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

TL;DR

This work establishes an L^2 smoothing estimate for quadratic images of Frostman measures: for a fixed non-degenerate quadratic f and α-Frostman μ on [0,1], the pushforward ν=f_{#}(μ×μ×μ) satisfies ∫ (φ_{δ}*ν)^2 ≲ δ^{α+ε−1} for small δ, with ε>0 depending on α and f. The authors connect the L^2 energy to a six-fold coincidence integral and reduce the main contribution to a planar incidence problem, proving a new incidence theorem for δ-separated lines interacting with bi-Lipschitz images of Cartesian products Φ(M×M). The key advance is a δ^ε gain in the incidence bound when M is δ-separated and non-concentrated, enabling the L^2 improvement for α≤1/2, and the results extend from AD-regular to Frostman measures via a weighted discretization; the paper also presents obstruction examples showing the necessity of hypotheses such as Frostman bounds and bounded support. Overall, the work advances the use of geometric incidence methods to control harmonic-analytic energies of nonlinear projections, with implications for Falconer-type problems involving quadratic forms.

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 $\varphi$ with $\int\varphi=1$ and $\varphi_δ(t)=δ^{-1}\varphi(t/δ)$, there exist $ε>0$ and $δ_{0}>0$ (depending only on $α$ and the coefficients of $f$) such that \[ \int_{\mathbb R}(\varphi_δ*ν(t))^{2}\,dt \ \lesssim\ δ^{α+ε-1} \qquad \text{for all } δ\in(0,δ_{0}]. \] 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.

Theorems & Definitions (40)

• Definition 1.1
• Definition 1.2
• Theorem 1.3: KPS, Theorem 1.2
• Theorem 1.4: quy, Theorem 3.1
• Definition 1.5
• Theorem 1.6
• Theorem 2.1
• Theorem 2.2: Theorem 3.2, PPS
• proof : Proof of Theorem \ref{['our_incidence_theorem']}
• Definition 3.1