#### Date of Award

2021

#### Document Type

Honors Thesis (Open Access)

#### Department

Colby College. Mathematics and Statistics Dept.

#### Advisor(s)

Evan Randles

#### Second Advisor

Leo Livshits

#### Abstract

In this thesis, we consider a class of function on $\mathbb{R}^d$, called positive homogeneous functions, which interact well with certain continuous one-parameter groups of (generally anisotropic) dilations. Generalizing the Euclidean norm, positive homogeneous functions appear naturally in the study of convolution powers of complex-valued functions on $\mathbb{Z}^d$. As the spherical measure is a Radon measure on the unit sphere which is invariant under the symmetry group of the Euclidean norm, to each positive homogeneous function $P$, we construct a Radon measure $\sigma_P$ on $S=\{\eta \in \mathbb{R}^d:P(\eta)=1\}$ which is invariant under the symmetry group of $P$. With this measure, we prove a generalization of the classical polar-coordinate integration formula and deduce a number of corollaries in this setting. We then turn to the study of convolution powers of complex functions on $\mathbb{Z}^d$ and certain oscillatory integrals which arise naturally in that context. Armed with our integration formula and the Van der Corput lemma, we establish sup-norm-type estimates for convolution powers; this result is new and partially extends results of (Randles, 2015) and (Randles, 2017). Much of this thesis is based on the article (Bui, 2021).

#### Keywords

Polar-coordinate Integration Formula, Spherical Measure, Oscillatory Integrals, Convolution Powers

#### Recommended Citation

Bui, Huan Q., "A Generalized Polar-coordinate Integration Formula, Oscillatory Integral Techniques, and Applications to Convolution Powers of Complex-valued Functions on $\mathbb{Z}^d$" (2021).*Honors Theses.*Paper 1293.

https://digitalcommons.colby.edu/honorstheses/1293

#### Included in

Analysis Commons, Ordinary Differential Equations and Applied Dynamics Commons, Partial Differential Equations Commons