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Perturbation of analytic operators and temporal regularity of discrete heat kernels

Sönke Blunck — 2000

Colloquium Mathematicae

In analogy to the analyticity condition A e t A C t - 1 , t > 0, for a continuous time semigroup ( e t A ) t 0 , a bounded operator T is called analytic if the discrete time semigroup ( T n ) n satisfies ( T - I ) T n C n - 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 ω n , n ∈ ℕ. As an application we generalize...

Maximal regularity of discrete and continuous time evolution equations

Sönke Blunck — 2001

Studia Mathematica

We consider the maximal regularity problem for the discrete time evolution equation u n + 1 - T u = f 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 ( T ) n 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...

A Hörmander-type spectral multiplier theorem for operators without heat kernel

Sönke Blunck — 2003

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Hörmander’s famous Fourier multiplier theorem ensures the L p -boundedness of F ( - Δ D ) whenever F ( s ) for some s > D 2 , where we denote by ( s ) the set of functions satisfying the Hörmander condition for s derivatives. Spectral multiplier theorems are extensions of this result to more general operators A 0 and yield the L p -boundedness of F ( A ) provided F ( s ) for some s sufficiently large. The harmonic oscillator A = - Δ + x 2 shows that in general s > D 2 is not sufficient even if A has a heat kernel satisfying gaussian estimates. In this paper,...

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