Lebesgue bounds for multilinear spherical and lacunary maximal averages

We establish $L^{p_1}({\mathbb{R}}^d)\times \cdots \times L^{p_n}({\mathbb{R}}^d)\to L^r({\mathbb{R}}^d)$ bounds for spherical averaging operators ${\mathcal{A}}^n$ in dimensions $d\ge 2$ for indices $1\le p_1,\dots ,p_n\le \mathrm{\infty }$ and $\frac{1}{p_1}+\cdots +\frac{1}{p_n}=\frac{1}{r}$. We obtain this result by first showing that ${\mathcal{A}}^n$ maps $L^1\times \cdots \times L^1\to L^1$. We also obtain similar estimates for lacunary maximal spherical averages in the largest possible open region of indices.