Variation and oscillation for harmonic operators in the inverse Gaussian setting

Víctor Almeida; Jorge J. Betancor

doi:10.3934/cpaa.2021183

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2022, Volume 21, Issue 2: 419-470. Doi: 10.3934/cpaa.2021183

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Variation and oscillation for harmonic operators in the inverse Gaussian setting

Víctor Almeida^, and
Jorge J. Betancor

Departamento de Análisis Matemático, Universidad de La Laguna, Campus de Anchieta, Avda. Astrofísico Sánchez, s/n, 38721 La Laguna (Sta. Cruz de Tenerife), Spain

^* Corresponding Author
^* Corresponding Author

Received: March 2021

Revised: September 2021

Early access: November 2021

Published: February 2022

This paper is partially supported by grant PID2019-106093GB-I00 from the Spanish Government

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We prove variation and oscillation $ L^p $-inequalities associated with fractional derivatives of certain semigroups of operators and with the family of truncations of Riesz transforms in the inverse Gaussian setting. We also study these variational $ L^p $-inequalities in a Banach-valued context by considering Banach spaces with the UMD-property and whose martingale cotype is fewer than the variational exponent. We establish $ L^p $-boundedness properties for weighted difference involving the semigroups under consideration.

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Mathematics Subject Classification: Primary: 43A85; Secondary: 42B20, 42B25.

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