Generalized Pizzetti's formula for Weinstein operator and its applications

This study examines various facets of harmonic analysis, with a specific emphasis on the Weinstein operator Delta nu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{\nu }$$\end{document} defined in Rn-1x(0,infinity)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}<<^>>{n-1}\times (0, \infty )$$\end{document}. The Weinstein operator is given by Delta nu=partial derivative 2 partial derivative x12+& ctdot;+partial derivative 2 partial derivative xn2+2 nu+1xn partial derivative partial derivative xn.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta _{\nu } = \frac{\partial <<^>>2}{\partial x_1<<^>>{2}} + \dots + \frac{\partial <<^>>2}{\partial x_n<<^>>{2}} + \frac{2\nu +1}{x_{n}}\frac{\partial }{\partial x_{n}}. \end{aligned}$$\end{document}The study begins with an exploration of the well-known Pizzetti's formula extended to accommodate the Weinstein operator, emphasizing the associated spherical mean and resulting in the derivation of asymptotic expansions. Subsequently, the investigation shifts its focus to the fractional power of the Weinstein operator, particularly exploring the regularized fractional Weinstein operator. Utilizing the Pezzetti formula related to the Weinstein operator, we construct a singular integral representation and establish a regularization scheme.