Browsing by Author "Tamir, Ella"

In this thesis, the Burkholder functional is considered from two different points of view, through the Beurling transform and Calculus of Variations. We aim to connect open problems of the two fields by the convexity properties of the functional. First we define quasiconformal maps, and examine their basic properties, giving the Beltrami equations as their characterizing PDE. We present the complex differential operator known as the Beurling transform, mapping the complex derivative with respect to the conjugate to the usual complex derivative. The exact L^p-bound of the potential is conjectured by Iwaniec to be max { p-1, 1/(p-1)}. Although the exact operator norm is yet unknown, the L^p-boundedness of the Beurling transform combined with other results of quasiconformal maps implies the measurable Riemann mapping theorem, which generalizes the Riemann mapping theorem of conformal maps to a result on existence of solutions to the Beltrami equation. We go through the classical setting of Calculus of Variations, transforming the problem of finding a solution for a PDE to finding a minimizer for a functional, denoted by I. A result on the implications of weak lower semicontinuity of I and a coercivity condition then ensures the existence of weak solutions to the original PDE. We are able to show a connection between a weaker convexity property known as quasiconvexity and weak lower semicontinuity, establishing the significance of the notion of convexity in Calculus of Variations. The second open problem the thesis presents is Morrey's problem, which asks whether another weaker convexity property called rank-one convexity is equivalent with quasiconvexity in the plane. Finally, we connect the two topics by presenting the Burkholder functional B_{p} , which is rank-one concave. Through a lower bound of the functional, we prove that the quasiconcavity of B_{p} would imply the Iwaniec conjecture. Therefore the functional either solves Morrey's problem, or gives the exact L^p-bound of the Beurling transform. We construct an original counterexample, which shows that -B_p is not weakly lower semicontinuous in W^{1,p}, but notice that its quasiconvexity would imply weak lower semicontinuity in W^{1,q} for 1 < q < p. In the last chapter, we give a recent result on the convexity properties of B_{p}(Df), namely that it satisfies a quasiconcavity property around the identity map, when sufficient additional assumptions are made on the function f and its quasiconformality constant.