Abstract
This article discusses several definitions of the fractional Laplace operator L = — (—Δ)α/2 in Rd, also known as the Riesz fractional derivative operator; here α ∈ (0,2) and d ≥ 1. This is a core example of a nonlocal pseudo-differential operator, appearing in various areas of theoretical and applied mathematics. As an operator on Lebesgue spaces ℒp (with p ∈ [1,∞)), on the space 𝒞0 of continuous functions vanishing at infinity and on the space 𝒞bu of bounded uniformly continuous functions, L can be defined, among others, as a singular integral operator, as the generator of an appropriate semigroup of operators, by Bochner’s subordination, or using harmonic extensions. It is relatively easy to see that all these definitions agree on the space of appropriately smooth functions. We collect and extend known results in order to prove that in fact all these definitions are completely equivalent: on each of the above function spaces, the corresponding operators have a common domain and they coincide on that common domain.
Similar content being viewed by others
References
R. Bañuelos T. Kulczycki, The Cauchy process and the Steklov problem. J. Funct. Anal. 211 No 2 (2004), 355–423
R. Bañuelos T. Kulczycki, Spectral gap for the Cauchy process on convex symmetric domains. Comm. Partial Diff. Equations. 31 (2006), 1841–1878
J. Bertoin, Lévy ProcessesCambridge University Press Melbourne-New York (1998
J. Bliedtner W. Hansen, Potential theory, an analytic and probabilistic approach to balayageSpringer-Verlag Berlin-Heidelberg-New York-Tokyo (1986
R.M. Blumenthal R. K. Getoor D. B. Ray, On the distribution of first hits for the symmetric stable processes. Trans. Amer. Math. Soc. 99 (1961), 540–554
K. Bogdan K. Burdzy Z.-Q. Chen, Censored stable processes. Probab. Theory Related Fields. 127 No 1 (2003), 89–152
K. Bogdan T. Byczkowski, Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains. Studia Math. 133 No 1 (1999), 53–92
K. Bogdan T. Byczkowski T. Kulczycki M. Ryznar R. Song Z. Vondraček, Potential Analysis of Stable Processes and its Extensions. Lecture Notes in Mathematics, (1980), Springer-Verlag Berlin-Heidelberg (2009)
K. Bogdan T. Kulczycki M. Kwaśnicki, Estimates and structure of αharmonic functions. Probab. Theory Related Fields. 140 No 3-4 (2008), 345–381
K. Bogdan T. Kumagai M. Kwaśnicki, Boundary Harnack inequality for Markov processes with jumps. Trans. Amer. Math. Soc. 367 No 1 (2015), 477–517
K. Bogdan T. Zak, On Kelvin transformation. J. Theor. Prob. 19 No 1 (2006), 89–120
C. Bucur E. Valdinoci, Non-local diffusion and applications. Lecture Notes of the Unione Matematica Italiana. 20, Springer (2016
L. Caffarelli L. Silvestre, An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations. 32 No 7 (2007), 1245–1260
L. Caffarelli S. Salsa L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Inventiones Math. 171 No 2 (2008), 425–461
A. Chechkin R. Metzler J. Klafter V. Conchar, Introduction to the Theory of Lévy Flights. In R. Klages G. Radons I.M. Sokolov Anomalous Transport: Foundations and Applications Wiley-VCH Weinheim (2008
R. D. DeBlassie, The first exit time of a two-dimensional symmetric stable process from a wedge. Ann. Probab. 18 (1990), 1034–1070
R. D. DeBlassie, Higher order PDE’s and symmetric stable processes. Probab. Theory Related Fields. 129 (2004), 495–536
R. D. DeBlassie P. J. Méndez-Hernández, α-continuity properties of the symmetric α-stable process. Trans. Amer. Math. Soc. 359 (2007), 2343–2359
E. Di Nezza G. Palatucci E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 No 5 (2012), 521–573
M. Dunford J. T. Schwartz, Linear Operators. General theory. Inter-science Publ New York (1953
B. Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian. Fract. Calc. Appl. Anal. 15 No 4 (2012), 536–55510.2478/sl3540-012-0038-8https://www.degruyter.com/view/j/fca.2012.15.issue-4/issue-files/fca.2012.15.issue-4.xml
B. Dyda A. Kuznetsov M. Kwaśnicki, Fractional Laplace operator and Meijer G-function. Constr. Approx First Online 20, April 2016, 22 pp.; 10.1007/s00365-016-9336-4
E. B. Dynkin, Markov processes. I and II, Springer-Verlag Berlin-Götingen-Heidelberg (1965
D. W. Fox J. R. Kuttler, Sloshing frequencies. Z. Angew. Math. Phys. 34 (1983), 668–696
K. O. Friedrichs H. Lewy, The dock problem. Commun. Pure Appl. Math. 1 (1948), 135–148
M. Fukushima Y. Oshima M. Takeda, Dirichlet Forms and Symmetric Markov ProcessesDe Gruyter Berlin-New York (2011
