In this paper, motivated by the analysis of the fractional Laplace equation on the unit disk in $\mathbb{R}^{2}$, we establish a characterization of the weighted Sobolev space $H_{\beta}^{s}(\Omega)$ in terms of the decay rate of Fourier-Jacobi coefficients. This framework is then used to give a precise analysis of the solution to the fractional Laplace equation on the unit disk.