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

Abstract

For $p>1$, we prove that, for a $p$-energy on a volume doubling metric measure space, the Poincar\'e inequality and the cutoff Sobolev inequality, both with $p$-walk dimension strictly larger than $p$, imply that the associated $p$-energy measure is singular with respect to the underlying measure. Under the slow volume regularity condition, we further prove that these two inequalities are equivalent to the resistance estimate; in particular, as part of the proof, we give a simple and direct derivation of the cutoff Sobolev inequality from the Poincar\'e inequality and the capacity upper bound. As a direct corollary, for a large family of fractals and metric measure spaces, including the Sierpi\'nski gasket and the Sierpi\'nski carpet, the $p$-energy measure is singular with respect to the underlying measure for any $p$ strictly greater than the Ahlfors regular conformal dimension.