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THE HARDY AND HEISENBERG INEQUALITIES IN MORREY SPACES

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  28 March 2018

HENDRA GUNAWAN Open the ORCID record for HENDRA GUNAWAN [Opens in a new window] ,
DENNY IVANAL HAKIM ,
EIICHI NAKAI  and
YOSHIHIRO SAWANO
HENDRA GUNAWAN*
Affiliation:
Department of Mathematics, Bandung Institute of Technology, Bandung 40132, Indonesia email hgunawan@math.itb.ac.id
DENNY IVANAL HAKIM
Affiliation:
Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1 Minami Ohsawa, Hachioji, Tokyo 192-0397, Japan email dennyivanalhakim@gmail.com
EIICHI NAKAI
Affiliation:
Department of Mathematics, Ibaraki University, Mito, Ibaraki 310-8512, Japan email eiichi.nakai.math@vc.ibaraki.ac.jp
YOSHIHIRO SAWANO
Affiliation:
Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1 Minami Ohsawa, Hachioji, Tokyo 192-0397, Japan RDUN, Peoples’ Friendship University of Russia, Miklukho-Maklaya str. 6, 117198 Moscow, Russia email ysawano@tmu.ac.jp
*
✉hgunawan@math.itb.ac.id
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Abstract

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We use the Morrey norm estimate for the imaginary power of the Laplacian to prove an interpolation inequality for the fractional power of the Laplacian on Morrey spaces. We then prove a Hardy-type inequality and use it together with the interpolation inequality to obtain a Heisenberg-type inequality in Morrey spaces.

Keywords

imaginary power of Laplace operatorsfractional power of Laplace operatorsinterpolation inequalityHardy’s inequalityHeisenberg’s inequalityMorrey spaces

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 42B35: Function spaces arising in harmonic analysis
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 3 , June 2018 , pp. 480 - 491
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author was supported by the ITB Research and Innovation Program 2017. The second and third authors were supported by a Grant-in-Aid for Scientific Research (B) No. 15H03621, Japan Society for the Promotion of Science. The fourth author is supported by a Grant-in-Aid for Scientific Research (C) No. 16K05209, Japan Society for the Promotion of Science, and Peoples’ Friendship University of Russia.

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