Abstract:

In this talk we will consider double Hilbert transforms $H$ along analytic surfaces defined by $$Hf(x,y,z) =\int_{|s|\le 1} \int_{|t|\le 1} f(x-t,y-t,z-P(s,t))\frac{ds dt}{st}$$ where $z=(z_1,...,z_n)$ and $P(s,t)=(P_1(s,t),...,P_n(s,t))$. $L^p$ boundedness of $H$ when $n=1$ and $P_1$ is a polynomial have been considered by Carbery, Wainger, and Wright in [1]. It turns out that $L^p$ boundedness of $H$ is related to the Newton Polygon of $P_1$. In this talk we will discuss about the extention of their result to the general dimension $2+n$. Necessary and sufficient conditions for $L^p$ boundedness of $H$ will be presented.

[1] A. Carbery, S. Wainger, and J. Wright, Double Hilbert transforms along polynomial surfaces in $R^3$, Duke Math. J. 101(2000), 499--513.