Maximal regularity of discrete and continuous time evolution equations

Sönke Blunck

Studia Mathematica, Tome 147 (2001), p. 157-176 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $l_{p}(X)$.

Détail

Publié le : 2001-01-01

Zbl 0981.39009

EUDML-ID : urn:eudml:doc:284448