## Duke Mathematical Journal

### Integrability of oscillatory functions on local fields: Transfer principles

#### Abstract

For oscillatory functions on local fields coming from motivic exponential functions, we show that integrability over ${\mathbb{Q}}_{p}^{n}$ implies integrability over ${\mathbb{F}}_{p}{((t))}^{n}$ for large $p$, and vice versa. More generally, the integrability only depends on the isomorphism class of the residue field of the local field, once the characteristic of the residue field is large enough. This principle yields general local integrability results for Harish-Chandra characters in positive characteristic as we show in other work. Transfer principles for related conditions such as boundedness and local integrability are also obtained. The proofs rely on a thorough study of loci of integrability, to which we give a geometric meaning by relating them to zero loci of functions of a specific kind.

#### Article information

Source
Duke Math. J., Volume 163, Number 8 (2014), 1549-1600.

Dates
First available in Project Euclid: 26 May 2014

https://projecteuclid.org/euclid.dmj/1401146372

Digital Object Identifier
doi:10.1215/00127094-2713482

Mathematical Reviews number (MathSciNet)
MR3210968

Zentralblatt MATH identifier
1327.14073

#### Citation

Cluckers, Raf; Gordon, Julia; Halupczok, Immanuel. Integrability of oscillatory functions on local fields: Transfer principles. Duke Math. J. 163 (2014), no. 8, 1549--1600. doi:10.1215/00127094-2713482. https://projecteuclid.org/euclid.dmj/1401146372

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