Fractional smoothness of distributions of polynomials and a fractional analog of the Hardy–Landau–Littlewood inequality
Fractional smoothness of distributions of polynomials and a fractional analog of the Hardy–Landau–Littlewood inequality
We prove that the distribution density of any non-constant polynomial $f(\xi_1,\xi_2,\ldots)$ of degree $d$ in independent standard Gaussian random variables $\xi$ (possibly, in infinitely many variables) always belongs to the Nikol'skii--Besov space $B^{1/d}(\mathbb{R}^1)$ of fractional order $1/d$ (and this order is best possible), and an analogous result holds for polynomial …