On $L^2$ estimates for quadratic images of product Frostman measures

Abstract

Let $f\in\mathbb R[x,y,z]$ be a fixed non-degenerate quadratic polynomial. Given an $\alpha$-Frostman probability measure $\mu$ supported on $[0,1]$ with $\alpha\in(0,1)$, consider the pushforward measure $\nu=f_{\#}(\mu\times\mu\times\mu)$ on $\mathbb R$. We prove the following $L^2$ energy estimate: for a fixed nonnegative Schwartz function $\varphi$ with $\int\varphi=1$ and $\varphi_\delta(t)=\delta^{-1}\varphi(t/\delta)$, there exist $\epsilon>0$ and $\delta_{0}>0$ (depending only on $\alpha$ and the coefficients of $f$) such that \[ \int_{\mathbb R}(\varphi_\delta*\nu(t))^{2}\,dt \ \lesssim\ \delta^{\alpha+\epsilon-1} \qquad \text{for all } \delta\in(0,\delta_{0}]. \] The proof expands the $L^2$ energy into a weighted six-fold coincidence integral and reduces the main contribution to a planar incidence problem after a controlled change of variables. The key new input is an incidence estimate for point sets that arise as bi-Lipschitz images of a Cartesian product $M\times M$ of a $\delta$-separated and non-concentrated set $M$, yielding a power saving beyond what is available from separation and non-concentration alone. We also give examples showing that bounded support and Frostman-type hypotheses are necessary for such $L^{2}$ control.