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 \(\Delta _{\nu }\) ${\mathrm{\Delta }}_{\nu }$ defined in \(\mathbb {R}^{n-1}\times (0, \infty )\) ${\mathbb{R}}^{n-1}\times (0,\infty )$ . The Weinstein operator is given by \(\begin{aligned} \Delta _{\nu } = \frac{\partial ^2}{\partial x_1^{2}} + \dots + \frac{\partial ^2}{\partial x_n^{2}} + \frac{2\nu +1}{x_{n}}\frac{\partial }{\partial x_{n}}. \end{aligned}\) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{\nu }=\frac{{\partial }^2}{\partial x_1^2}+\cdots +\frac{{\partial }^2}{\partial x_n^2}+\frac{2\nu +1}{x_n}\frac{\partial }{\partial x_n}.\end{array}$$ 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.