On smooth behavior of probability distributions under polynomial mappings

Let $X$ be a random variable with probability distribution $PX$ concentrated on $[-1,1]$ and let $Q(x)$ be a polynomial of degree $k\ge 2$. The characteristic function of a random variable $Y=Q(X)$ is of order $O(1/|t|1/k)$ as $|t|\to\infty$ if $PX$ is sufficiently smooth. In addition, for every $1/k>\varepsilon>0$ there exists a singular distribution $PX$ such that every convolution $P^{n\star}_X$ is also singular while the characteristic function of $Y$ is of order $O(1/|t|^{1/k-\varepsilon})$. While the characteristic function of $X$ is small when “averaged” the characteristic function of the polynomial transformation $Y$ of $X$ is uniformly small.