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 $n = 3$.


The results stated in your post are improved in the recent work of Bourgain and Guth, in dimensions 5 and higher. The numerology is the same as for the restriction problem for the sphere (see the statement of theorem 1 in that paper). In the case of the restriction problem for the sphere Bourgain and Guth improved the 3 dimensional result to $p > 3 + 3/10=3.3$, which is slightly better than the $p>10/3=3.333...$ from Tao's bilinear estimates (which, combined with Lee's work, gives the corresponding result for Bochner-Riesz.) This is the "same" $p>10/3$ you mention in your post. It seems likely that Bourgain and Guth's argument will give a similar improvement for the 3 dimensional Bochner-Riesz problem, but they do not work this out in the paper. They do write (page 5) "Thus in principle, one could expect the proof of Theorem 2 to carry over and lead to the validity of the Bochner-Riesz conjecture for $max(p, p′) ≥ 3 +3/10$ , if n = 3. We do not pursue the details of this matter here."


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