On singularity of $p$-energy measures on metric measure spaces

Abstract: For $p\in(1,+\infty)$, we prove that for a $p$-energy on a metric measure space, under the volume doubling condition, the conjunction of the Poincar\'e inequality and the cutoff Sobolev inequality both with $p$-walk dimension strictly larger than $p$ implies the singularity of the associated $p$-energy measure with respect to the underlying measure. We also prove that under the slow volume regular condition, the conjunction of the Poincar\'e inequality and the cutoff Sobolev inequality is equivalent to the resistance estimate. As a direct corollary, on a large family of fractals and metric measure spaces, including the Sierpi\'nski gasket and the Sierpi\'nski carpet, we obtain the singularity of the $p$-energy measure with respect to the underlying measure for all $p$ strictly great than the Ahlfors regular conformal dimension.