# The Bochner-Riesz problem: an old approach revisited

## Location

Abstract: The Bochner-Riesz conjecture arose from the classical problem of convergence of Fourier series and Fourier integrals in higher dimensions. One may ask whether the integrals \[ \int_{B(0,R)} \widehat{f}(\xi) e^{2\pi i x\cdot \xi } d\xi \] converge in $L^p$ as $R\rightarrow \infty$. This was disproved by Fefferman for $p\neq 2$. The Bochner-Riesz problem asks whether the above convergence holds if we replace $\chi_{B(0,R)}$ by a smoother family of functions (multipliers).

Tao proved in 1999 that the Bochner-Riesz conjecture implies the restriction conjecture. We show that the current techniques developed to study the restriction problem apply equally well to the Bochner-Riesz problem. This is joint work with Guo, Oh, Wu and Zhang.