The Inhomogeneous Wave Equation with L^p Data
The Fourier transform associates a polynomial to each linear differential operator with constant coefficients, and a formal calculation shows that elements in the kernel of such a differential operator have their Fourier transforms supported on the vanishing set of that polynomial. For example, the Helmholtz, wave, and linear Schrödinger operators correspond to polynomials whose vanishing sets are the sphere, cone, and paraboloid respectively; thus, we have a duality between such operators and certain geometric subsets of Euclidean space. When studying partial differential equations with inhomogeneous data, taking the Fourier transform gives a formal solution whose Fourier transform is instead potentially singular on that geometric region. Assuming the inhomogeneous data lies in a suitable L^p space with a vanishing condition on its Fourier transform, Michael Goldberg showed that this Fourier-analytic solution operator is bounded for the Helmholtz equation by exploiting the compactness and curvature properties of the associated geometric region, the sphere. In this talk, I'll discuss how to extend the result to wave operators, for which the associated geometric object is a noncompact cone, and how the geometry of the cone affects the possible estimates for the operator.