Norms and essential norms of the singular integral operator with Cauchy kernel on weighted Lebesgue spaces
Let \(\mathbb T\) be the unit circle, \(W\) some weight function on \(\mathbb T\), \(L^2(W)\) the Lebesgue space on \(\mathbb T\) with norm \(\left\| f\right\|_{W}^2= \int_{\mathbb T}| f|^2W\,dm\), and \[ Sf(\zeta)=\frac1{\pi i}\int_{\mathbb T}\frac{f(\eta)}{\eta-\zeta}d\eta \] be the singular integral operator on \(L^2(W)\), understood in the sense of Cauchy's principal value. The paper deals with calculation/estimates for the norm and essential norm of the operator \(S_{\alpha,\beta}f=\alpha Pf+\beta Qf\) in \(L^2(W)\) where \(P=(I+S)/2\) and \(Q=(I-S)/2\) are the analytic and co-analytic projections, respectively, and \(\alpha,\beta\in L^\infty\). Recall that the essential norm of \(S_{\alpha,\beta}\) is the infimum of the norms of all compact (linear) perturbations of \(S_{\alpha,\beta}\). For the case that \(\alpha\) and \(\beta\) are constant, explicit formulas for the norm and essential norm of \(S_{\alpha,\beta}\) had been obtained by \textit{I. Gohberg} and \textit{N. Krupnik} [One-dimensional linear singular integral equations. II. Basel: Birkhäuser (1992; Zbl 0781.47038)]. Morever, for the case \(W=1\), explicit formulas for the norm and essential norm of \(S_{\alpha,\beta}\) had been obtained by \textit{T. Nakazi} and \textit{T. Yamamoto} [J. Oper. Theory 40, 185--207 (1998; Zbl 0992.45009)] and by \textit{T. Nakazi} [Arch. Math. 73, 439--441 (1999; Zbl 0948.45006)], respectively. It is shown that -- up to at most constant factors for which explicit estimates are given -- the norm and essential norm of \(S_{\alpha,\beta}\) in \(L^2(W)\) relates to the norm and essential norm of \(S_{\varphi\alpha,\beta}\) in \(L^2(1)\), respectively, if \(W=| h|^2\) and \(\varphi=\overline h/h\) with an outer function \(h\).
scientific article; zbMATH DE number 5808419
scientific article