dorsal/arxiv
View SchemaOn $L^2$ estimates for quadratic images of product Frostman measures
| Authors | Sung-Yi Liao, Thang Pham, Chun-Yen Shen |
|---|---|
| Categories | |
| ArXiv ID | 2601.09582vv1 |
| URL | https://arxiv.org/abs/2601.09582 |
| License | http://creativecommons.org/licenses/by/4.0/ |
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.
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"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 $\\epsilon\u003e0$ and $\\delta_{0}\u003e0$ (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.",
"arxiv_id": "2601.09582",
"authors": [
"Sung-Yi Liao",
"Thang Pham",
"Chun-Yen Shen"
],
"categories": [
"math.CA",
"math.CO"
],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "On $L^2$ estimates for quadratic images of product Frostman measures",
"url": "https://arxiv.org/abs/2601.09582",
"version": "v1"
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