On the Global Integrability of Non-Negative Harmonic Functions

Let ℋ+(D) be the set of all non-negative harmonic functions on a domain D⊂Rd. Let q>0 and define Lq(D) to be the set of all Borel functions f such that |f|q is Lebesgue-integrable on D. Let x0∈D. N. Suzuki established the following: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{H}^{+}(D)\subset L^{q}(D)\Leftrightarrow \sup\biggl\{\int_{D}h^{q}(x)\,\mathrm{d}x:h\in\mathcal{H}^{+}(D),h(x_{0})=1\biggr\}<\infty.$$\end{document}