Bilinear generalized Radon transforms in the plane

Let $σ$ be arc-length measure on $S^1subset mathbb R^2$ and $Θ$ denote rotation by an angle $θin (0, π]$. Define a model bilinear generalized Radon transform, $$B_θ(f,g)(x)=int_{S^1} f(x-y)g(x-Θ y), dσ(y),$$ 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 $B_θ$ are motivated by problems in geometric measure theory and combinatorics. For $θu003cπ$, we show that $B_θ: L^p({Bbb R}^2) times L^q({Bbb R}^2) to L^r({Bbb R}^2)$ 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 $θ=π$, 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 $$ B(f,g)(x)=int int δ(φ_1(x,y)-t_1)δ(φ_2(x,z)-t_2) δ(φ_3(y,z)-t_3) f(y)g(z) ψ(y,z) , dy, dz, $$ where $δ$ denotes the Dirac distribution, $t_1,t_2,t_3inmathbb R$, $ψ$ is a smooth cut-off and the defining functions $φ_j$ satisfy some natural geometric assumptions.