Toeplitz products and two-weight inequalities on spaces of vector-valued functions

QA Mathematics

Kerr, Robert ; Pott, Sandra

University:University of Glasgow

Department:School of Mathematics and Statistics

关键词: Toeplitz Operators, Hilbert Transform, Bergman Space, Vector-Valued Function Spaces;   

Others  :  http://theses.gla.ac.uk/2469/1/2011kerrphd.pdf

来源: University of Glasgow

【 摘 要 】

This thesis is concerned with operators on certain vector-valued function spaces. Namely, Bergman spaces of \mathbb{C}^n$-valued functions and L^2(\mathbb{R},\mathbb{C}^n,V)$, where $V$ is a matrix weight. We will study products of Toeplitz operators on the vector Bergman space $L^2_a(\mathbb{C}^n)$. We also study various operators, including the dyadic shift and the Hilbert transform, between $L^2(\mathbb{R},\mathbb{C}^n,V)$ and $L^2(\mathbb{R},\mathbb{C}^n,U)$. These function spaces are generalizations of normed vector spaces of functions which take values in $\mathbb{C}$.The thesis is split into two distinct areas of function space theory: analytic function spaces and harmonic analysis. There is, however, a common theme of matrix weights, particularly the reverse Hölder condition on matrix weights and a generalization of the $A_p$ conditions on matrix weights for $p=2$ and $p=\infty$.