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 $\phi$ with $\int \phi =1$ and ${\phi }_{\delta }(t)={\delta }^{-1}\phi (t/\delta )$, there exist $\epsilon >0$ and ${\delta }_0>0$ (depending only on $\alpha$ 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 $\delta$-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.

V2: 48 pages; Ready for submission