多项式推前测度的非退化性与正则性
Nondegeneracy and regularity of polynomial pushforwards
浏览论文内容
中文总结 AI 辅助
该研究在对数凹测度下证明了向量值多项式推前测度的测度界与Besov正则性，构造了合适的非退化参数，推广了相关不等式与准则，并得到两类收敛的关联估计。
中文摘要 AI 辅助
设$\mu$是$\mathbb{R}^n$上的对数凹概率测度，$f\colon\mathbb{R}^n\to\mathbb{R}^k$是次数不超过$d$的多项式映射。我们证明，当像测度$\mu\circ f^{-1}$绝对连续时，对每个Borel集$A\subset\mathbb{R}^k$，都有$\mu(f\in A) \le C\bigl(\lambda_k(A)\bigr)^{\frac{1}{k(d-1)+1}}$，其中常数$C$与维数$n$无关，且指数$\frac{1}{k(d-1)+1}$是最优的。这一结果推广了标量情形的Carbery--Wright不等式，并在对数凹框架下解答了Avni、Glazer和Larsen提出的问题。此外，我们证明，当$\mu\circ f^{-1}$的密度存在时，它属于Nikolskii--Besov空间$B^{\frac{1}{k(d-1)+1}}_{1,\infty}(\mathbb{R}^k)$，且对应范数存在与维数无关的界。\n从标量多项式过渡到向量值多项式映射的核心难点在于，缺乏像标量情形中方差那样的合适非退化参数，来量化$\mu\circ f^{-1}$的绝对连续性。协方差矩阵或雅可比矩阵这类自然候选者，要么无法刻画该性质，要么无法得到与维数无关的估计。我们找到了这样一个参数，将其定义为$f$的标准化分量中次数不超过$d^{k-1}$的单项式构成的向量的协方差矩阵。\n我们结果的无维数特性，使得我们能将Kusuoka针对高斯多项式随机向量的绝对连续性准则推广到对数凹情形。此外，在该情形下，我们得到了多项式随机向量依分布收敛与全变差收敛之间关联的估计。
英文摘要
Let $μ$ be a log-concave probability measure on $\mathbb R^n$ and let $f\colon\mathbb R^n\to\mathbb R^k$ be a polynomial mapping of degree at most $d$. We show that \[ μ(f\in A) \le C\bigl(λ_k(A)\bigr)^{\frac{1}{k(d-1)+1}} \] for every Borel set $A\subset\mathbb R^k$ whenever the image measure $μ\circ f^{-1}$ is absolutely continuous. The constant $C$ is independent of the dimension $n$, and the exponent $\frac{1}{k(d-1)+1}$ is sharp. This extends the scalar Carbery--Wright inequality and answers, in the log-concave setting, a question raised by Avni, Glazer, and Larsen. In addition, we show that the density of $μ\circ f^{-1}$, whenever it exists, belongs to the Nikolskii--Besov space $B^{\frac{1}{k(d-1)+1}}_{1,\infty}(\mathbb R^k)$, with a dimension-free bound for the corresponding norm. A central difficulty in passing from scalar polynomials to vector-valued polynomial mappings is the lack of a suitable nondegeneracy parameter quantifying absolute continuity of $μ\circ f^{-1}$, as the variance does in the scalar case. Natural candidates such as the covariance matrix or the Jacobian matrix either fail to characterize this property or do not lead to dimension-free estimates. We identify such a parameter and define it to be the covariance matrix of the vector formed by the monomials of degree up to $d^{k-1}$ in the normalized components of $f$. The dimension-free nature of our results allows us to extend Kusuoka's absolute continuity criterion for Gaussian polynomial random vectors to the log-concave setting. Moreover, in this setting, we obtain estimates relating convergence in distribution to convergence in total variation for polynomial random vectors.
