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Global \(L^q\)-Gevrey Functions and Their Applications

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Abstract

The goal of this paper is to introduce a class of \(C^\infty \) functions derivatives of which satisfy quantitative size estimates. The estimates, called global \(L^q\) Gevrey estimates, first arose in the work of Boggess and Raich ( J. Fourier Anal. Appl. 19:180–224, 2013) when they investigated how to capture a particular type of exponential decay through estimates on the Fourier transform. In the present work, we refine the notion of global \(L^q\)-Gevrey functions and include a discussion of the function theory as well as the relationship to Gevrey classes and known function spaces. In addition, we present explicit examples of global \(L^q\)-Gevrey functions and ways to generate new global \(L^q\)-Gevrey functions from old ones. We conclude with three applications: The first is solving a Carleman-type problem for constructing functions derivatives of which are a given sequence of global \(L^q\)-Gevrey functions. The other two applications concern extensions of a given global \(L^q\)-Gevrey function: the first is constructing an almost analytic extension, and the second is building an approximate solution to a first-order complex vector field coefficients of which are global \(L^q\)-Gevrey functions.

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Acknowledgments

The second author was partially supported by FAPESP and CNPq. The third author was partially supported by NSF Grant DMS-1405100. He also gratefully acknowledges the support provided by Erwin Schrödinger Institute during his stay there.

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Authors and Affiliations

  1. American University in Dubai, P.O. Box 28282, Dubai, UAE

    Ziad Adwan

  2. Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP, 13565-905, Brazil

    Gustavo Hoepfner

  3. Department of Mathematical Sciences, 1 University of Arkansas, SCEN 327, Fayetteville, AR, 72701, USA

    Andrew Raich

Authors
  1. Ziad Adwan
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  2. Gustavo Hoepfner
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  3. Andrew Raich
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Correspondence to Andrew Raich.

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Adwan, Z., Hoepfner, G. & Raich, A. Global \(L^q\)-Gevrey Functions and Their Applications. J Geom Anal 27, 1874–1913 (2017). https://doi.org/10.1007/s12220-016-9743-6

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  • DOI: https://doi.org/10.1007/s12220-016-9743-6

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