Monotonicity Formulas for Diffusion Operators on Manifolds and Carnot Groups, Heat Kernel Asymptotics and Wiener's Criterion on Heisenberg-type Groups
Date of Award
Doctor of Philosophy (PhD)
Committee Member 1
Committee Member 2
Nung Kwan Yip
The contents of this thesis are an assortment of results in analysis and subRiemannian geometry, with a special focus on the Heisenberg group Hn, Heisenbergtype (H-type) groups, and Carnot groups.
As we wish for this thesis to be relatively self-contained, the main definitions and background are covered in Chapter 1. This includes basic information about Carnot groups, Hn, H-type groups, diffusion operators, and the curvature dimension inequality.
Chapter 2 incorporates excerpts from a paper by N. Garofalo and the author, . In it, we propose a generalization of Almgren’s frequency function N : (0, 1) → R for solutions to the sub-elliptic Laplace equation ΔHu = 0 in the unit ball of a Carnot group of arbitrary step. If the function u has vanishing discrepancy, then the frequency is monotonically non-decreasing, and we are able to prove a form of strong unique continuation for such functions.
Chapter 3 grew out of the author seeking parabolic montonicity formulas in the same vein as Almgren’s frequency. These include two types of monotonicity formulas, those of Struwe- and Poon-type , . If a diffusion operator L on a complete manifold M satisfies the curvature dimension inequality CD(ρ, n), then we are able to prove that for solutions to L u = ut in M × (0, T), Struwe’s energy monotonicity holds, at least for time values close enough to T. We introduce a new condition, C(ω) where ω ∈ C1(0, T), related to the Hessian of the heat kernel, and are able to prove a Poon-type frequency monotonicity formula when taking into account a weighting factor depending on ω. We also give examples of manifolds satisfying C(ω), the most interesting of which includes the Ornstein-Uhlenbeck operator. Monotonicity of the weighted frequency also implies a form of strong-unique continuation.
In Chapter 4, we derive asymptotics for the heat kernel on H-type groups and generalize a gradient bound from a paper of Garofalo and Segala  to these groups. This gradient bound in turn implies a strong Harnack inequality and Wiener criterion similar to those found in  and .
Rotz, Kevin L., "Monotonicity Formulas for Diffusion Operators on Manifolds and Carnot Groups, Heat Kernel Asymptotics and Wiener's Criterion on Heisenberg-type Groups" (2016). Open Access Dissertations. 700.