Maximal decay inequalities for trilinear oscillatory integrals of convolution type

In this paper we prove sharp L∞L∞–L∞L∞–L∞L∞ decay for certain trilinear oscillatory integral forms of convolution type on R2R2. These estimates imply earlier L2L2–L2L2–L2L2 results obtained by the second author as well as corresponding sharp, stable sublevel set estimates of the form studied by Christ [3] and Christ, Li, Tao, and Thiele [5]. New connections to the multilinear results of Phong, Stein, and Sturm [13] are also considered.