The strong type endpoint estimate for the local spherical maximal function

Let $p_d=(d^2+1)/(d(d-1))$ and $q_d=(d^2+1)/(d-1)$. We prove that the local spherical maximal operator $\sup_{1\leq t\leq 2}|f*dσ_t|$ is bounded from $L^{p_d}(\mathbb{R}^d)$ to $L^{q_d}(\mathbb{R}^d)$ for every $d\geq 2$. Previously only the restricted weak type estimate was known at this endpoint.