Potential Analysis

, Volume 22, Issue 2, pp 171–181 | Cite as

On the Global Integrability of Non-Negative Harmonic Functions

  • Khalifa El Mabrouk


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 x0D. N. Suzuki established the following:
$$\mathcal{H}^{+}(D)\subset L^{q}(D)\Leftrightarrow \sup\biggl\{\int_{D}h^{q}(x)\,\mathrm{d}x:h\in\mathcal{H}^{+}(D),h(x_{0})=1\biggr\}<\infty.$$

In this paper, we prove results of this kind in a general setting of harmonic spaces covering the elliptic case and the parabolic one as well. The last section deals with some applications of these results.


\(\mathcal{P}\) -harmonic Bauer spaces reference measures integral representation of non-negative harmonic functions 


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© Springer 2005

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany

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