The thin-annuli formulation of the Bochner–Riesz conjecture

Let $\chi$ be a Schwartz function satisfying

$$\mathbf 1_{[-1/4,1/4]}\leq\chi\leq\mathbf 1_{[-1/2,1/2]},$$

and let $S_\tau$ be the Fourier multiplier with symbol $\chi((|\xi|-1)/\tau)$, where $0<\tau<1$. Thin-annuli formulation of the Bochner–Riesz conjecture. If

$$n\left\lvert\frac{1}{p}-\frac{1}{2}\right\rvert<\frac{1}{2},$$

then

$$\lVert S_\tau\rVert_{L^p\mapsto L^p}\lesssim_\epsilon 1,$$

uniformly for $0<\tau<1$. This is presented immediately as an equivalent formulation of the Bochner–Riesz conjecture, expressing uniform $L^p$ bounds for Fourier multipliers supported near a thin annulus. Since the source supplies no resolution evidence, the formulation is recorded as open.