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 $α$-Frostman probability measure $μ$ supported on $[0,1]$ with $α\in(0,1)$, consider the pushforward measure $ν=f_{\#}(μ\timesμ\timesμ)$ on $\mathbb R$. We prove the following $L^2$ energy estimate: for a fixed nonnegative Schwartz function $φ$ with $\intφ=1$ and $φ_δ(t)=δ^{-1}φ(t/δ)$, there exist $ε>0$ and $δ_{0}>0$ (depending only on $α$ and the coefficients of $f$) such that \[ \int_{\mathbb R}(φ_δ*ν(t))^{2}\,dt \ \lesssim\ δ^{α+ε-1} \qquad \text{for all } δ\in(0,δ_{0}]. \] 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 $δ$-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.