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Bernstein-type inequalities for shift-coinvariant subspaces and their applications to Carleson embeddings. (English) Zbl 1082.46019
Let \(\Theta\) be an inner function in the upper half-plane \(\mathbb C^+\). The model subspace \(K_\Theta^p\) of the Hardy space \(H^p=H^p(\mathbb C^+)\) is then defined as \(K^p_\Theta=H^p\cap \Theta\overline{H^p}\). The author obtains Bernstein-type inequalities for functions from \(K_\Theta^p\), in other words, he estimates the differentiation operator \(\frac d{dz}\) as an operator from \(K_\Theta^p\) to \(L^p(\mu)\) for measures \(\mu\) satisfying certain conditions.
One of the main results of the paper is as follows. Let \(k_z(\zeta)=\frac i{2\pi}(1-\overline{\Theta(z)}\Theta(\zeta))/(\zeta-\bar z)\) and \(w_p(z)=\| k_z^2\| _{L^q}^{-p/(p+1)}\) for \(p\in[1,+\infty)\) (where \(q=p/(p-1)\)). Then for any Carleson measure \(\mu\) the operator \(T_p\,:\,f\mapsto f'(z)w_p(z)\) is of weak type \((p,p)\) as an operator from \(K_\Theta^p\) to \(L^p(\mu)\) and bounded as an operator from \(K_\Theta^r\) to \(L^r(\mu)\) for any \(r>p\). The author obtains also a number of estimates for the weight \(w_p\) in terms of the function \(\Theta\). As an application, new Carleson-type embedding theorems for functions from \(K^p_\Theta\) are obtained, generalizing earlier results by S. R. Treil and A. L. Volberg [J. Sov. Math. 42, No. 2, 1562–1572 (1988; Zbl 0654.30027); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 149, 38–51 (1986; Zbl 0612.30032)] and B. Cohn [Pac. J. Math. 103, 347–364 (1982; Zbl 0509.30026)].

MSC:
46E15 Banach spaces of continuous, differentiable or analytic functions
47B38 Linear operators on function spaces (general)
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