A proof of the Bochner-Riesz conjecture

Abstract

For $f\in {\frak S}({\Bbb R}^d)$, we consider the Bochner-Riesz operator ${\frak R}^{\delta}$ of index $\delta>0$ defined by $$\hat {{\frak R}^{\delta}f}(\xi)=(1-|\xi|^2)^{\delta}_+ \hat f (\xi).$$ Then we prove the Bochner-Riesz conjecture which states that if $\delta>\max\{d|1/p-1/2|-1/2,0\}$ and $p>1$ then ${\frak R}^{\delta}$ is a bounded operator from $L^p({\Bbb R}^d)$ into $L^p({\Bbb R}^d)$; moreover, if $\delta(p)=d(1/p-1/2)-1/2$ and $1<p<2d/(d+1)$, then ${\frak R}^{\delta(p)}$ is a bounded operator from $L^p({\Bbb R}^d)$ into $L^{p,\infty}({\Bbb R}^d)$.