Toeplitz operators and arguments of analytic functions (Q1014217)

For a Toeplitz operator \(T_\varphi\) with symbol \(\varphi\in L^\infty({\mathbb T})\), let \(\text{Ker}_pT_\varphi\) denote its kernel on the Hardy space \(H^p\), \(1\leq p\leq\infty\), of the unit circle \({\mathbb T}\). Relations between \(\text{Ker}_pT_\varphi\) and the smoothness of \(\varphi\) are studied. Let \(\Lambda^\alpha\) (resp., \(A^\alpha\)), \(0<\alpha<\infty\), denote the Lipschitz (Hölder-Zygmund) space (resp., its subspace of analytic functions). Suppose that \(\beta=\alpha-1/p\) and \(N\) is the integral part of \(1/(\alpha p)\). Let us formulate two sample results for unimodular symbols \(\varphi\in\Lambda^\alpha\). If \(1/p<\alpha<\infty\), then \(\text{Ker}_pT_\varphi\subset A^\beta\) and \(\|f\|_{\Lambda^\beta}\leq\text{const}\,\|\varphi\|_{\Lambda^\alpha}\|f\|_p\) for each \(f\in\text{Ker}_pT_\varphi\). If \(0<\alpha<\infty\), then \(\text{Ker}_pT_\varphi\subset A^\alpha\) and \(\|f\|_{\Lambda^\alpha}\leq\text{const}\,\|\varphi\|_{\Lambda^\alpha}^{N+2} \|f\|_p\) for every \(f\in\text{Ker}_pT_\varphi\). The constants in the above inequalities are independent of \(f\) and \(\varphi\). Similar and more general results for various smoothness classes are obtained, and several approaches are discussed.