Almost everywhere convergence of inverse Fourier transforms

Abstract:

We show that if $\log (2-\Delta )f\in L^2({\mathbb R}^d)$, then the inverse Fourier transform of $f$ converges almost everywhere. Here the partial integrals in the Fourier inversion formula come from dilates of a closed bounded neighbourhood of the origin which is star shaped with respect to $0$. Our proof is based on a simple application of the Rademacher-Menshov Theorem. In the special case of spherical partial integrals, the theorem was proved by Carbery and Soria. We obtain some partial results when $\sqrt {\log (2-\Delta )}f\in L^2({\mathbb R}^d)$ and $\log \log (4-\Delta )f\in L^2({\mathbb R}^d)$. We also consider sequential convergence for general elements of $L^2({\mathbb R}^d)$.