  In this article, we consider a class of functions 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 [20] and [21].

Classical Analysis and ODEs