Santi Tasena

Abstract: This thesis is concerned with heat kernel estimates on weighted Dirichlet spaces.

The Dirichlet forms considered here are strongly local and regular. They are defined on a complete locally compact separable metric space. The associated heat equation is similar to that of local divergence form differential operators.

The weight functions studied have the form of a function of the distance from a closed set Σ, that is, x → a(d(x, Σ)). We place conditions on the geometry of the set Σ and the growth rate of function a itself. The function a can either blow up at 0 or ∞ or both. Some results include the case where Σ separates the whole spaces. It can also apply to the case where Σ do not separate the space, for example, a domain Ω and its boundary Σ = ∂Ω. The condition on Σ is rather mild and do not assume differentiability condition.