AI 中文总结
研究谢尔宾斯基垫片上非常数调和函数能量测度间拉东 - 尼科迪姆密度的$L^p$可积性，通过证明密度比幂和一致有界，得出在特定子区间及特定条件下的$L^p$可积性结论。
AI 中文摘要
已知标准谢尔宾斯基垫片上任意两个非常数调和函数的能量测度相互绝对连续。Strichartz和Tse报告了在\[1 < p < \frac{\log 15}{\log 9}\]范围内相应拉东 - 尼科迪姆密度的$L^p$可积性的数值证据。对于任意非常数调和函数的有序对，我们证明了相关密度比幂和的一致有界性，从而在子区间\[1 < p < \frac{\log(35/3)}{\log 9}\]内证明了$L^p$可积性。当分母调和方向由边界值\((0, -1, 1)\)表示时，我们证明在整个推测区间内有界性。
英文摘要
It is known that the energy measures of any two nonconstant harmonic functions on the standard Sierpiński 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.
