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• # Artykuł - szczegóły

## Studia Mathematica

2001 | 146 | 2 | 157-176

## Maximal regularity of discrete and continuous time evolution equations

EN

### Abstrakty

EN
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 𝕋 providing conditions on a symbol $M ∈ L_{∞}(𝕋;𝔏(X))$ such that the associated Fourier multiplier $T_{M}$ is bounded on $l_{p}(X)$.

157-176

wydano
2001

### Twórcy

autor
• Département des Mathématiques, Université de Cergy-Pontoise, 2, avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France