Cauchy problem for one class of degenerate parabolic equations of Kolmogorov type with positive genus

We investigate properties of a fundamental solution and establish the correct solvability of the Cauchy problem for one class of degenerate Kolmogorov-type equations with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{ {\overrightarrow p, \overrightarrow h } \right\} $$\end{document}-parabolic part with respect to the main group of variables and with positive vector genus in the case where solutions are infinitely differentiable functions and their initial values may be generalized functions of Gevrey ultradistribution type.