Multilinear singular integrals. (Q1861101)

In this survey the author describes some recent results (and new proofs of existing results) concerning multilinear singular integrals such as the bilinear Hilbert transform, as well as related objects such as the Carleson maximal operator. In particular, the method of phase space decomposition (using truncations in both frequency space and physical space, not necessarily centered around the frequency origin, and thus generalizing Littlewood-Paley theory) to pass from these operators to a dyadically decomposed model operator is emphasized as playing an important role for the multilinear \(L^p\) theory of operators which obey a certain type of modulation symmetry. Further topics discussed include the change in phase space decompositions necessary to handle the bilinear Hilbert transform. The treatment of the Carleson operator is done in some detail, but most of the other operators are discussed in heuristic terms only. Finally, a discussion of the relationship between these operators and the eigenfunctions of one-dimensional Schrödinger operators is discussed.