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Contractive inequalities for Bergman spaces and multiplicative Hankel forms. (English) Zbl 1417.30049

Two classical results of the theory of spaces of holomorphic functions on the unit disk \(\mathbb{D}\subset \mathbb{C}\) are the Carleman inequality \(\|f\|_{A^2(\mathbb{D})}\le \|f\|_{H^1(\mathbb{D})}\) and the Weissler inequality \(\|f(r\cdot)\|_{H^q(\mathbb{D})} \le \|f\|_{H^p(\mathbb{D})}\), only valid for \(0\le r\le\sqrt{p/q}\le 1\). Since both inequalities are contractive, they carry on to the infinite polydisk, and this yields results for Hardy spaces of Dirichlet series.
In this paper the authors prove Carleman- and Weissler-type inequalities for weighted Bergman spaces \(A^p_\alpha=A^p_\alpha(\mathbb{D},dA_\alpha)\), \( 0 < p < \infty\) and \(\alpha\ge 1\). Here, for \(\alpha> 1\), \(dA_\alpha\) is the probabilistic measure \(dA_\alpha=(\alpha-1)(1-|z|^2)^{\alpha-2} dA(z)\), where \(dA\) denotes the normalized area measure on \(\mathbb{D}\), and for \(\alpha=1\), \(dA_1\) is the normalized arc length measure on the unit circle. Hence, \(A^p_1=H^p\). The Carleman-type inequality states that if \(\alpha\ge (1+\sqrt{17})/4\) then \( \|f\|_{A^{p(\alpha+1)/\alpha}_{\alpha+1}} \le \|f\|_{A^{p}_{\alpha}}. \) The Weissler-type inequality states that if \(0< p \le q< \infty\) and \(\alpha=(n+1)/2\) for some \(n\in\mathbb{N}\), the operator \(P_r: A^p_\alpha\to A^q_\alpha\), defined by \(P_rf(z)=f(rz)\), \(0 < r \le 1\), is contractive if and only if \(r\le\sqrt{p/q}\le 1\). By contractivity and a tensorization procedure, these inequalities can be extended to the Bergman spaces \(A^p_\alpha(\mathbb{D}^\infty)\).
As a consequence of these results the authors obtain a relationship between multiplicative Hankel forms on weighted spaces of sequences \(\ell^2_d\), small Hankel bilinear forms on Bergman spaces of Dirichlet series and small Hankel bilinear forms on Bergman spaces \(A^p_\alpha(\mathbb{D}^\infty)\).
In addition, the authors give some counterexamples which prove that some well-known results on Carleson measures for Hardy spaces on finite-dimensional polydisks cannot be extended to the infinite-dimensional case.

MSC:

30H20 Bergman spaces and Fock spaces
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
30B50 Dirichlet series, exponential series and other series in one complex variable
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