Multipliers and Transference on Noncommutative Lp-spaces: and the Relative Haagerup Property

This thesis deals with multipliers and transference on noncommutative Lp-spaces. It falls within the realm of noncommutative harmonic analysis, i.e. harmonic analysis on noncommutative spaces. Such spaces appear in several areas of mathematics, such as non-abelian groups, quantum groups, noncommutative geometry and the general theory of operator...

Multilinear transference of Fourier and Schur multipliers acting on non-commutative L_p-spaces

Let G be a locally compact unimodular group, and let φ be some function of n variables on G. To such a φ, one can associate a multilinear Fourier multiplier, which acts on some n-fold product of the noncommutative L _p-spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on an n...

Martingales and stochastic calculus in Banach spaces

In this thesis we study martingales and stochastic integration of processes with<br/>values in UMD Banach spaces.

Rescaled extrapolation for vector-valued functions

We extend Rubio de Francia's extrapolation theorem for functions valued in UMD Banach function spaces, leading to short proofs of some new and known results. In particular we prove Littlewood-Paley-Rubio de Francia-type estimates and boundedness of variational Carleson operators for Banach function spaces with UMD concavifications.

Harmonic analysis and BMO-spaces of free Araki-Woods factors

We consider semigroup BMO-spaces associated with arbitrary von Neumann algebras and prove interpolation theorems. This extends results by Junge-Mei for the tracial case. We give examples of multipliers on free Araki-Woods algebras and in particular we L_∞ → BMO multipliers. We also provide L_p-bounds for a natural...

Vector-Valued Extensions of Operators Through Multilinear Limited Range Extrapolation

We give an extension of Rubio de Francia’s extrapolation theorem for functions taking values in UMD Banach function spaces to the multilinear limited range setting. In particular we show how boundedness of an m-(sub)linear operator T:Lp1(w1p1)×⋯×Lpm(wmpm)→Lp(wp) for a certain class of Muckenhoupt weights yields an extension of the operator...

Stability theory for semigroups using  (L^p,L^q)  Fourier multipliers

We study polynomial and exponential stability for C₀-semigroups using the recently developed theory of operator-valued (L^p,L^q) Fourier multipliers. We characterize polynomial decay of orbits of a C₀-semigroup in terms of the (L^p,L^q) Fourier multiplier properties of its...

Fourier multipliers and weak differential subordination of martingales in UMD Banach spaces

We introduce the notion of weak differential subordination for martingales, and show that a Banach space X is UMD if and only if for all p ∈ (1, ∞) and all purely discontinuous X-valued martingales M and N such that N is weakly differentially subordinated to M, one has the estimate E || N_∞ ||^p ≤ C_pE|| M<sub>∞<...

Maximal regularity for parabolic equations with measurable dependence on time and applications

The subject of this thesis is the study of maximal L^p-regularity of the Cauchy problem u'(t)+A(t)u(t)=f(t), t∈ (0,T), u(0)=x. <br/>We assume (A(t))_{t∈ (0,T)} to be a family of closed operators on a Banach space X₀, with constant domain D(A(t))=X₁ for every t∈ (0,T). Maximal L^p-regularity means that...