Tao's maximal Bochner–Riesz conjecture

For $n\geq 1$, let $T_t^\lambda$ be the Bochner–Riesz means and define the maximal Bochner–Riesz operator by

$$T^\lambda_\ast f(x):=\sup_{t>0}|T_t^\lambda f(x)|.$$

Tao's conjecture. For any $1\leq p<2$, the operator $T^\lambda_\ast$ extends boundedly on $L^p(\mathbb R^n)$ whenever

$$\lambda>\max\left\{0,\frac{2n-1}{2p}-\frac n2\right\},$$

that is,

$$\big\|T^\lambda_\ast f\big\|_{L^p(\mathbb R^n)}\leq C\|f\|_{L^p(\mathbb R^n)}.$$

This conjecture gives the expected maximal-operator criterion for almost-everywhere convergence of Bochner–Riesz means in the lower range $p<2$; its status is not resolved by the supplied text.