Oscillatory Loomis—Whitney and Projections of Sublevel Sets

We consider an oscillatory integral operator with Loomis—Whitney multilinear form. The phase is real analytic in a neighborhood of the origin ℝd and satisfies nondegeneracy condition related to its Newton polyhedron. Maximal decay obtained for this certain cases, depending on polyhedron given Lebesgue exponents. Our estimates imply volumes sublevel sets such functions are small relative product areas projections onto coordinate hyperplanes.