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 $B_{\theta }:L^p({\mathbb{R}}^2)\times L^q({\mathbb{R}}^2)\to L^r({\mathbb{R}}^2)$ 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 $$B(f,g)(x)=\int \int \delta ({\varphi }_1(x,y)-t_1)\delta ({\varphi }_2(x,z)-t_2)\delta ({\varphi }_3(y,z)-t_3)f(y)g(z)\psi (y,z)dydz,$$ 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 ${\varphi }_j$ satisfy some natural geometric assumptions.

arXiv comment18 pages, 3 figures

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