The strong type endpoint estimate for the local spherical maximal function

Let $p_d=(d^2+1)\mathrm{/}(d(d-1))$ and $q_d=(d^2+1)\mathrm{/}(d-1)$. We prove that the local spherical maximal operator ${\sup }_{1\le t\le 2}\mathrm{\mid }f\ast d{\sigma }_t\mathrm{\mid }$ is bounded from $L^{p_d}({\mathbb{R}}^d)$ to $L^{q_d}({\mathbb{R}}^d)$ for every $d\ge 2$. Previously only the restricted weak type estimate was known at this endpoint. Together with known results, this completes the characterization of the strong type $L^p\to L^q$ bounds for the local spherical maximal operator. We also prove the maximal estimate $$\parallel \underset{t\in \mathbb{R}}{\sup }\mathrm{\mid }{e}^{it\sqrt{-\mathrm{\Delta }}}f\mathrm{\mid }{\parallel }_{L^{2(d+1)\mathrm{/}(d-1)}({\mathbb{R}}^d)}\lesssim \mathrm{\parallel }f{\mathrm{\parallel }}_{{\dot{H}}^{d\mathrm{/}(d+1)}({\mathbb{R}}^d)}$$ for the wave equation at the critical exponent. Both results are obtained by exploiting the gain in bilinear estimates that comes from the interaction between high and low frequencies.