Generalized Hardy spaces on tube domains over cones (Q2773265)

Let \(\Omega\) be an irreducible symmetric cone in \({\mathbf R}^n\) and let NEWLINE\[NEWLINE T_\Omega={\mathbf R}^n+i\Omega\subset {\mathbf C}^n NEWLINE\]NEWLINE be the tube domain based on \(\Omega\). In this paper, the author first studies a general family of spaces \(H^p_\mu(T_\Omega)\) of holomorphic functions in \(T_\Omega\) satisfying an integrability condition of Hardy type: NEWLINE\[NEWLINE \|F\|_{H^p_\mu}:=\sup_{y\in\Omega}\left [ \int_{\overline\Omega}\int_{{\mathbf R}^n} |F(x+i(y+t))|^pdxd\mu(t)\right]^{1/p}<\infty. NEWLINE\]NEWLINE Here \(0<p<\infty\) and \(\mu\) is a positive measure in \({\mathbf R}^n\) with the following two geometric assumptions: NEWLINENEWLINENEWLINE\(\mu\) is locally finite in \({\mathbf R}^n\), with supp\((\mu)\subset \overline\Omega\); NEWLINENEWLINENEWLINE\(\mu\) is quasi-invariant with respect to a subgroup \(H\) of \(G(\Omega)\), acting simply transitively on the cone; i.e., NEWLINE\[NEWLINE \int f(h^{-1} y)d\mu(y)=\chi(h)\int f(y)d\mu(y) NEWLINE\]NEWLINE for all \(f\in L^1(d\mu)\), \(h\in H\). Here \(G(\Omega)\) is the group of linear transformations of the cone. The author proves the following main result of the paper.NEWLINENEWLINENEWLINETheorem. Let \(0<p<\infty\) and \(\mu\) be a measure as above. Then for every \(F\in H^p_\mu(T_\Omega)\) there exists \(F^{(b)}\in L^p_\mu\) such that NEWLINE\[NEWLINE \lim_{y\rightarrow 0, y\in \Omega_0}\|F(\cdot+iy)-F^{(b)}\|_{L^p_\mu}=0, NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \lim_{y\rightarrow 0, y\in \Omega_0} F(z+iy)=F^{(b)}(z) NEWLINE\]NEWLINE for almost every \(z\in T_\mu\), for every proper subcone \(\Omega_0\) of \(\Omega\). When \(p\geq 1\), the first limit holds as well as with \(\Omega_0\) replaced by \(\Omega\).