A Generalized Polar-Coordinate Integration Formula with Applications to the Study of Convolution Powers of Complex-Valued Functions on $${\mathbb {Z}}^d$$ Z d
Author(s)
Bui, Huan Q.; Randles, Evan
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Abstract
In this article, we consider a class of functions on
$${\mathbb {R}}^d$$
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
$${\mathbb {Z}}^d$$
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
$$\sigma _P$$
σ
P
on
$$S=\{\eta \in {\mathbb {R}}^d:P(\eta )=1\}$$
S
=
{
η
∈
R
d
:
P
(
η
)
=
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
$${\mathbb {Z}}^d$$
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 and Saloff-Coste (J Fourier Anal Appl 21(4):754–798, 2015; Rev Mat Iberoam 33(3):1045–1121, 2017).
Date issued
2022-03-04Department
Massachusetts Institute of Technology. Department of PhysicsPublisher
Springer US
Citation
Journal of Fourier Analysis and Applications. 2022 Mar 04;28(2):19
Version: Author's final manuscript