Double Hilbert Transforms along Polynomial Surfaces in R3

where P(s, t) is a polynomial in s and t with P(0,0)= 0, and ∇P(0,0)= 0. We call H the (local) double Hilbert transform along the surface (s, t,P (s, t)). The operator may be precisely defined for a Schwartz function f by integrating where ≤ |s| ≤ 1 and η ≤ |t | ≤ 1, and then taking the limit as ,η→ 0. The corresponding 1-parameter problem has been extensively studied (see [RS1], [RS2], and [S], for example). The type of operator that we are concerned with in this paper has been previously studied in [NW], [RS3], and [V]. In those works, operators that are in some ways more general than ours are considered, but only under an appropriate dilation invariance, which in our setting would force P to be a monomial. If P(s, t) = st, then according to [RS3] (see Section 5 below for the precise statement), for any p, 1 < p < ∞, H is bounded in L if and only if at least one of m and n is even. Our present result is stated in terms of the Newton diagram of P , which we describe below. Recently Phong and Stein have shown how the Newton diagram also plays a decisive role in describing the mapping properties of certain degenerate Fourier integral operators (see [PS]). Main theorem. For any p, 1< p <∞, ‖Hf ‖Lp ≤ Ap‖f ‖Lp