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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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In this work we investigate the construction of Carleson measures from families of multilinear integral operators applied to tuples of L ∞ and BM O functions. We show that if the family R t of multilinear operators possesses cancellation in each variable, then for BM O functions b 1 ,. .. , b m , the measure |R t (b 1 ,. .. , b m)(x)| 2 dxdt/t is Carleson. However, if the family of multilinear operators has cancellation in all variables combined this result is still valid if b j are L ∞ functions, but it may fail if b j are unbounded BM O functions, as we indicate via an example. As an application of our results we obtain a multilinear quadratic T (1) type theorem and a multilinear version of a quadratic T (b) theorem analogous to those in Semmes [23].

Abstract. In the present work we extend a local Tb theorem for square functions of Christ [3] and Hofmann [17] to the multilinear setting. We also present a new BM O type interpolation result for square functions associated to multilinear operators. These square function bounds are applied to prove a multilinear local Tb theorem for singular integral operators.