Characterization of the weighted Sobolev space $H_{\beta }^s(\mathrm{\Omega })$ in ${\mathbb{R}}^2$ in terms of the decay rate of Fourier-Jacobi coefficients

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 $H_{\beta }^s(\mathrm{\Omega })$ 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.