Thesis Defense: "Weighted Inequalities for Three Operators"

This thesis deals with weighted inequalities for three operators. In the most general formulation, we are given a family of operators $\{T_\alpha\}_{\alpha\in\mathcal{A}}$ and two measure spaces $(\mu,\mathcal{X})$ and $(\nu,\mathcal{Y})$ we would like to estimate the norm $\norm{T_\alpha:L^p(\mu,\mathcal{X})\to L^q(\nu,\mathcal{Y})}$ in terms of ``geometric'' properties of the measure $\mu$ and $\nu$. In the case that $\mu\neq\nu$, these are called ``two--weight inequalities'' and in the case $\mu=\nu$ these are called ``one--weight inequalities''.

 

In this thesis we consider sharp one--weight inequalities for the Bergman projection, two--weight inequalities for the fractional integral operator and two--weight inequalities for the commutator of a fractional integral operator and a function.