A proof of the Bochner-Riesz conjecture

For $f\in \mathfrak{S}({\mathbb{R}}^d)$, we consider the Bochner-Riesz operator ${\mathfrak{R}}^{\delta }$ of index $\delta >0$ defined by $$\widehat{{\mathfrak{R}}^{\delta }f}(\xi )=(1-\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}_{+}^{\delta }\widehat{f}(\xi )\mathrm{.}$$ Then we prove the Bochner-Riesz conjecture which states that if $\delta >\max \{d\mathrm{\mid }1\mathrm{/}p-1\mathrm{/}2\mathrm{\mid }-1\mathrm{/}2,0\}$ and $p>1$ then ${\mathfrak{R}}^{\delta }$ 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${\frak R}^{\delta(p)}$isaboundedoperatorfrom$L^p({\Bbb R}^d)$into$L^{p,\infty}({\Bbb R}^d)$.