Skip to main content
Article
Licensed
Unlicensed Requires Authentication

On the harmonic extension approach to fractional powers in Banach spaces

  • EMAIL logo and
Published/Copyright: September 11, 2020

Abstract

We show that fractional powers of general sectorial operators on Banach spaces can be obtained by the harmonic extension approach. Moreover, for the corresponding second order ordinary differential equation with incomplete data describing the harmonic extension we prove existence and uniqueness of a bounded solution (i.e., of the harmonic extension).

References

[1] NIST Digital Library ofMathematical Functions. Version 1.0.22, visited 09.05.2019, http://dlmf.nist.gov/.Search in Google Scholar

[2] W. Arendt, A. F. M. ter Elst, and M. Warma, Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator. Comm. Partial Differential Equations43, No 1 (2018), 1–24.10.1080/03605302.2017.1363229Search in Google Scholar

[3] A. V. Balakrishnan, An operational calculus for infinitesimal generators of semigroups. Trans. Amer. Math. Soc. 91 (1959), 330–353.Search in Google Scholar

[4] A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them. Pacific J. Math. 10, No 2 (1960), 419–437.10.2140/pjm.1960.10.419Search in Google Scholar

[5] S. Bochner, Diffusion equation and stochastic processes. Proc. Nat. Acad. Sciences35 (1949), 368–370.10.1073/pnas.35.7.368Search in Google Scholar PubMed PubMed Central

[6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations32, No 8 (2007), 1245–1260.10.1080/03605300600987306Search in Google Scholar

[7] R. deLaubenfels, Unbounded holomorphic functional calculus and abstract Cauchy problems for operators with polynomial bounded resolvents. J. Funct. Anal. 114, No 2 (1993), 348–394.10.1006/jfan.1993.1070Search in Google Scholar

[8] X.T. Duong and A. McIntosh, Functional calculi of second-order elliptic partial differential operators with bounded measurable coefficients. J. Geom. Anal. 6, No 2 (1996), 181–205.10.1007/BF02921599Search in Google Scholar

[9] K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations. Volume 194 of Grad. Texts in Math., Springer, New York-Berlin-Heidelberg (1999).Search in Google Scholar

[10] W. Feller, On a generalization of Marcel Riesz' potentials and the semi-groups generated by them. Comm. Sém. Mathém. Université de Lund (1952), 73–81.Search in Google Scholar

[11] J. E. Galé, P. J. Miana, and P. R. Stinga, Extension problem and fractional operators: Semigroups and wave equations. J. Evol. Equ. 13, No 2 (2013), 343–368.10.1007/s00028-013-0182-6Search in Google Scholar

[12] M. Haase, The Functional Calculus for Sectorial Operators. Ser. Operator Theory: Advances and Applications, Birkhäuser, Basel (2006).Search in Google Scholar

[13] M. Haase, Lectures on Functional Calculus. Lecture Notes of the 21st Internet Seminar (2018), https://www.math.uni-kiel.de/isem21/en/course/phase1/isem21-lectures-on-functional-calculus.Search in Google Scholar

[14] E. Hille, Functional Analysis and Semigroups. Volume 31 of Amer. Math. Soc. Colloquium Publications, de Gruyter (1948).Search in Google Scholar

[15] T. Kato, Note on fractional powers of linear operators. Proc. Japan Acad. 36 (1960), 94–96.10.3792/pja/1195524082Search in Google Scholar

[16] T. Kato, Perturbation Theory for Linear Operators. Volume 132 of Grundlagen der mathematischen Wissenschaften, Springer, 2nd Ed. (1980).Search in Google Scholar

[17] H. Komatsu, Fractional powers of operators, III negative powers. J. Math. Soc. Japan21 (1969), 205–220.10.2969/jmsj/02120205Search in Google Scholar

[18] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems. Modern Birkhäuser Classics, Springer, Basel (1995).10.1007/978-3-0348-9234-6Search in Google Scholar

[19] C. Martinez and M. Sanz, The Theory of Fractional Powers of Operators. North-Holland Mathematics Studies, Elsevier Sci. (2001).Search in Google Scholar

[20] A. McIntosh, Operators which have an H functional calculus. Miniconference on Operator Theory and Partial Differential Equations (1986), 210–231.Search in Google Scholar

[21] J. Meichsner and C. Seifert, Fractional powers of non-negative operators in Banach spaces via the Dirichlet-to-Neumann operator. arXiv Preprint, https://arxiv.org/abs/1704.01876 (2017).Search in Google Scholar

[22] S. A. Molchanow and E. Ostrovskii, Symmetric stable processes as traces of degenerate diffusion processes. Theory Probab. Appl. 14, No 1 (1968), 128–131.10.1137/1114012Search in Google Scholar

[23] E. Nelson, A functional calculus using singular Laplace integrals. Trans. Amer. Math. Soc. 88, No 2 (1958), 400–413.10.1090/S0002-9947-1958-0096136-8Search in Google Scholar

[24] R. S. Phillips, On the generation of semigroups of linear operators. Pacific J. Math. 2 (1952), 343–369.10.2140/pjm.1952.2.343Search in Google Scholar

[25] R. L. Schilling, R. Song, and Z. Vondracek, Bernstein Functions: Theory and Applications. Volume 37 of de Gruyter Stud. Math., de Gruyter, 2nd Ed. (2012).Search in Google Scholar

[26] P. R. Stinga and J. L. Torrea, Extension problem and Harnack’s inequality for some fractional operators. Comm. Partial Differential Equations35, No 11 (2010), 2092–2122.10.1080/03605301003735680Search in Google Scholar

[27] G. Teschl, Ordinary Differential Equations and Dynamical Systems. Graduate Studies in Math., American Mathematical Society, 10th Ed. (2011).10.1090/gsm/140Search in Google Scholar

[28] I. Wood, Maximal Lp-regularity for the Laplacian on Lipschitz domains. Math. Z. 255, No 4 (2007), 855–875.10.1007/s00209-006-0055-6Search in Google Scholar

Received: 2019-05-24
Revised: 2020-07-09
Published Online: 2020-09-11
Published in Print: 2020-08-26

© 2020 Diogenes Co., Sofia

Downloaded on 22.4.2026 from https://www.degruyterbrill.com/document/doi/10.1515/fca-2020-0055/html
Scroll to top button