Isoperimetric-type inequalities for pluriharmonic functions on the polydisc

Abstract

We prove isoperimetric-type inequalities for complex-valued pluriharmonic functions in the unit polydisc ${\mathbb{U}}^n\subset {\mathbb{C}}^n$. Denote by $h^p({\mathbb{U}}^n)$ and $b_{\mathbf{q}}^p({\mathbb{U}}^n)$, respectively, the pluriharmonic Hardy space and the pluriharmonic weighted Bergman space in ${\mathbb{U}}^n$, where $\mathbf{q}=(q_1,\dots ,q_n)>-1$. For $m\in \mathbb{N}$, $m\ge 2$, write $\mathbf{m}-2=(m-2,\dots ,m-2)$ and let \[ dμ_{\mathbf{m-2}}(z) =\frac{(m-1)^n}{π^n} \prod_{k=1}^n \left[(1-|z_k|^2)^{m-2}\,dx_kdy_k\right], \qquad z_k=x_k+iy_k. \] We prove that if $1<p_1,\dots ,p_m<\infty$ and $f_j\in h^{p_j}({\mathbb{U}}^n)$, then \[ \int_{\mathbb{U}^n}\prod_{j=1}^m|f_j(z)|^{p_j}\, dμ_{\mathbf{m-2}}(z) \leq \prod_{j=1}^m \left[ \frac{\sqrt2\cos\left(\fracπ{2mp_j}\right)} {\sqrt{1-|\cos(π/p_j)|}} \right]^{p_j} \prod_{j=1}^m \|f_j\|_{h^{p_j}(\mathbb{U}^n)}^{p_j}. \] In particular, \[ \|f\|_{b^{mp}_{\mathbf{m-2}}(\mathbb{U}^n)} \leq \frac{\sqrt2\cos\left(\fracπ{2mp}\right)} {\sqrt{1-|\cos(π/p)|}} \|f\|_{h^p(\mathbb{U}^n)}. \] For $p>2$, we refine this in the form \[ \|f\|_{b^{mp}_{\mathbf{m-2}}(\mathbb{U}^n)} \leq \left[ \sqrt2\cos\left(\fracπ{4m}\right) \right]^{2n/p} \|f\|_{h^p(\mathbb{U}^n)}. \] Consequently, the obtained constant in the diagonal inclusion tends to $1$ as $p\to \infty$, for fixed $m$ and $n$. When $m=2$ and $n=1$, the latter estimate coincides with best-known planar estimate. Explicit lower bounds at $p=2$, together with the dimension-free upper estimate, show that the optimal diagonal constants converge to $\sqrt{2}$ as $m\to \infty$, uniformly in the dimension.

27 pages