Isomorphisms of Royden Type Algebras over $mathbb S^1$

Let $mathbb S^1$ and $mathbb D$ be the unit circle and the unit disc in the plane and let us denote by $mathcal A(mathbb S^1)$ the algebra of the complex valued continuous functions on $mathbb S^1$ which are traces of functions in the Sobolev class $mathscr W^{1,2}(mathbb D)$. On $mathcal A(mathbb S^1)$ we define the following norm $|u|= |u|_{L^{infty}(mathbb S^1)}+ left(int int_{mathbb D}| abla ilde{u}|^2 ight)^{1/2}$ where $ ilde{u}$ is the harmonic extension of $u$ to $mathbb D$. We prove that every isomorphism of the functional algebra $mathcal A(mathbb S^1)$ is a quasisymmetric change of variables on $mathbb S^1$.