A proof of the Bochner-Riesz conjecture

Abstract

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