UC Berkeley

For trilinear functionals $\int_B e^{i\lambda \phi(x)}\prod_{j = 1}^3 f_j(x_j)dx$, where $B$ is a ball, $f_j$ are $L^\infty$ functions and $\phi$ is a real-valued analytic function satisfying certain auxiliary and nondegeneracy conditions, we prove a bound in terms of a negative power of the large parameter $\lambda$ and Lebesgue norms of $f_j$. The main goal is to relax the auxiliary hypothesis in a previous result. Inequalities are also established for certain multilinear oscillatory integrals in odd dimensions. A key component of the proof is the estimate of related sublevel sets.

Lebesgue space bounds $L^{p_1} (\reals) \times L^{p_2} (\reals) \rightarrow L^q(\reals)$ are established for certain singular maximal bilinear operators. The proof combines a trilinear smoothing inequality with Calder on-Zygmund theory.