its \(L_2\)-regularity for some recent examples of unbounded operators \(A\) on \(L_p\), \(1< p\neq 2< +\infty\). We give a characterization of the \(L_1\)-regularity. We characterize the regularity of the problems associated with diagonal operators on classical sequence spaces \(\ell_p\) and \(c_0\).
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 47A05
Mathematics Subject Classification ID: 47A50
Mathematics Subject Classification ID: 46E40
zbMATH DE Number: 878712
zbMATH Keywords: \(L_ p\)-regularity
zbMATH Keywords: generator of analytic semi-groups
zbMATH Keywords: unconditional basis
zbMATH Keywords: convolution operator-multiplier
zbMATH Keywords: regularity
zbMATH Keywords: vector-valued Cauchy problem
zbMATH Keywords: diagonal operators
author: Sylvie Guerre-Delabriere
MaRDI profile type: Publication
Recommended article: Q4840128
Recommended article: Q4316234
Recommended article: On the geometry of Banach spaces of the form 𝐿𝑖𝑝₀(𝐶(𝐾))
Recommended article: Arens-regularity of Banach algebras and the geometry of Banach spaces
Recommended article: A geometry characteristic of Banach spaces with \(c^1\)-norm
Recommended article: Maximal regularity in \(L^p\) spaces for an abstract Cauchy problem
Recommended article: On a Cauchy problem in Banach spaces
Recommended article: On the regularity of the \(L_p\) Minkowski problem
Recommended article: The 𝐿^{𝑝} regularity problem on Lipschitz domains
Recommended article: Q3991311
description: scientific article; zbMATH DE number 878712