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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.

We discuss L p (R n ) boundedness for Fourier multiplier operators that satisfy the hypotheses of the Hörmander multiplier theorem in terms of an optimal condition that relates the distance | 1 p − 1 2 | to the smoothness s of the associated multiplier measured in some Sobolev norm. We provide new counterexamples to justify the optimality of the condition | 1 p − 1 2 | < s n and we discuss the endpoint case | 1 p