Vector-valued harmonic analysis with applications to SPDE

In this dissertation we develop vector-valued harmonic analysis methods. Particular emphasis is put on the study of stochastic singular integral operators, which arise naturally in the study of SPDE.

Singular stochastic integral operators

We introduce Calderón-Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove L ^p-extrapolation results under a Hörmander condition on the kernel. Sparse domination and sharp weighted bounds are obtained under a Dini condition on the kernel, leading to a stochastic version of the...

On Pointwise ℓ^r -Sparse Domination in a Space of Homogeneous Type

We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the usual ℓ¹-sum in the sparse operator is replaced by an ℓ<sup>r<...

The ℓ ^s-boundedness of a family of integral operators on UMD banach function spaces

We prove the ℓ^s-boundedness of a family of integral operators with an operator-valued kernel on UMD Banach function spaces. This generalizes and simplifies earlier work by Gallarati, Veraar and the author, where the ℓ^s-boundedness of this family of integral operators was shown on Lebesgue spaces. The proof is based on a...

On the ℓs-boundedness of a family of integral operators

In this paper we prove an ℓs-boundedness result for integral operators with operator-valued kernels. The proofs are based on extrapolation techniques with weights due to Rubio de Francia. The results will be applied by the first and third author in a subsequent paper where a new approach to maximal Lp-regularity for parabolic problems with time...

Maximal functions, factorization, and the R-boundedness of integral operators

We generalize a recent result on the ℓs-boundedness of a family of integral operators from the weighted vector-valued Lp(Rd,w;Lq(Ω)) setting to the weighted vector-valued Lp(Rd,w;X) setting for a large class of Banach function spaces X. For this we first extensively study the first part of an unpublished manuscript on the factorization of ℓ2...