Martingales, the Beurling-Ahlfors transform, and lower bounds for ground state eigenfunctions

We study the n-dimensional Beurling-Ahlfors transform S via probabilistic methods. In particular, we estimate the $L\sp {p}$ operator norms by representing the operator as a martingale transform and using a martingale inequality. This representation is not unique, however, and we also analyze the various choices. The methods presented here provide new estimates and, in particular, we show that when S is restricted to k-forms, its norm is independent of dimension. In Chapter 7, we discuss some of the analytical issues raised by this approach. The remaining part of the thesis discusses sharp lower bounds for the ground state eigenfunction on certain horn-shaped and cusped domains