Improved resolvent estimates for constant-coefficient elliptic operators in three dimensions

Abstract

We prove new L p -L q -estimates for solutions to elliptic differential operators with constant coefficients in R 3 . We use the estimates for the decay of the Fourier transform of particular surfaces in R 3 with vanishing Gaussian curvature due to Erdős-Salmhofer to derive new Fourier restrictionextension estimates. These allow for constructing distributional solutions in L q (R 3 ) for L p -data via limiting absorption by well-known means.