#### Date of Award

4-2016

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Nicola Garofalo

#### Second Advisor

Donatella Danielli

#### Committee Chair

Nicola Garofalo

#### Committee Co-Chair

Donatella Danielli

#### Committee Member 1

Fabrice Baudoin

#### Committee Member 2

Nung Kwan Yip

#### Abstract

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, [42]. In it, we propose a generalization of Almgren’s frequency function *N* : (0, 1) → R for solutions to the sub-elliptic Laplace equation Δ*Hu* = 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 [72], [67]. 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* = *u*t 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 ∈ *C*1(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 [43] to these groups. This gradient bound in turn implies a strong Harnack inequality and Wiener criterion similar to those found in [31] and [43].

#### Recommended Citation

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.

https://docs.lib.purdue.edu/open_access_dissertations/700