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Maximal regularity of discrete and continuous time evolution equations

100%

Blunck S.

Studia Mathematica

2001

tom 146

nr 2

157-176

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 $l_{p}(X)$.

Perturbation of analytic operators and temporal regularity of discrete heat kernels

100%

Blunck S.

Colloquium Mathematicum

2000

tom 86

nr 2

189-201

In analogy to the analyticity condition $∥ Ae^{tA}∥ ≤ Ct^{-1}$, t > 0, for a continuous time semigroup $(e^{tA})_{t ≥ 0}$, a bounded operator T is called analytic if the discrete time semigroup $(T^n)_{n ∈ ℕ}$ satisfies $∥ (T-I)T^{n}∥ ≤ Cn^{-1}$, n ∈ ℕ. We generalize O. Nevanlinna's characterization of powerbounded and analytic operators T to the following perturbation result: if S is a perturbation of T such that $∥ R(λ_0,T)-R(λ_0,S)∥$ is small enough for some $λ_{0} ∈ ϱ(T) ∩ ϱ(S)$, then the type $ω$ of the semigroup $(e^{t(S-I)})$ also controls the analyticity of S in the sense that $∥(S-I)S^{n}∥ ≤ C(ω+n^{-1})e^{ωn}$, n ∈ ℕ. As an application we generalize and give a simple proof of a result by M. Christ on the temporal regularity of random walks T on graphs of polynomial volume growth. On arbitrary spaces Ω of at most exponential volume growth we obtain this regularity for any powerbounded and analytic operator T on $L_{2}(Ω)$ with a heat kernel satisfying Gaussian upper bounds.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Studia Mathematica

2 Blunck S.

1 2001

1 2000

Wyniki wyszukiwania

Maximal regularity of discrete and continuous time evolution equations

Perturbation of analytic operators and temporal regularity of discrete heat kernels