AI 中文总结
该研究推广了二维维纳混沌向量的二分性，证明了分量次数至多为d的多项式随机向量的绝对连续性，得到相关下界并在高斯测度下恢复了猜想估计的松弛版本。
AI 中文摘要
设$X=\{X_j\}_{j=1}^\infty$为独立随机变量序列，其密度函数与2d阶矩均一致有界。对分量次数至多为d的多项式泛函构成的随机向量$f(X)=(f_1(X),f_2(X))$，我们证明：对任意Borel集$A\subset\mathbb{R}^2$，有$[[f]]_{\mu,\infty}^{\frac{1}{2d-1}}\mu(f\in A) \le C\bigl(\lambda_2(A)\bigr)^{\frac{1}{2d-1}}$，其中常数C仅依赖于d以及密度与矩的一致界，$\lambda_2$为$\mathbb{R}^2$上的勒贝格测度；$[[f]]_{\mu,\infty}$衡量$f_1$与$f_2$关于$X$的分布$\mu$的最高阶正交混沌分量比例关系的偏离程度。由此，当这些分量不成比例时，$f$的分布在弱洛伦兹空间$L^{\frac{2d-1}{2d-2},\infty}(\mathbb{R}^2)$中存在密度。该结果恢复了Nualart与Tudor针对二维维纳混沌向量建立的二分性，并将其推广至高斯分布之外的场景。我们还得到下界$\int_{\mathbb{R}^\infty}\Delta_f\\,d\mu \ge C[[f]]_{\mu,\infty}^2$，其中$\Delta_f$是$\nabla f_1$与$\nabla f_2$的格拉姆矩阵的行列式；在高斯测度的特殊情形下，该式给出了Nourdin、Nualart与Poly所猜想估计的松弛版本。
英文摘要
Let $X=\{X_j\}_{j=1}^\infty$ be a sequence of independent random variables whose densities and moments of order $2d$ are uniformly bounded. For a random vector $f(X)=(f_1(X),f_2(X))$ whose components are polynomial functionals of degree at most $d$, we prove that \[ [[f]]_{μ,\infty}^{\frac1{2d-1}}μ(f\in A) \le C\bigl(λ_2(A)\bigr)^{\frac1{2d-1}} \] for every Borel set $A\subset\mathbb R^2$, where $C$ depends only on $d$ and the uniform density and moment bounds, and $λ_2$ denotes the Lebesgue measure on $\mathbb R^2$. Here $[[f]]_{μ,\infty}$ measures the failure of proportionality of the highest-order orthogonal-chaos components of $f_1$ and $f_2$ with respect to the law $μ$ of $X$. Consequently, whenever these components are not proportional, the law of $f$ admits a density in the weak Lorentz space $L^{\frac{2d-1}{2d-2},\infty}(\mathbb R^2)$. This recovers the dichotomy established by Nualart and Tudor for two-dimensional Wiener chaos vectors and extends it beyond the Gaussian setting. We also obtain the lower bound \[ \int_{\mathbb R^\infty}Δ_f\,dμ\ge C[[f]]_{μ,\infty}^2, \] where $Δ_f$ is the determinant of the Gram matrix of $\nabla f_1$ and $\nabla f_2$. In the special case of Gaussian measures, this gives a relaxed version of the estimate conjectured by Nourdin, Nualart, and Poly.
