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Egor D. Kosov

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Preprint Aug 2026

Nondegeneracy and regularity of polynomial pushforwards

Let $\mu$ 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 \[ \mu(f\in A) \le C\bigl(\lambda_k(A)\bigr)^{\frac{1}{k(d-1)+1}} \] for every Borel set $A\subset\mathbb R^k$ whenever the image measure $\mu\circ f^{-1}$ 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 $\mu\circ f^{-1}$, 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 $\mu\circ f^{-1}$, 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.

Egor D. Kosov, A. Zhukova · 1 citation · ⚡1
Preprint Aug 2026

Absolute continuity of two-dimensional polynomial random vectors

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 \[ [[f]]_{\mu,\infty}^{\frac1{2d-1}}\mu(f\in A) \le C\bigl(\lambda_2(A)\bigr)^{\frac1{2d-1}} \] for every Borel set $A\subset\mathbb R^2$, where $C$ depends only on $d$ and the uniform density and moment bounds, and $\lambda_2$ denotes the Lebesgue measure on $\mathbb R^2$. Here $[[f]]_{\mu,\infty}$ measures the failure of proportionality of the highest-order orthogonal-chaos components of $f_1$ and $f_2$ with respect to the law $\mu$ 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 \[ \int_{\mathbb R^\infty}\Delta_f\,d\mu \ge C[[f]]_{\mu,\infty}^2, \] where $\Delta_f$ 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.

Egor D. Kosov · 0 citations