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

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.