Resumen de A Microlocal F. and M. Riesz Theorem with Applications

R. G. M. Brummelhuis

• Consider, by way of example, the following F. and M. Riesz theorem for Rn: Let µ be a finite measure on Rn whose Fourier transform µ* is supported in a closed convex cone which is proper, that is, which contains no entire line. Then µ is absolutely continuous (cf. Stein and Weiss [SW]). Here, as in the sequel, absolutely continuous means with respect to Lebesque measure. In this theorem one can replace the condition on the support of µ* by a similar condition on the wave front set WF(µ) of µ, while keeping the same conclusion. The resulting microlocal F. and M. Riesz theorem can be applied with great flexibility to derive F. and M. Riesz theorems for measures on Lie groups, measures satisfying partial differential equations, etc. This is, essentially, the program of this paper.