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 $b^p_{\mathbf q}(\mathbb{U}^n)$, respectively, the pluriharmonic Hardy space and the pluriharmonic weighted Bergman space in $\mathbb{U}^n$, where $\mathbf q=(q_1,\ldots,q_n)>-\mathbf1$. For $m\in\mathbb{N}$, $m\geq2$, write $\mathbf{m-2}=(m-2,\ldots,m-2)$ and let \[ d\mu_{\mathbf{m-2}}(z) =\frac{(m-1)^n}{\pi^n} \prod_{k=1}^n \left[(1-|z_k|^2)^{m-2}\,dx_kdy_k\right], \qquad z_k=x_k+iy_k. \] We prove that if $1 2$, we refine this in the form \[ \|f\|_{b^{mp}_{\mathbf{m-2}}(\mathbb{U}^n)} \leq \left[ \sqrt2\cos\left(\frac{\pi}{4m}\right) \right]^{2n/p} \|f\|_{h^p(\mathbb{U}^n)}. \] 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 $\sqrt2$ as $m\to\infty$, uniformly in the dimension.