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On the Global Integrability of Non-Negative Harmonic Functions

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Abstract

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:

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

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.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501, Bielefeld, Germany

    Khalifa El Mabrouk

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  1. Khalifa El Mabrouk
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Correspondence to Khalifa El Mabrouk.

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Mathematics Subject Classifications (2000)

31D05, 31B05, 31A05.

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El Mabrouk, K. On the Global Integrability of Non-Negative Harmonic Functions. Potential Anal 22, 171–181 (2005). https://doi.org/10.1007/s11118-004-0586-6

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