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 $$\mathbb {R}^{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , we establish a characterisation of the weighted Sobolev space $$H_{\gamma }^{s}(\Omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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.