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Ten equivalent definitions of the fractional Laplace operator. (English) Zbl 1375.47038
Summary: This article discusses several definitions of the fractional Laplace operator $$L=-(-\Delta)^{\alpha/2}$$ in $$\mathbb{R}^d$$, also known as the Riesz fractional derivative operator; here $$\alpha\in (0,2)$$ and $$d\geq 1$$. 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 $$\mathcal L^p$$ (with $$p\in [1,\infty))$$, on the space $$\mathcal C_0$$ of continuous functions vanishing at infinity and on the space $$\mathcal C_{bu}$$ 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.

##### MSC:
 47G30 Pseudodifferential operators 35S05 Pseudodifferential operators as generalizations of partial differential operators 60J35 Transition functions, generators and resolvents
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##### References:
 [1] R. Bañuelos, T. Kulczycki, The Cauchy process and the Steklov problem. J. Funct. Anal. 211, No 2 (2004), 355-423. · Zbl 1055.60072 [2] R. Bañuelos, T. Kulczycki, Spectral gap for the Cauchy process on convex symmetric domains. Comm. Partial Diff. Equations31 (2006), 1841-1878. · Zbl 1123.60029 [3] J. Bertoin, Lévy Processes. Cambridge University Press, Melbourne-New York (1998). [4] J. Bliedtner, W. Hansen, Potential theory, an analytic and probabilistic approach to balayage. Springer-Verlag, Berlin-Heidelberg-New York-Tokyo (1986). · Zbl 0706.31001 [5] 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. · Zbl 0118.13005 [6] K. Bogdan, K. Burdzy, Z.-Q. Chen, Censored stable processes. Probab. Theory Related Fields127, No 1 (2003), 89-152. · Zbl 1032.60047 [7] K. Bogdan, T. Byczkowski, Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains. Studia Math. 133, No 1 (1999), 53-92. · Zbl 0923.31003 [8] 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). [9] K. Bogdan, T. Kulczycki, M. Kwaśnicki, Estimates and structure of αharmonic functions. Probab. Theory Related Fields140, No 3-4 (2008), 345-381. · Zbl 1146.31004 [10] K. Bogdan, T. Kumagai, M. Kwaśnicki, Boundary Harnack inequality for Markov processes with jumps. Trans. Amer. Math. Soc. 367, No 1 (2015), 477-517. · Zbl 1309.60080 [11] K. Bogdan, T. Zak, On Kelvin transformation. J. Theor. Prob. 19, No 1 (2006), 89-120. · Zbl 1105.60057 [12] C. Bucur, E. Valdinoci, Non-local diffusion and applications. Lecture Notes of the Unione Matematica Italiana 20, Springer (2016). · Zbl 1377.35002 [13] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations32, No 7 (2007), 1245-1260. · Zbl 1143.26002 [14] 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. · Zbl 1148.35097 [15] A. Chechkin, R. Metzler, J. Klafter, V. Conchar, Introduction to the Theory of Lévy Flights. In: R. Klages, G. Radons, I.M. Sokolov (Eds), Anomalous Transport: Foundations and Applications, Wiley-VCH, Weinheim (2008). [16] R. D. DeBlassie, The first exit time of a two-dimensional symmetric stable process from a wedge. Ann. Probab. 18 (1990), 1034-1070. · Zbl 0709.60075 [17] R. D. DeBlassie, Higher order PDE’s and symmetric stable processes. Probab. Theory Related Fields129 (2004), 495-536. · Zbl 1060.60077 [18] R. D. DeBlassie, P. J. Méndez-Hernández, α-continuity properties of the symmetric α-stable process. Trans. Amer. Math. Soc. 359 (2007), 2343-2359. · Zbl 1115.60072 [19] E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, No 5 (2012), 521-573. · Zbl 1252.46023 [20] M. Dunford, J. T. Schwartz, Linear Operators. General theory. Inter-science Publ., New York (1953). · Zbl 0084.10402 [21] B. Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian. Fract. Calc. