Parabolic Harnack inequality on fractal-type metric measure Dirichlet spaces

Lierl, Janna

doi:10.4171/RMI/1001

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Revista Matemática Iberoamericana

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Volume 34, Issue 2, 2018, pp. 687–738

DOI: 10.4171/RMI/1001

Published online: 2018-05-28

Parabolic Harnack inequality on fractal-type metric measure Dirichlet spaces

Janna Lierl^[1]

(1) University of Connecticut, Storrs, USA

This paper proves the strong parabolic Harnack inequality for local weak solutions to the heat equation associated with time-dependent (nonsymmetric) bilinear forms. The underlying metric measure Dirichlet space is assumed to satisfy the volume doubling condition, the strong Poincaré inequality, and a cutoff Sobolev inequality. The metric is not required to be geodesic. Further results include a weighted Poincaré inequality, as well as upper and lower bounds for non-symmetric heat kernels.

Keywords: Parabolic Harnack inequality, Moser iteration, weighted Poincaré inequality, heat kernel estimates, non-symmetric Dirichlet form, fractals

Lierl Janna: Parabolic Harnack inequality on fractal-type metric measure Dirichlet spaces. Rev. Mat. Iberoam. 34 (2018), 687-738. doi: 10.4171/RMI/1001