Title: Radial Fourier Multipliers in $R^3$
Abstract: The “Radial Fourier Multiplier Conjecture” in its simplest form states that for radial multipliers supported in a compact subset away from the origin, the multiplier operator $T_m$ is bounded on $L^p(R^d)$ if and only if the kernel $K=\hat{m}$ is in $L^p(R^d)$, for the range $1<p<2d/(d+1)$. This conjecture belongs near the top of the tree of a number of important related conjectures in harmonic analysis, including the Local Smoothing, Bochner-Riesz, Restriction, and Kakeya conjectures. We discuss new partial progress towards the Radial Fourier Multiplier Conjecture in three dimensions. Our method of proof relies on a geometric argument involving sizes of multiple intersections of three-dimensional annuli.