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Fractional Gaussian estimates and holomorphy of semigroups

  • Valentin Keyantuo
  • Fabian Seoanes
  • Mahamadi WarmaEmail author
Article
  • 16 Downloads

Abstract

Let \(\Omega \subset {\mathbb {R}}^N\) be an arbitrary open set, \(0<s<1\) and denote by \((e^{-t(-\Delta )_{{{\mathbb {R}}}^N}^s})_{t\ge 0}\) the semigroup on \(L^2({{\mathbb {R}}}^N)\) generated by the fractional Laplace operator. In the first part of the paper, we show that if T is a self-adjoint semigroup on \(L^2(\Omega )\) satisfying a fractional Gaussian estimate in the sense that \(|T(t)f|\le Me^{-bt(-\Delta )_{{{\mathbb {R}}}^N}^s}|f|\), \(0\le t \le 1\), \(f\in L^2(\Omega )\), for some constants \(M\ge 1\) and \(b\ge 0\), then T defines a bounded holomorphic semigroup of angle \(\frac{\pi }{2}\) that interpolates on \(L^p(\Omega )\), \(1\le p<\infty \). Using a duality argument, we prove that the same result also holds on the space of continuous functions. In the second part, we apply the above results to the realization of fractional order operators with the exterior Dirichlet conditions.

Keywords

Fractional Laplace operator Fractional heat equation Semigroup Fractional Gaussian estimates Holomorphy 

Mathematics Subject Classification

35R11 47D06 47D03 

Notes

Acknowledgements

The work of the authors is partially supported by the Air Force Office of Scientific Research (AFOSR) under Award No.: FA9550-18-1-0242

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Authors and Affiliations

  1. 1.Department of Mathematics Faculty of Natural SciencesUniversity of Puerto Rico Rio Piedras CampusSan JuanUSA

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