Jan van Neerven

Conical square function estimates in UMD Banach spaces and applications to H∞-functional calculi

We study conical square function estimates for Banach-valued functions and introduce a vector-valued analogue of the Coifman-Meyer-Stein tent spaces. Following recent work of Auscher-McIntosh-Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator A with certain off-diagonal bounds such that A always has a bounded H∞-functional calculus on these spaces. This provides a new way of proving functional calculus of A on the Bochner spaces Lp(ℝn; X) by checking appropriate conical square function estimates and also a conical analogue ofBourgain’s extension of the Littlewood-Paley theory to the UMD-valued context. Even when X = ℂ, our approach gives refined p-dependent versions of known results.

Stochastic maximal Lp-regularity

In this article we prove a maximal Lp-regularity result for stochastic convolutions, which extends Krylov’s basic mixed Lp(Lq)-inequality for the Laplace operator on ℝd to large classes of elliptic operators, both on ℝd and on bounded domains in ℝd with various boundary conditions. Our method of proof is based on McIntosh’s H∞-functional calculus, R-boundedness techniques and sharp Lp(Lq)-square function estimates for stochastic integrals in Lq-spaces. Under an additional invertibility assumption on A, a maximal space–time Lp-regularity result is obtained as well.

Tauberian theorems and stability of solutions of the Cauchy problem

Let f : R+ → X be a bounded, strongly measurable function with values in a Banach space X, and let iE be the singular set of the Laplace transform f in iR. Suppose that E is countable and α ∥∥∫∞ 0 e −(α+iη)uf(s + u) du ∥∥ → 0 uniformly for s ≥ 0, as α↘ 0, for each η in E. It is shown that ∥∥∥∥∫ t 0 e−iμuf(u) du− f(iμ) ∥∥∥∥→ 0 as t → ∞, for each μ in R \ E; in particular, ‖f(t)‖ → 0 if f is uniformly continuous. This result is similar to a Tauberian theorem of Arendt and Batty. It is obtained by applying a result of the authors concerning local stability of bounded semigroups to the translation semigroup on BUC(R+,X), and it implies several results concerning stability of solutions of Cauchy problems.

Pathwise Hölder convergence of the implicit-linear Euler scheme for semi-linear SPDEs with multiplicative noise

In this article we prove pathwise Hölder convergence with optimal rates of the implicit Euler scheme for the abstract stochastic Cauchy problem 1.1

Whitney coverings and the tent spaces T1, q(γ) for the Gaussian measure

\begin{aligned} \left\{ \begin{aligned} dU(t)&= AU(t)\,dt + F(t,U(t))\,dt + G(t,U(t))\,dW_H(t);\quad t\in [0,T],\\ U(0)&=x_0. \end{aligned}\right. \end{aligned}

Stochastic Integration in Banach Spaces - a Survey

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Gradient Estimates and Domain Identification for Analytic Ornstein-Uhlenbeck Operators

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