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 R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}<^>{2}$$\end{document}, we establish a characterisation of the weighted Sobolev space H gamma s(Omega)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\gamma }<^>{s}(\Omega )$$\end{document} 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.