Harmonic and superharmonic majorants on the disk

Details Harmonic and superharmonic majorants on the disk Harmonic and superharmonic majorants on the disk

2003-04-29

arxiv.org/abs/math/0304482v2

Mathematics

Complex Variables

13 pages ; this expands and improves on the previous version, due to the two authors named last

Scientific paper

We prove that a positive function on the unit disk admits a harmonic majorant if and only if a certain logarithmic Lipschitz upper envelope of it (relevant because of the Harnack inequality) admits a superharmonic majorant. We discuss the logarithmic Lipschitz regularity of this superharmonic majorant, and show that in general it cannot be better than that of the Poisson kernel. We provide examples to show that mere superharmonicity of the data does not help with the problem of existence of a harmonic majorant.

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