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We obtain boundedness for the bilinear spherical maximal function in a range of exponents that includes the Banach triangle and a range of L p with p < 1. We also obtain counterexamples that are asymptotically optimal with our positive results on certain indices as the dimension tends to infinity.

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Abstract. In this paper we obtain an upper bound for the L 2 norm for maximal operators associated to arbitrary finite sets of directions in R 2 .

Abstract. We prove a weak type estimate for operators of the form f → s∈S f, ϕs ϕs for certain collections of Schwartz functions {ϕs} s∈S . This extends some of the orthogonality issues involved in the study of the bilinear Hilbert transform by Lacey and Thiele.