Consider the family of operators $T_\delta$, $\delta \geq 0$, defined on $\mathbb{R}^n$ by

$ \widehat{T_\delta f}(\xi) = (1-|\xi|^2)_+^\delta \widehat{f}(\xi). $

($(1-|\xi|^2)_+^\delta$ are known as *Bochner-Riesz multipliers*.) We are interested in the $L^p$ boundedness of $T_\delta$. The case $\delta = 0$ has been solved since 1971, once Fefferman provided a proof that $L^p$ boundedness fails in dimension $n \geq 2$ when $p \neq 2$. For general $\delta > 0$, a Theorem due to Herz shows that a necessary condition for boundedness is that

$ |\frac{1}{p}-\frac{1}{2}| < \frac{2\delta + 1}{2n}. $

It's thus natural for one to conjecture that this is also a sufficient condition.

Here's what I think I know about current progress, from jotted down notes:

- Holds for $n \leq 2$
- Holds for $p = 2$
- $T_\delta$ is bounded when $\delta > \frac{n-1}{2}$ (by Young's inequality)
- $T_\delta$ is bounded when $|\frac{1}{p}-\frac{1}{2}| < \frac{\delta}{n-1}$
- Holds if $\delta > \frac{n-1}{2(n+1)}$ (Can't remember the reference)
- Holds for $n \geq 3$ when $p \geq \frac{2(n+2)}{n}$ or $p \leq \frac{2(n+2)}{n+4}$ (Found in Tao's
*Recent progress on the Restriction conjecture*)

Question: What is the most recent progress on this conjecture? I'm curious about the general case and also specific values of $n$, such as $n = 3$.