Multilinear spherical maximal function

In dimensions n ≥ 2 n\ge 2 we obtain alttext="upper L Superscript p 1 Baseline left-parenthesis double-struck upper R n right-parenthesis times midline-horizontal-ellipsis Super Subscript m right-parenthesis"> L p 1 ( R stretchy="false">) ×<!-- × <mml:mo>⋯<!-- ⋯ <mml:mi>m encoding="application/x-tex">L^{p_1}(\mathbb R^n) \times \dots L^{p_m}(\mathbb R^n) to encoding="application/x-tex">L^p(\mathbb boundedness for the multilinear spherical maximal function in largest possible open set of indices and provide counterexamples that indicate optimality our results. Moreover, weak type Lorentz space estimates as well endpoint cases.