In dimensions n [greater than or equal to] 2 we obtain Lp1(Rn) x ... x Lpm(Rn) to Lp(Rn) boundedness for the multilinear spherical maximal function in the largest possible open set of indices and we provide counterexamples that indicate the optimality of our results. Moreover, we obtain weak type and Lorentz space estimates as well as counterexamples in the endpoint cases. We also study a family of maximal operators that provides a continuous link connecting the Hardy-Littlewood maximal function to the spherical maximal function. Our theorems are proved in the multilinear setting but may contain new results even in the linear case. For this family of operators we obtain bounds between Lebesgue spaces in the optimal range of exponents. Moreover, we provide multidimensional versions of the Kakeya, Nikodym, and Besicovitch constructions associated with a fixed rectifiable set. These yield counterexamples indicating that maximal operators given by translations of spherical averages are unbounded on all Lp(Rn) for p [less than] [infinity]. For lower-dimensional sets of translations, we obtain Lp boundedness for the associated maximally translated spherical averages and for the uncentered spherical maximal functions for a certain range of p that depends on the upper Minkowski dimension of the set of translations. This implies that the Nikodym sets associated with spheres have full Hausdorff dimension.Includes bibliographical references (pages 91-96)
