Lebesgue bounds for multilinear spherical and lacunary maximal averages

We establish $L^{p_1}(\mathbb R^d) \times \cdots \times L^{p_n}(\mathbb R^d) \rightarrow L^r(\mathbb R^d)$ bounds for spherical averaging operators $\mathcal A^n$ in dimensions $d \geq 2$ for indices $1\le p_1,\dots , p_n\le \infty$ 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 $L^1 \times \cdots \times L^1 \rightarrow L^1$. We also obtain similar estimates for lacunary maximal spherical averages in the largest possible open region of indices.