In dimension n = 1, we obtain $L^{p_1}(\mathbb R) \times\dots\times L^{p_m}(\mathbb R)$ to $L^p(\mathbb R)$ 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.

An iteration technique for characterizing boundedness of certain types of multilinear operators is presented, reducing the problem to a corresponding linear-operator case. The method gives a simple proof of a characterization of validity of the weighted bilinear Hardy inequality

for all non-negative f, g on (a, b), for 1 < p 1, p 2, q < ∞. More equivalent characterizing conditions are presented.

The same technique is applied to various further problems, in particular those involving multilinear integral operators of Hardy type.

We characterize bounded mean oscillation in terms of the boundedness of commutators of various bilinear singular integral operators with pointwise multiplication. In particular, we study commutators of a wide class of bilinear operators of convolution type, including bilinear Calderón–Zygmund operators and the bilinear fractional integral operators.

We show that if the general real method ${{\left( \cdot \,,\,\cdot \right)}_{\Gamma }}$ preserves the Banach-algebra structure, then a bilinear interpolation theorem holds for ${{\left( \cdot \,,\,\cdot \right)}_{\Gamma }}$.