This project concerns fundamental mathematical questions in harmonic analysis. Among the principal research directions will be the study of multidimensional convolutions with measures supported on curves. The object is to expand on E. Stein's observation that the convolution of singular measures with functions in Lebesgue spaces may result in functions in related Lebesgue spaces. This research will focus on those measures concentrated on smooth curves. The entire question of convolution along cuves has its roots in the theory of differentiation of integrals, which in turn is a formulation of integral solutions of partial differential equations. A second line of investigation will focus on obtaining sharp Sobolev inequalities for non-elliptic, constant coefficient differential operators which reflect the symmetry of certain orthogonal groups. Sobolev-type inequalities are crucial in obtaining existence results for partial differential operators. They form a class of inequalities in which the norm of a function is dominated by the norm (possibly a different norm) of its gradient. Two particular cases will be addressed in this work, the classical wave operator and the Klein-Gordon operator.
