On the harmonic extension approach to fractional powers in Banach spaces

References

[1] NIST Digital Library ofMathematical Functions. Version 1.0.22, visited 09.05.2019, http://dlmf.nist.gov/.Suche 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), 1–24.10.1080/03605302.2017.1363229Suche in Google Scholar

[3] A. V. Balakrishnan, An operational calculus for infinitesimal generators of semigroups. Trans. Amer. Math. Soc. 91 (1959), 330–353.Suche 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.1960.10.419Suche in Google Scholar

[5] S. Bochner, Diffusion equation and stochastic processes. Proc. Nat. Acad. Sciences35 (1949), 368–370.10.1073/pnas.35.7.368Suche 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), 1245–1260.10.1080/03605300600987306Suche 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.1993.1070Suche 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/BF02921599Suche 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).Suche 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.Suche 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/s00028-013-0182-6Suche in Google Scholar

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

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

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

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

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

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

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

[20] A. McIntosh, Operators which have an H_∞ functional calculus. Miniconference on Operator Theory and Partial Differential Equations (1986), 210–231.Suche 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).Suche 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), 128–131.10.1137/1114012Suche 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/S0002-9947-1958-0096136-8Suche 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.1952.2.343Suche 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).Suche 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), 2092–2122.10.1080/03605301003735680Suche 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/140Suche in Google Scholar

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