Small Hankel operators on Dirichlet-type spaces and applications (Q2794815)

Let \(\mathbb D\) be the unit disk in the complex plane. For a right-continuous and nondecreasing function \(\rho:[0,\infty)\to[0,\infty)\), denote by \(L^{2}_{\rho}\) the space of functions on \(\mathbb D\) such that NEWLINE\[NEWLINE \| f \|_{L^{2}_{\rho}}^{2} = \int_{\mathbb D}|f(z)|^{2}\rho(1-|z|^{2})dA(z)<\infty, NEWLINE\]NEWLINE where \(dA\) is the plane Lebesgue measure on \(\mathbb D\). The Dirichlet-type space \(D_{\rho}\) is the space of analytic functions on \(\mathbb D\) such that NEWLINE\[NEWLINE \| f \|_{D_{\rho}}^{2} = |f'(0)|^{2} + \| f'\|_{L^{2}_{\rho}}^{2} < \infty. NEWLINE\]NEWLINE For a function \(\rho\) satisfying some conditions, the function \(f\in L_{\rho}^{2}\) being analytic on \(\mathbb D\) and \(\alpha\geq0\), the small Hankel-type operator \(h_{\alpha,f}\) is defined on the set of polynomials on \(\mathbb D\) by the formula \(h_{\alpha,f}:g\mapsto\overline{P_{\alpha}(f\bar{g})}\), where NEWLINE\[NEWLINE (P_{\alpha}f)(z) = \int_{\mathbb D} \frac{f(w)}{(1-\bar{w}z)^{2+\alpha}}(1-|w|^{2})^{\alpha}\,dA(w). NEWLINE\]NEWLINENEWLINENEWLINENecessary and sufficient conditions for boundedness and compactness of \(h_{\alpha,f}:D_{\rho}\to L^{2}_{\rho}\) are obtained.