The Fejér-Riesz inequality for Siegel domains
The classical Fejér-Riesz inequality was extended from the unit disc of the complex plane \({\mathbb{C}}\) to balls and polydiscs of \({\mathbb{C}}^ n\). For unbounded domains, an analogous inequality on the upper half-plane \({\mathbb{R}}^ 2_+=\{z\in {\mathbb{C}}| \quad Im z>0\}\) is \(\int_{{\mathbb{R}}_+}| f(x+iy)|^ pdy\leq 2^{- 1}\sup_{y>0}\int_{{\mathbb{R}}}| f(x+iy)|^ pdx,\) where \({\mathbb{R}}_+\) denotes the positive numbers and \(f\in H^ p({\mathbb{R}}^ 2_+)\), \(1\leq p<\infty\). In this paper, the author generalizes this inequality to several complex variables and a wider class of functions. An n-polygonal cone \(\Omega\) is an open cone in \({\mathbb{R}}^ n\) which is the interior of the convex hull of n linearly independent half-lines starting from the origin. The Siegel domain considered here is \(D=\{(Z,W)\in {\mathbb{C}}^ n\times {\mathbb{C}}^ m| \quad Im Z-\Phi (W,W)\in \Omega \},\) where \(\Phi\) is an \(\Omega\)-hermitian form. A real values function u on D is called a log. plurisubharmonic function if \(u\geq 0\) and log u is plurisubharmonic. Let the class \(LH^ p(D)\), \(0<p<\infty\), be the totality of log. plurisubharmonic functions u on D satisfying \[ M(u,p;D)=\sup_{Y\in \Omega}\int_{{\mathbb{R}}^ n\times {\mathbb{C}}^ m}u(X+i(Y+\Phi (W,W)),W)^ pdXdW<\infty, \] where dW denotes the volume element of \({\mathbb{R}}^{2m}={\mathbb{C}}^ m\). The Hardy space \(H^ p(D)\) is the class of holomorphic functions f on D such that \(M(| f|,p;D)<\infty.\) The main result of this paper is the following theorem. Let \(u\in LH^ p(D)\), \(0<p<\infty\). Then for any \(X\in {\mathbb{R}}^ n\), \[ \int_{\Omega \times {\mathbb{C}}^ m}u(X+i(Y+\Phi (W,W)),W)^ pdYdW\leq 2^{-n}M(u,p;D). \] The proof consists of a sequence of lemmas and frequent use of a result from a subharmonic function on the half plane \({\mathbb{R}}^ 2_+\). The paper closes with a few remarks on some special cases and results on Hardy space.
scientific article; zbMATH DE number 3933434