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.

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<...