Maximal regularity of discrete and continuous time evolution equations

Sönke Blunck. "Maximal regularity of discrete and continuous time evolution equations." Studia Mathematica 146.2 (2001): 157-176. <http://eudml.org/doc/284448>.

@article{SönkeBlunck2001,
abstract = {We consider the maximal regularity problem for the discrete time evolution equation $u_\{n+1\} - Tuₙ = fₙ$ for all n ∈ ℕ₀, u₀ = 0, where T is a bounded operator on a UMD space X. We characterize the discrete maximal regularity of T by two types of conditions: firstly by R-boundedness properties of the discrete time semigroup $(Tⁿ)_\{n∈ℕ₀\}$ and of the resolvent R(λ,T), secondly by the maximal regularity of the continuous time evolution equation u’(t) - Au(t) = f(t) for all t > 0, u(0) = 0, where A:= T - I. By recent results of Weis, this continuous maximal regularity is characterized by R-boundedness properties of the continuous time semigroup $(e^\{t(T-I)\})_\{t≥0\}$ and again of the resolvent R(λ,T). As an important tool we prove an operator-valued Mikhlin theorem for the torus providing conditions on a symbol $M ∈ L_\{∞\}(;(X))$ such that the associated Fourier multiplier $T_\{M\}$ is bounded on $l_\{p\}(X)$.},
author = {Sönke Blunck},
journal = {Studia Mathematica},
keywords = {maximal regularity; multipliers; semigroup; discrete time evolution equation},
language = {eng},
number = {2},
pages = {157-176},
title = {Maximal regularity of discrete and continuous time evolution equations},
url = {http://eudml.org/doc/284448},
volume = {146},
year = {2001},
}

TY - JOUR
AU - Sönke Blunck
TI - Maximal regularity of discrete and continuous time evolution equations
JO - Studia Mathematica
PY - 2001
VL - 146
IS - 2
SP - 157
EP - 176
AB - We consider the maximal regularity problem for the discrete time evolution equation $u_{n+1} - Tuₙ = fₙ$ for all n ∈ ℕ₀, u₀ = 0, where T is a bounded operator on a UMD space X. We characterize the discrete maximal regularity of T by two types of conditions: firstly by R-boundedness properties of the discrete time semigroup $(Tⁿ)_{n∈ℕ₀}$ and of the resolvent R(λ,T), secondly by the maximal regularity of the continuous time evolution equation u’(t) - Au(t) = f(t) for all t > 0, u(0) = 0, where A:= T - I. By recent results of Weis, this continuous maximal regularity is characterized by R-boundedness properties of the continuous time semigroup $(e^{t(T-I)})_{t≥0}$ and again of the resolvent R(λ,T). As an important tool we prove an operator-valued Mikhlin theorem for the torus providing conditions on a symbol $M ∈ L_{∞}(;(X))$ such that the associated Fourier multiplier $T_{M}$ is bounded on $l_{p}(X)$.
LA - eng
KW - maximal regularity; multipliers; semigroup; discrete time evolution equation
UR - http://eudml.org/doc/284448
ER -