Isoperimetric-type inequalities for pluriharmonic functions on the polydisc

We prove isoperimetric-type inequalities for pluriharmonic functions in the unit polydisc $\mathbb{U}^n$. Let $h^p(\mathbb{U}^n)$ and $b^p_{\mathbf{q}}(\mathbb{U}^n)$ denote, respectively, the pluriharmonic Hardy space and the pluriharmonic weighted Bergman space in $\mathbb{U}^n$. We prove that if $m\in\mathbb{N}$, $m\geq2$, $1<p_1,\ldots,p_m<\infty$, and $f_j\in h^{p_j}(\mathbb{U}^n)$, then \[
\int_{\mathbb{U}^n}\prod_{j=1}^m |f_j(z)|^{p_j}\,d\mu_{\mathbf{m-2}}(z)
\leq
\prod_{j=1}^m
\left[
\frac{\sqrt2\cos\left(\frac{\pi}{2mp_j}\right)}
{\sqrt{1-|\cos(\pi/p_j)|}}
\right]^{p_j}
\prod_{j=1}^m
\|f_j\|_{h^{p_j}(\mathbb{U}^n)}^{p_j}. \] In particular, \[
\|f\|_{b^{mp}_{\mathbf{m-2}}(\mathbb{U}^n)}
\leq
\frac{\sqrt2\cos\left(\frac{\pi}{2mp}\right)}
{\sqrt{1-|\cos(\pi/p)|}}
\|f\|_{h^p(\mathbb{U}^n)}. \] We also prove the following inclusion theorem: If $f\in h^2(\mathbb{U}^n)$, then \[
\|f\|_{h^{2n}(\mathbb{B}_n)}
\leq
\sqrt2\cos\left(\frac{\pi}{4n}\right)
\|f\|_{h^2(\mathbb{U}^n)}, \] where $\mathbb{B}_n$ is the unit ball in $\mathbb{C}^n$. A corresponding ball-volume inequality is obtained as well. The constants are explicit and are obtained from sharp Riesz-type estimates. In the planar case, they coincide with the best available constants in the literature, although sharpness of the resulting pluriharmonic inclusions remains open.