Equivalence of the Weighted Fractional Sobolev Space on a Disk with Characterization by the Decay Rate of Fourier–Jacobi Coefficients and K-Interpolation

In this paper, motivated by the analysis of the fractional Laplace equation on the unit disk in \(\mathbb {R}^{2}\) ${\mathbb{R}}^2$ , we establish a characterisation of the weighted Sobolev space \(H_{\gamma }^{s}(\Omega )\) $H_{\gamma }^s(\mathrm{\Omega })$ 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.