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.

Sharp Estimates and Extrapolation for Multilinear Weight Classes

The subject of this thesis is the study of the multilinear Muckenhoupt weight classes and the quantitative boundedness of operators with respect to these weights in both the scalar-valued and the vector-valued setting. This includes the study of multisublinear Hardy-Littlewood maximal operators, sparse forms, and multilinear Rubio de Francia...

Maximal Operators Defined by Rearrangement Invariant Banach Function Spaces

In this thesis we study the boundedness of a generalization of the Hardy-Littlewood maximal operator, involving rearrangement invariant Banach function space and indices of the spaces.<br/> We first consider a classical proof of boundedness of the Hardy-Littlewood maximal operator on rearrangement invariant Banach function spaces. After...

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<...

Sparse domination for the lattice Hardy–Littlewood maximal operator

We study the domination of the lattice Hardy–Littlewood maximal operator by sparse operators in the setting of general Banach lattices. We prove that the admissible exponents of the dominating sparse operator are determined by the q-convexity of the Banach lattice.

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...