Local times of linear multifractional stable sheets

Let XH(u)(u)={XH(u)(u),u∈ℝ+N}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X^{H(u)}(u)=\left\{X^{H(u)}(u), u \in \mathbb{R}_{+}^{N}\right\}$$\end{document} be linear multifractional stable sheets with index functional H(u), where H(u) = (H1(u),⋯, HN(u)) is a function with values in (0, 1)N. Based on some assumptions of H(u), we obtain the existence of the local times of XH(u)(u) and establish its joint continuity and the Hölder regularity. These results generalize the corresponding results about fractional stable sheets to multifractional stable sheets.