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This paper continues the analysis, started in [2, 3], of a class ofdegenerate elliptic operators defined on manifolds with corners, which arise inPopulation Biology. Using techniques pioneered by J. Moser, and extended andrefined by L. Saloff-Coste, Grigoryan, and Sturm, we show that weak solutionsto the parabolic problem defined by a sub-class of these operators, whichconsists of those that can be defined by Dirichlet forms and have non-vanishingtransverse vector field, satisfy a Harnack inequality. This allows us toconclude that the solutions to these equations belong, for positive times, tothe natural anisotropic Holder spaces, and also leads to upper and, in somecases, lower bounds for the heat kernels of these operators. These resultsimply that these operators have a compact resolvent when acting on $C^0$ or$L^2.$ The proof relies upon a scale invariant Poincare inequality that weestablish for a large class of weighted Dirichlet forms, as well as estimatesto handle certain mildly singular perturbation terms. The weights that weconsider are neither Ahlfors regular, nor do they generally belong to theMuckenhaupt class $A_2.$

Harnack Inequalities and Heat-kernel Estimates for Degenerate Diffusion Operators Arising in Population Biology