DOUBLE HILBERT TRANSFORMS ALONG POLYNOMIAL SURFACES IN R3

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]...

Polynomial Interpolation in R3

K e y w o r d s L a g r a n g e interpolation, Simultaneous approximation, Freud weights 1. I N T R O D U C T I O N In this paper , we are concerned with construct ing in terpola t ing subspaces of polynomials of several variables of relat ively small dimension as well as the corresponding in terpola t ion formulae a l a Lagrange. DEFINITION 1. A (linear) subspace G C C(Rn) is called k-interpol...

Bilinear Hilbert Transforms along Curves I. the Monomial Case

We establish an L2×L2 to L estimate for the bilinear Hilbert transform along a curve defined by a monomial. Our proof is closely related to multilinear oscillatory integrals.

The Conway Polynomial in R3 and in Thickened Surfaces: A New Determinant Formulation

Two-dimensional Hilbert Transforms'