Decay Estimates of High Dimensional Adjoint Radon Transforms

In this paper we prove an optimal $L^2-L^{2d}$ decay estimate of the adjoint Radon transform of compactly supported data in $d$-dimensional space via a geometric method. A similar problem in dimension $3$ has be considered in the author's previous work. This work deals with all higher dimensional case $d\geq 4$. As an application we give the decay of Strichartz norms of $5$-dimensional non-radiative free waves. The general idea is similar to the lower dimensional case but we introduce a new method to prove the corresponding geometric inequality because the old method becomes too complicated in higher dimensions.