The essential norm of Hankel operators on the weighted Bergman spaces with exponential type weights (Q2494164)

Let \(f\in L^2({\mathbb D})\), \(\phi:{\mathbb D}\to {\mathbb R}\) be a subharmonic function satisfying certain natural conditions [first considered by \textit{V.\ L.\ Oleinik}, J.\ Sov.\ Math.\ 9, 228--243 (1978; Zbl 0396.31001)] and \(H_f\) stand for the Hankel operator \(H_f(g)= fg-P_\phi(fg)\), where \(P_\phi\) is the orthogonal projection from \(L^2({\mathbb D}, e^{-2\phi}d\lambda)\) onto the subspace of analytic functions \(AL^2_\phi({\mathbb D})\). The author shows that the essential norm \(\| H_f\| _e\) is comparable to the distance of the operator \(H_f\) to the compact Hankel operators, or equivalently to the distance (in a certain space of functions with bounded distance to analytic) of the symbol \(f\) to the space of functions with vanishing distance to analytic, which extends the result proved for potential weights by \textit{P.~Lin} and \textit{R.~Rochberg} [Integral Equations Oper.\ Theory 17, No.~3, 361--372 (1993; Zbl 0812.47021)]. The proof uses some modifications of ideas from the same authors, using extremal functions and others of the usual Hörmander estimates for the \(\bar\partial\) operator.