Structure of algebras of singular integral operators in weighted \(L_ p\)-spaces on the circle
Let \(N_{\omega^{1/p}}: (L_p(\mathbb{T},\omega(t)dt)\to L_p(\mathbb{T})\) be the operator of multiplication by the function \(\omega^{1/p}\). It is easy to show that \(N_{\omega^{1/p}}\) is an isometric isomorphism. Obviously, its inverse operator \(N_{\omega^{1/p}}^{- 1}\) is the operator of multiplication by the function \(\omega^{-1/p}\). Such an operator is denoted by \(N_{\omega^{- 1/p}}\). Let us consider the similarity isomorphism \[ \alpha_\omega: \text{End } L_p(\mathbb{T},\omega(t)dt)\to \text{End } L_p(\mathbb{T}), \] where \(\alpha_\omega(A)= N_{\omega^{1/p}} AN_{\omega^{-1/p}}\) for \(A\in \text{End }L_p (\mathbb{T},\omega(t)dt)\). In the present article the operator \(\alpha_\omega (S_{p,\omega})\) is described in terms of the operator \(S_p\) and the operators of multiplication by the functions \(u,v\) from the representation \(\ln(\omega)= u+Sv\); \(u,v\in L_\infty(\mathbb{T})\) of \(\omega\), \(\omega\) the Hunt-Muckenhoupt weight. This description enables us to obtain by the given coefficient algebra \(\Omega\) such conditions on the weight \(\omega\) that under these conditions the mapping \(\alpha_\omega\) realizes the similarity isomorphism of the algebra \({\mathfrak A}(S_{p,\omega}, \Omega)\) on the algebra \({\mathfrak A}(S_p, \Omega)\).
scientific article; zbMATH DE number 819975