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

Abstract

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