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Sharp inequalities over the unit polydisc. (English) Zbl 1322.32006

Given a positive integer \(n\), the unit polydisc \(U^n\) in \({\mathbb C}^n\) is the Cartesian product of \(n\) copies of the unit disc \(U\). For \(0<p<\infty\), let \(H^p(U^n)\) be the classical Hardy space over \(U^n\) with “norm” \(||\cdot||_p\).
For a positive measurable function \(g\) on \(U\), let \(d\mu:= gdA\), where \(A\) is the area measure on \(U\), and put \(\nu_n := \mu\times \cdots\times \mu\) (\(n\) times). Given a positive integer \(m\), let \(\Phi: {\mathbb R}_+^m \to {\mathbb R}_+\) (\({\mathbb R}_+\) denotes the set of all nonnegative real numbers) be continuous and strictly increasing in each variable separately, and moreover assume that \(\Phi(x_1, \dots, x_j, \dots, x_m)=0\) if \(x_j=0\) for some \(j\). Suppose that \(\Phi\) and \(\nu_n\) satisfy the following condition:
(\(†\))
There are positive real numbers \(\tilde p_1, \dots, \tilde p_m\) such that \[ \int_{U^n} \Phi\big(|f_1|^{\tilde p_1}, \dots, |f_m|^{\tilde p_m}\big)\, d\nu_n \leq \Phi \big(||f_1||_{\tilde p_1}^{\tilde p_1}, \dots, ||f_1||_{\tilde p_m}^{\tilde p_m}\big) \] for all \(f_j\in H^{\tilde p_j}(U^n)\), \(j=1, \dots, m\), where the equality holds if and only if either
(i)
\(f_j\equiv 0\) for some \(j\) or
(ii)
each \(f_j\) (\(j=1, \dots, m\)) is nontrivial and equal to \(\Psi_j^n\), where \((\dots, \Psi_j^n, \dots)\) belongs to a class denoted by \({\mathcal E} (\Phi, \nu_n)\).

Condition \((\dagger)\) is motivated by an inequality due to J. Burbea [Ill. J. Math. 31, 248–264 (1987; Zbl 0614.32003)] for the case \(\Phi(x_1, \dots, x_m) = x_1 \cdots x_m\) and \(\tilde p_1 = \dots = \tilde p_m = 2\).
The main theorem of the paper is the following: Suppose that \((\dagger)\) holds for \(\Phi\) and \(\nu_n\). Then
\[ \int_{U^n} \Phi\big(|f_1|^{p_1}, \dots, |f_m|^{p_m}\big)\, d\nu_n \leq \Phi \big(||f_1||_{p_1}^{p_1}, \dots, ||f_1||_{p_m}^{p_m}\big) \]
for all \(f_j\in H^{p_j}(U^n)\) with \(0<p_j<\infty\) \((j=1, \dots, m)\). Moreover,
(a) each extremal function \(\Psi_j^n\) in (ii) vanishes nowhere on \(U^n\) and
(b) equality is attained in the above inequality if and only if either some \(f_j\equiv 0\) or each \(f_j\) is of the form \((\Psi_j^n)^{\tilde p_j / p_j}\) for some \((\dots, \Psi_j^n, \dots)\in {\mathcal E} (\Phi, \nu_n)\).
For the proof, the author begins with the one-dimensional case which is rather easy by means of the Riesz factorization theorem. Such one-dimensional proof certainly does not extend to the setting of higher-dimensional polydiscs, except for some special cases mentioned in Section 3 of the paper. To overcome such difficulty the author introduces and studies the logarithmically subharmonic Hardy space over \(U^n\).

MSC:

32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
32A36 Bergman spaces of functions in several complex variables
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))

Citations:

Zbl 0614.32003
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References:

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