On Abel summation for Laplace transform of the homogeneous functions in $R^n$

The article discusses homogeneous functions of the form $\theta \left(\tau \right)\left|t\right|^{\alpha }$, which are determined by the order of homogeneity $\alpha >-n$, as well as by the function $\theta \left(\tau \right)$ on the unit sphere $S^{n-1}=\left\{t{\in}R^n,\left|t\right|=1\right\}$. When calculating the Laplace transform of these functions supported in a sharp cone, it is necessary to obtain an explicit representation. This is achieved by summing integrals according to Abel, as well as by applying Fourier analysis on the sphere, which will allow to bring calculations to transformations of hypergeometric functions necessary to calculate the limits as $\varepsilon \rightarrow 0$. The article presents formulas for the Laplace transform of homogeneous functions for various function spaces on the unit sphere.