Vector-valued Laplace transforms and Cauchy problems 1 Wolfgang Arendt ... [et al.]. p. cm. -(Monographs in mathematics ; voi. 96) lncludes bibliographical references and index.

Vector-valued Laplace transforms and Cauchy problems 1 Wolfgang Arendt ... [et al.]. p. cm. -(Monographs in mathematics ; voi. 96) lncludes bibliographical references and index.

Given a closed linear operator on a UMD-space, we characterize maximal regularity of the non-homogeneous problem u + Au = f with periodic boundary conditions in terms of R-boundedness of the resolvent. Here A is not necessarily generator of a C 0 -semigroup. As main tool we prove an operator-valued discrete multiplier theorem. We also characterize maximal regularity of the second order problem for periodic, Dirichlet and Neumann boundary conditions. Classical theorems on L p -multipliers are no longer valid for operator-valued functions unless the underlying space is isomorphic to a Hilbert space (see Sect. 1 for precise statements of this result). However, recent work of Clément-de Pagter-Sukochev-Witvliet [CPSW], Weis [W1], [W2] and Clément-Prüss [CP] show that the right notion in this context is R-boundedness of sets of operators. This condition is strictly stronger than boundedness in operator norm (besides in the Hilbert space) and may be defined with help of the Rademacher functions. And indeed, Weis [W1] showed that Mikhlin's classical theorem on Fourier multipliers on L p (R; X) holds if boundedness is replaced by R-boundedness (see [CP] for another proof based on results of Clément-de Pagter-Sukochev and Witvliet [CPSW]).

The main result is the following stability theorem: Let T = ( T ( t ) ) t ⩾ 0 \mathcal {T} = {(T(t))_{t \geqslant 0}} be a bounded C 0 {C_0} -semigroup on a reflexive space X X . Denote by A A the generator of T \mathcal {T} and by σ ( A ) \sigma (A) the spectrum of A A . If σ ( A ) ∩ i R \sigma (A) \cap i{\mathbf {R}} is countable and no eigenvalue of A A lies on the imaginary axis, then lim t → ∞ T ( t ) x = 0 {\lim _{t \to \infty }}T(t)x = 0 for all x ∈ X x \in X .

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Let 1 p, q ∞, s ∈ R and let X be a Banach space. We show that the analogue of Marcinkiewicz's Fourier multiplier theorem on L p (T) holds for the Besov space B s p,q (T; X) if and only if 1 < p < ∞ and X is a UMD-space. Introducing stronger conditions we obtain a periodic Fourier multiplier theorem which is valid without restriction on the indices or the space (which is analogous to Amann's result (Math. Nachr. 186 (1997), 5-56) on the real line). It is used to characterize maximal regularity of periodic Cauchy problems.

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Vector-valued Laplace transforms and Cauchy problems 1 Wolfgang Arendt ... [et al.]. p. cm. -(Monographs in mathematics ; voi. 96) lncludes bibliographical references and index.