Bochner-Riesz Summarbility

Bochner-Riesz Summarbility

For $\alpha >0$, $R>0$, $f\in L^p({\mathbb{R}}^n)$ we define $S_R^{\alpha }f(x)={\int }_{|\xi |\le R}{\left(1-\frac{|\xi {|}^2}{R^2}\right)}^{\alpha }\hat{f}(\xi )e^{2\pi ix\cdot \xi }d\xi$

$n\ge 2$, $1\le p\le \frac{2n}{n+1}$,$\Vert S_{1}f{\Vert }_{L^p}/\lesssim \Vert f{\Vert }_{L^p}$(Herz (1954)) and for $n\ge 2$, $\frac{2n}{n+1}<p<2$, $S_{R}f\nrightarrow f$ in $L^p({\mathbb{R}}^n)$(Fefferman (1971) and the theorem called Ball multiplier theorem).

The reason we want to define Bochner-Riesz Summarbility is form the fact for

Conjecture

If $n\ge 2$, $\alpha >0$, then the following inequalities holds for $p$ such that $\frac{2n}{n+1+2\alpha }<p<\frac{2n}{n-1-2\alpha }$ $\Vert S_R^{\alpha }f{\Vert }_{L^p}\le \Vert f{\Vert }_{L^p}.$ Equivalently let ${\phi }_{\delta }(\xi )$ be a smooth function supported on a $\delta$-neighbourhood of $S^{n-1}$ normalized s.t. ${\phi }_{\delta }(\xi )\sim 1$ near $|\xi |=1$. Then the conjecture is ${\Vert {\int }_{{\mathbb{R}}^n}{\phi }_{\delta }(\xi )\hat{f}(\xi )e^{2\pi ix\cdot \xi }d\xi \Vert }_{L^p({\mathbb{R}}^n)}\lesssim {\delta }^{-(n-1)((1/2-1/p)-1/p)}\Vert f{\Vert }_{L^p({\mathbb{R}}^n)}\forall p>\frac{2n}{n-1}$