Hölder Continuity and Boundedness of Composition Operators on Pluriharmonic $\alpha$-Bloch Spaces

Abstract

In this paper, we study the metric properties of pluriharmonic $\alpha$-Bloch-type mappings $f=h+\overline{g}\in \mathcal{H}({\mathbb{B}}^n,{\mathbb{C}}^n)$ on the unit ball ${\mathbb{B}}^n\subset {\mathbb{C}}^n$. By synthesizing local real-part estimates with the subharmonicity of $\log (|\det Dh(z)|+|\det Dg(z)|)$ along hyperbolic geodesics, we establish a globalized non-trivial lower bound across the entire unit ball ${\mathbb{B}}^n$. Using this structural decay estimate, we prove that the invariant weight quantity $D_f^{n,\alpha }(z)$ satisfies a global Hölder continuity estimate of order $1/n$ with respect to the Bergman metric $\beta$. Furthermore, we investigate composition operators $C_{\phi }(f)=f\circ \phi$ induced by holomorphic self-mappings $\phi \in \mathcal{A}({\mathbb{B}}^n,{\mathbb{B}}^n)$. We establish both sharp sufficient and necessary sampling conditions for $C_{\phi }$ to be bounded from below on the pluriharmonic $\alpha$-Bloch space ${\mathcal{B}}_{P(n,\alpha )}$.

24 pages, 0 figures