A Generalized Polar-Coordinate Integration Formula with Applications to the Study of Convolution Powers of Complex-Valued Functions on $${\mathbb {Z}}^d$$ Z d
Abstract In this article, we consider a class of functions on $${\mathbb {R}}^d$$ R d...
| Main Authors: | , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer US
2022
|
| Online Access: | https://hdl.handle.net/1721.1/141026 |
| _version_ | 1811091837261185024 |
|---|---|
| author | Bui, Huan Q. Randles, Evan |
| author2 | Massachusetts Institute of Technology. Department of Physics |
| author_facet | Massachusetts Institute of Technology. Department of Physics Bui, Huan Q. Randles, Evan |
| author_sort | Bui, Huan Q. |
| collection | MIT |
| description | 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). |
| first_indexed | 2024-09-23T15:08:50Z |
| format | Article |
| id | mit-1721.1/141026 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T15:08:50Z |
| publishDate | 2022 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1410262023-12-06T21:11:17Z A Generalized Polar-Coordinate Integration Formula with Applications to the Study of Convolution Powers of Complex-Valued Functions on $${\mathbb {Z}}^d$$ Z d Bui, Huan Q. Randles, Evan Massachusetts Institute of Technology. Department of Physics 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). 2022-03-07T13:27:11Z 2022-03-07T13:27:11Z 2022-03-04 2022-03-05T04:40:21Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/141026 Journal of Fourier Analysis and Applications. 2022 Mar 04;28(2):19 en https://doi.org/10.1007/s00041-022-09905-x Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature application/pdf Springer US Springer US |
| spellingShingle | Bui, Huan Q. Randles, Evan A Generalized Polar-Coordinate Integration Formula with Applications to the Study of Convolution Powers of Complex-Valued Functions on $${\mathbb {Z}}^d$$ Z d |
| title | A Generalized Polar-Coordinate Integration Formula with Applications to the Study of Convolution Powers of Complex-Valued Functions on $${\mathbb {Z}}^d$$ Z d |
| title_full | A Generalized Polar-Coordinate Integration Formula with Applications to the Study of Convolution Powers of Complex-Valued Functions on $${\mathbb {Z}}^d$$ Z d |
| title_fullStr | A Generalized Polar-Coordinate Integration Formula with Applications to the Study of Convolution Powers of Complex-Valued Functions on $${\mathbb {Z}}^d$$ Z d |
| title_full_unstemmed | A Generalized Polar-Coordinate Integration Formula with Applications to the Study of Convolution Powers of Complex-Valued Functions on $${\mathbb {Z}}^d$$ Z d |
| title_short | A Generalized Polar-Coordinate Integration Formula with Applications to the Study of Convolution Powers of Complex-Valued Functions on $${\mathbb {Z}}^d$$ Z d |
| title_sort | generalized polar coordinate integration formula with applications to the study of convolution powers of complex valued functions on mathbb z d z d |
| url | https://hdl.handle.net/1721.1/141026 |
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