Bilinear generalized Radon transforms in the plane

Abstract

Let $\sigma$ be arc-length measure on $S^1\subset \mathbb R^2$ and $\Theta$ denote rotation by an angle $\theta \in (0, \pi]$. Define a model bilinear generalized Radon transform, $$B_{\theta}(f,g)(x)=\int_{S^1} f(x-y)g(x-\Theta y)\, d\sigma(y),$$ an analogue of the linear generalized Radon transforms of Guillemin and Sternberg \cite{GS} and Phong and Stein (e.g., \cite{PhSt91,St93}). Operators such as $B_\theta$ are motivated by problems in geometric measure theory and combinatorics. For $\theta<\pi$, we show that $B_{\theta}: L^p({\Bbb R}^2) \times L^q({\Bbb R}^2) \to L^r({\Bbb R}^2)$ if $\left(\frac{1}{p},\frac{1}{q},\frac{1}{r}\right)\in Q$, the polyhedron with the vertices $(0,0,0)$, $(\frac{2}{3}, \frac{2}{3}, 1)$, $(0, \frac{2}{3}, \frac{1}{3})$, $(\frac{2}{3},0,\frac{1}{3})$, $(1,0,1)$, $(0,1,1)$ and $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$, except for $\left( \frac{1}{2},\frac{1}{2},\frac{1}{2} \right)$, where we obtain a restricted strong type estimate. For the degenerate case $\theta=\pi$, a more restrictive set of exponents holds. In the scale of normed spaces, $p,q,r \ge 1$, the type set $Q$ is sharp. Estimates for the same exponents are also proved for a class of bilinear generalized Radon transforms in $\mathbb R^2$ of the form $$ B(f,g)(x)=\int \int \delta(\phi_1(x,y)-t_1)\delta(\phi_2(x,z)-t_2) \delta(\phi_3(y,z)-t_3) f(y)g(z) \psi(y,z) \, dy\, dz, $$ where $\delta$ denotes the Dirac distribution, $t_1,t_2,t_3\in\mathbb R$, $\psi$ is a smooth cut-off and the defining functions $\phi_j$ satisfy some natural geometric assumptions.