Skip to main content
Log in

Fractional Gaussian estimates and holomorphy of semigroups

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abadias, L., Miana, P.J.: A subordination principle on Wright functions and regularized resolvent families. J. Funct. Spaces, Art. ID 158145, 9 pages (2015)

  2. Amann, H.: Dual semigroups and second order linear elliptic boundary value problems. Israel J. Math. 45(2–3), 225–254 (1983)

    Article  MathSciNet  Google Scholar 

  3. Arendt, W.: Heat kernels—manuscript of the 9th internet seminar 20. http://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/arendt/downloads/internetseminar.pdf (2006)

  4. Arendt, W., Batty, C.J.K.: L’holomorphie du semi-groupe engendré par le Laplacien Dirichlet sur \(L^1(\Omega )\). C. R. Acad. Sci. Paris Sér. I Math. 315(1), 31–35 (1992)

    MathSciNet  MATH  Google Scholar 

  5. Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems, volume 96 of Monographs in Mathematics. Birkhäuser/Springer Basel AG, Basel, second edition (2011)

    Book  Google Scholar 

  6. Arendt, W., Grabosch, A., Greiner, G., Groh, U., Lotz, H.P., Moustakas, U., Nagel, R., Neubrander, F., Schlotterbeck, U.: One-Parameter Semigroups of Positive Operators, volume 1184 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1986)

  7. Biccari, U., Warma, M., Zuazua, E.: Local elliptic regularity for the Dirichlet fractional Laplacian. Adv. Nonlinear Stud. 17(2), 387–409 (2017)

    Article  MathSciNet  Google Scholar 

  8. Blumenthal, R.M., Getoor, R.K.: Some theorems on stable processes. Trans. Am. Math. Soc. 95, 263–273 (1960)

    Article  MathSciNet  Google Scholar 

  9. Caffarelli, L.A., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)

    Article  MathSciNet  Google Scholar 

  10. Chen, Z.-Q., Kim, P., Song, R.: Heat kernel estimates for the Dirichlet fractional Laplacian. J. Eur. Math. Soc. (JEMS) 12(5), 1307–1329 (2010)

    Article  MathSciNet  Google Scholar 

  11. Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Relat. Fields 140(1–2), 277–317 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Claus, B., Warma, M.: Realization of the fractional Laplacian with nonlocal exterior conditions via forms method. arXiv preprint arXiv:1904.13312 (2019)

  13. Davies, E.B.: Heat Kernels and Spectral Theory, volume 92 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1990)

  14. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)

    Article  MathSciNet  Google Scholar 

  15. Dubkov, A.A., Spagnolo, B., Uchaikin, V.V.: Lévy flight superdiffusion: an introduction. Int. J. Bifur. Chaos Appl. Sci. Eng. 18(9), 2649–2672 (2008)

    Article  Google Scholar 

  16. Duong, X.T., Robinson, D.W.: Semigroup kernels, Poisson bounds, and holomorphic functional calculus. J. Funct. Anal. 142(1), 89–128 (1996)

    Article  MathSciNet  Google Scholar 

  17. Gal, C.G., Warma, M.: Fractional in time semilinear parabolic equations and applications. HAL Id: hal-01578788 (2017)

  18. Gal, C.G., Warma, M.: Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces. Comm. Partial Differ. Equ. 42(4), 579–625 (2017)

    Article  MathSciNet  Google Scholar 

  19. Gorenflo, R., Mainardi, F., Vivoli, A.: Continuous-time random walk and parametric subordination in fractional diffusion. Chaos Solitons Fractals 34(1), 87–103 (2007)

    Article  MathSciNet  Google Scholar 

  20. Grisvard, P.: Elliptic Problems in Nonsmooth Domains, volume 69 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Reprint of the 1985 original. With a foreword by Susanne C. Brenner (2011)

  21. Mandelbrot, B.B., Van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437 (1968)

    Article  MathSciNet  Google Scholar 

  22. Ouhabaz, El-M: Gaussian estimates and holomorphy of semigroups. Proc. Am. Math. Soc. 123(5), 1465–1474 (1995)

    Article  MathSciNet  Google Scholar 

  23. Ouhabaz, El-M: Invariance of closed convex sets and domination criteria for semigroups. Potential Anal. 5(6), 611–625 (1996)

    Article  MathSciNet  Google Scholar 

  24. Ouhabaz, E.-M.: Analysis of Heat Equations on Domains, volume 31 of London Mathematical Society Monographs Series. Princeton University Press, Princeton (2005)

  25. Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. (9) 101(3), 275–302 (2014)

    Article  MathSciNet  Google Scholar 

  26. Schneider, W.R.: Grey noise. In: Stochastic Processes, Physics and Geometry (Ascona and Locarno, 1988), pp. 676–681. World Scientific Publishing, Teaneck (1990)

  27. Servadei, R., Valdinoci, E.: On the spectrum of two different fractional operators. Proc. R. Soc. Edinb. Sect. A 144(4), 831–855 (2014)

    Article  MathSciNet  Google Scholar 

  28. Stewart, H.B.: Generation of analytic semigroups by strongly elliptic operators. Trans. Am. Math. Soc. 199, 141–162 (1974)

    Article  MathSciNet  Google Scholar 

  29. Stewart, H.B.: Generation of analytic semigroups by strongly elliptic operators under general boundary conditions. Trans. Am. Math. Soc. 259(1), 299–310 (1980)

    Article  MathSciNet  Google Scholar 

  30. Valdinoci, E.: From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Mat. Apl. SeMA 49, 33–44 (2009)

    MathSciNet  MATH  Google Scholar 

  31. Warma, M.: The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets. Potential Anal. 42(2), 499–547 (2015)

    Article  MathSciNet  Google Scholar 

  32. Yosida, K.: Functional Analysis. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the sixth (1980) edition

    Chapter  Google Scholar 

Download references

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

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mahamadi Warma.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-019-01381-y

Keywords

Mathematics Subject Classification

Navigation