The Bochner-Riesz conjecture in three dimensions (strict-order range)

Theorem 1.1: for every $\delta>0$, $\|T_\delta f\|_{L^3(\mathbb R^3)}\le C_\delta\|f\|_{L^3(\mathbb R^3)}$. By complex interpolation and duality this gives boundedness on $L^p(\mathbb R^3)$ for all $1\le p\le\infty$ and $\delta>\max\{3|1/p-1/2|-1/2,0\}$, the full strict-order range in three dimensions, hence $L^p$ convergence of Bochner-Riesz means for $3/2\le p\le3$ at every positive order. Via Tao's implication it yields the spherical extension estimate $\|Eg\|_{L^p(\mathbb R^3)}\le C_p\|g\|_{L^\infty(S^2)}$ for $p>3$. Not shown: the endpoint order, the maximal Bochner-Riesz operator, or any dimension $n\ge4$; it is a partial result on the $n$-dimensional conjecture.

Theorem 1.1: for every $\delta>0$, $\|T_\delta f\|_{L^3(\mathbb R^3)}\le C_\delta\|f\|_{L^3(\mathbb R^3)}$. By complex interpolation and duality this gives boundedness on $L^p(\mathbb R^3)$ for all $1\le p\le\infty$ and $\delta>\max\{3|1/p-1/2|-1/2,0\}$, the full strict-order range in three dimensions, hence $L^p$ convergence of Bochner-Riesz means for $3/2\le p\le3$ at every positive order. Via Tao's implication it yields the spherical extension estimate $\|Eg\|_{L^p(\mathbb R^3)}\le C_p\|g\|_{L^\infty(S^2)}$ for $p>3$. Not shown: the endpoint order, the maximal Bochner-Riesz operator, or any dimension $n\ge4$; it is a partial result on the $n$-dimensional conjecture.