Lutz Weis

Operator–valued Fourier multiplier theorems and maximal

Abstract. We prove a Mihlin–type multiplier theorem for operator–valued multiplier functions on UMD–spaces. The essential assumption is R–boundedness of the multiplier function. As an application we give a characterization of maximal

L_p

L_p

-regularity

–regularity for the generator of an analytic semigroup

The H ∞ −calculus and sums of closed operators

T_t

Stochastic integration in UMD Banach spaces

in terms of the R–boundedness of the resolvent of A or the semigroup

Maximal Lp-regularity for Parabolic Equations, Fourier Multiplier Theorems and \(H^\infty\)-functional Calculus

T_t

Functional analytic methods for evolution equations

.

Operator-valued Fourier multiplier theorems on Lp(X) and geometry of Banach spaces

We develop a very general operator-valued functional calcu- lus for operators with an H 1 −calculus. We then apply this to the joint functional calculus of two commuting sectorial operators when one has an H 1 calculus. Using this we prove theorem of Dore-Venni type on sums of commuting sectorial operators and apply our results to the problem of Lp−maximal regularity. Our main assumption is the R-boundedness of certain sets of operators, and therefore methods from the geometry of Ba- nach spaces are essential here. In the final section we exploit the special Banach space structure of L1−spaces and C(K)−spaces, to obtain some more detailed results in this setting.

Itô's formula in UMD Banach spaces and regularity of solutions of the Zakai equation

In this paper we develop a stochastic integration theory for processes with values in a quasi-Banach space. The integrator is a cylindrical Brownian motion. The main results give sufficient conditions for stochastic integrability. They are natural extensions of known results in the Banach space setting. We apply our main results to the stochastic heat equation where the forcing terms are assumed to have Besov regularity in the space variable with integrability exponent

Stochastic maximal Lp-regularity

p\in (0,1]