Walk dimension and vanishing curve modulus in metric measure spaces

2026/06/26 by Aobo Chen
#math.MG

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Abstract

On a regular local p-Dirichlet space supporting a Poincaré inequality and a capacity upper bound, we characterize the existence of curve families of positive p-modulus in terms of the small-scale behaviour of the scaling function. As an application, we show that, on an Ahlfors-regular metric measure space carrying a strongly local regular Dirichlet form with sub-Gaussian heat kernel estimates, the comparison between the walk dimension and 2 determines whether the given metric attains its Ahlfors-regular conformal dimension.

Citations

• Modulus and the Poincar� inequality on metric measure spaces
• Brownian Motion and Harmonic Analysis on Sierpinski Carpets
• Quasiconformal maps in metric spaces with controlled geometry
• On the parabolic kernel of the Schrödinger operator

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