J. E. Gale P. J. Miana P. R. Stinga, Extension problem and fractional operators: semigroups and wave equations. J. Evol. Equations. 13 (2013), 343–368
R. K. Getoor, First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc. 101 No 1 (1961), 75–90
I. S. Gradshteyn I. M. Ryzhik, Table of Integrals, Series and Products Academic Press (2007
Z. Hao Y. Jiao, Fractional integral on martingale hardy spaces with variable exponents. Fract. Calc. Appl. Anal. 18 No 5 (2015), 1128-114510.1515/fca-2015-0065
R. Hilfer, Experimental implications of Bochner-Lévy-Riesz diffusion. Fract. Calc. Appl. Anal. 18 No 2 (2015), 333–34110.1515/fca-2015-0022https://www.degruyter.eom/view/j/fca.2015.18.issue-5/issue-files/fca.2015.18.issue-5.xml
R. L. Holford, Short surface waves in the presence of a finite dock. I, II. Proc. Cambridge Philos. Soc. 60 (1964), 957-983 985–1011
N. Jacob, Pseudo Differential Operators and Markov Processes. 1, Imperial College Press London (2001
M. Kac, Some remarks on stable processes. Publ. Inst. Statist. Univ. Paris. 6 (1957), 303–306
V. Kokilashvili A. Meskhi H. Rafeiro S. Samko, Integral Operators in Non-standard Function Spaces. I, II, Springer-Birkhäuser (2016
V. Kozlov N. G. Kuznetsov, The ice-fishing problem: The fundamental sloshing frequency versus geometry of holes. Math. Meth. Appl. Sci. 27 (2004), 289–312
T. Kulczycki M. Kwaśnicki J. Małecki A. Stós, Spectral properties of the Cauchy process on half-line and interval. Proc. London Math. Soc. 30 No 2 (2010), 353–368
M. Kwaśnicki, Spectral analysis of subordinate Brownian motions on the half-line. Studia Math. 206 No 3 (2011), 211–271
N. S. Landkof, Foundations of Modern Potential TheorySpringer New York-Heidelberg (1972
W. Luther, Abelian and Tauberian theorems for a class of integral transforms. J. Math. Anal. Appl. 96 No 2 (1983), 365–387
C. Martínez M. Sanz, The Theory of Fractional Powers of Operators. North-Holland Math. Studies. 187, Elsevier Amsterdam (2001
T. M. Michelitsch G. A. Maugin A. F. Nowakowski F. C. G. A. Nicol-leau M. Rahman, The fractional Laplacian as a limiting case of a self-similar spring model and applications to η-dimensional anomalous diffusion. Fract. Calc. Appl. Anal. 16 No 4 (2013), 827–85910.2478/sl3540-013-0052-5 }rs https://www.degruyter.eom/ url }view/j/fca.2013.16.issue-4/issue-files/fca.2013.16.issue-4.xml
S. A. Molchanov E. Ostrowski, Symmetric stable processes as traces of degenerate diffusion processes. Theor. Prob. Appl. 14 No 1 (1969), 128–131
M. Riesz, Intégrales de Riemann-Liouville et potentiels. Acta Sci. Math. Szeged. 9 (1938), 1–42
M. Riesz, Rectification au travail “Intégrales de Riemann-Liouville et potentiels”. Acta Sci. Math. Szeged. 9 (1938), 116–118
X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey. Publ. Mat. 60 No 1 (2016), 3–26
B. Rubin, Fractional Integrals and Potentials. Monographs and Surveys in Pure and Applied Mathematics. 82 Chapman and Hall/CRC (1996).
B. Rubin, On some inversion formulas for Riesz potentials and k-plane transforms. Fract. Calc. Appl. Anal. 15 No 1 (2012), 34–4310.2478/sl3540-012-0004-5https://www.degruyter.eom/view/j/fca.2012.15.issue-l/issue-files/fca.2012.15.issue-l.xml
S. Samko, Hyper singular Integrals and Their ApplicationsCRC Press London-New York (2001
S. Samko, A note on Riesz fractional integrals in the limiting case. a(x)p(x) = n. Fract. Calc. Appl. Anal 16, No 2 (2013), 370–377 10.2478/sl3540-013-0023-xhttps://www.degruyter.eom/view/j/fca.2013.16.issue-2/issue-files/fca.2013.16.issue-2.xml
K. Sato, Lévy Processes and Infinitely Divisible DistributionsCambridge Univ. Press Cambridge (1999
R. Schilling R. Song Z. Vondraček, Bernstein Functions: Theory and Applications. Studies in Math. 37, De Gruyter Berlin (2012
K. Soni R. P. Soni, Slowly varying functions and asymptotic behavior of a class of integral transforms: I, II, III. J. Anal. Appl. 49 (1975), 166–179 477–495-612–628
F. Spitzer, Some theorems concerning 2-dimensional Brownian motion. Trans. Amer. Math. Soc. 87 (1958), 187–197
E. M. Stein, Singular Integrals And Differentiability Properties Of FunctionsPrinceton University Press Princeton (1970
P. R. Stinga J. L. Torrea, Extension problem and Harnack’s inequality for some fractional operators. Comm. Partial Diff. Equations. 35 (2010), 2092–2122
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Kwaśnicki, M. Ten Equivalent Definitions of the Fractional Laplace Operator. FCAA 20, 7–51 (2017). https://doi.org/10.1515/fca-2017-0002
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1515/fca-2017-0002