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arXiv · 2607.22784

$L^p$-Integrability of Radon-Nikodym Densities Between Harmonic Energy Measures on the Sierpinski Gasket

Konstantinos Tsougkas
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Abstract

It is known that the energy measures of any two nonconstant harmonic functions on the standard Sierpi\'nski gasket are mutually absolutely continuous. Strichartz and Tse reported numerical evidence for $L^p$-integrability of the corresponding Radon--Nikodym densities in the range \[ 1<p<\frac{\log 15}{\log 9}. \] For arbitrary ordered pairs of nonconstant harmonic functions, we prove uniform boundedness of the associated density-ratio power sums, and hence $L^p$-integrability, in the subinterval \[ 1<p<\frac{\log(35/3)}{\log 9}. \] When the denominator harmonic direction is represented by the boundary values $(0,-1,1)$, we prove boundedness throughout the full conjectured interval.

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BibTeXRIS

Konstantinos Tsougkas. 2026-07-24. $L^p$-Integrability of Radon-Nikodym Densities Between Harmonic Energy Measures on the Sierpinski Gasket. https://arxiv.org/abs/2607.22784

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