Title: Functions whose Fourier transform vanishes on the sphere
Abstract: For the Helmholtz equation $(-\Delta - 1)u = f$ in $R^n$, the Fredholm condition for solvability in $L^2$ is that the Fourier transform of $f$ must vanish on the unit sphere. Agmon showed this is a sufficient condition if $(1+|x|)f \in L^2$, with significant implications to the spectral theory of Schrödinger operators with a pointwise bounded short-range potential. Ionescu and Schlag expanded the results further by obtaining mapping estimates for the solution operator $(-\Delta - 1)^{-1}$ on a broad family of weighted spaces with local behavior in $W^{k,p}$.
We show that an unweighted version is also possible, with the Fredholm condition being sufficient provided $f \in L^{\frac{2n+2}{n+5}}(R^n)$, $n \geq 3$. This can be interpreted as an $L^p \to L^2$ bound for a Bochner-Riesz multiplier of order $-1$. Similar methods yield a family of $L^p \to L^q$ estimates for Bochner-Riesz multipliers of other orders, some of them sharp, which are valid whenever $\hat{f}$ vanishes on the unit sphere.