On singularity of energy measures on self-similar sets

By M. Hino. 2005

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Energy inequalities for cutoff functions and some applications (2012, background): “= R |∇f|2 dµ, then is just the Riemannian metric. The pseudo-metric is not always useful. For some fractal sets such as the Sierpinski carpet the measures Γ(f,f) and m are mutually singular – see [Hi]. In these cases the only functions f satisfying the conditions of (1.18) are constants, and so i”
Energy measures and indices of Dirichlet forms, with applications to derivatives on some fractals (2009, result): “Since self-similar measures and energy measures are mutually singular in many cases (see [5, 7]), their result and ours are not comparable.”
Stability of parabolic Harnack inequalities on metric measure spaces (2006, methodology): “(In fact, we can also verify (O) in [Hin], so using Theorem 2.4 in [Hin] we can further conclude the singularity with respect to a certain class of Bernoulli type measures.) Note that this was not proved in [ Hin ] for higher dimensional carpets since at that point the elliptic Harnack inequality ha”