Bounds for Multilinear Sublevel Sets

The question in the form (1.2) was posed by Li, Tao, Thiele, and this author in [5]. Nonoscillatory inequalities of the form ∫ ∏ j |fj◦`j | . ∏ j ‖fj‖Lpj have been studied in [1],[2]. Oscillatory integral inequalities of this type have been extensively studied for n = 2, where one is dealing with bilinear forms 〈Tλ(f1), f2〉. The associated linear operators Tλ are commonly known as oscillatory integrals of the second type, and a simple necessary and sufficient condition for (1.2) to hold (with some unspecified exponent) is known [7]. There is an extensive literature dealing with more specific inequalities involving Lp norms, in which one seeks the optimal exponent δ as a function of exponents p. For n ≥ 3, however, there arises a class of singular oscillatory integrals which have no direct analogues in the bilinear case. These singular cases arise when d < ∑ j dj . Generic ordered n-tuples of points (x1, · · · , xn) ∈