Phragmén-Lindelöf conditions for graph varieties
Let \(V\) be an algebraic variety in \(\mathbb{C}^n\) and \(\omega\) a weight function. \(V\) satisfies the condition \(\text{PL}(\mathbb{R}^n,\omega)\) of Phragmén-Lindelöf type, if the following holds: There exists \(A\geq 1\) so that for each \(\rho>0\) there exists \(B_\rho>0\) so that each \(u\in \text{PSH}(V)\) which satisfies (\(\alpha\)) and (\(\beta\)) also satisfies (\(\gamma\)), where [\((\alpha)\)] \(u(z)\leq|\operatorname {Im} z|+O(\omega(z))\), \(z\in V\), [\((\beta)\)] \(u(z)\leq\rho|\operatorname {Im} z|\), \(z\in V\), [\((\gamma)\)] \(u(z)\leq A|\text{Im} z|+B_\rho\omega(z)\), \(z\in V \). If \(\omega(t)=\log(2+t)\) then they write \(\text{PL}(\mathbb{R}^n,\log)\). The class \(\text{PSH}(V)\) of plurisubharmonic functions on \(V\) is defined. For \(P_m\in\mathbb{C}[z_1,\dots,z_n]\), homogeneous of degree \(m\) the authors investigate when the graph of \(P_m\) in \(\mathbb{C}^{n+1}\) satisfies the condition \(\text{PL}(\mathbb{R}^{n+1},\log)\). This is shown to happen if the varieties \(V_{\pm}:=\{z\in\mathbb{C}^n:P_m(z)=\pm 1\}\) satisfy the strong version of the Phragmén-Lindelöf condition (SPL). Necessary as well as sufficient conditions for \(V_{\pm}\) to satisfy (SPL) are derived and sevaral examples are given. Furthermore, in the present paper, the authors investigate the following perturbation problem. Let \(P(z)=P_m(z_1,\dots,z_n)-z_{n+1}\), for some \(P_m\in\mathbb{C}[z_1,\dots,z_n]\), homogeneous of degree \(m\geq 2\) . Then \(\text{PL}(\mathbb{R}^{n+1},\omega)\) for \(V(P)\) and for some \(\omega\) satisfying \(\omega(t)=o(t^{1/m})\) is equivalent to the fact that \(V=V(P)\) as well as \(V=V_{\pm}:=\{z\in\mathbb{C}^n:P_m(z)=\pm 1\}\) satisfy the (SPL). Because of the connections between the Phragmén-Lindelöf conditions and partial differential equations \((\text{PL}(\mathbb{R}^{n+1},\log)\) for the graph \(P_m\) is equivalent to the fact that the operator \(i\frac\partial{\partial x_{n+1}}+P_m(D)\) admits a continuous solution operator on \(C^\infty(\mathbb{R}^{n+1})\)), the results proved in this paper imply new facts about fundamental solutions for certain evolution equations.
scientific article; zbMATH DE number 1346919