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Fractional Gaussian estimates and holomorphy of semigroups

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.

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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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Correspondence to Mahamadi Warma.

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Keyantuo, V., Seoanes, F. & Warma, M. Fractional Gaussian estimates and holomorphy of semigroups. Arch. Math. 113, 629–647 (2019). https://doi.org/10.1007/s00013-019-01381-y

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Keywords

  • Fractional Laplace operator
  • Fractional heat equation
  • Semigroup
  • Fractional Gaussian estimates
  • Holomorphy

Mathematics Subject Classification

  • 35R11
  • 47D06
  • 47D03