Bochner–Riesz means

Summary

The Bochner–Riesz means of a function $f$ on $\R^n$ are the operators $S^\delta_R f$ that multiply the Fourier transform of $f$ by $(1 - \abs{\xi}^2/R^2)_+^\delta$. They are a smoothed version of the spherical partial sums of the Fourier integral, with smoothing order $\delta \ge 0$. The central question is for which $\delta$ and $p$ they are bounded on $L^p(\R^n)$, which decides whether $S^\delta_R f \to f$ in $L^p$ as $R \to \infty$. This question is tied to the restriction and Kakeya problems.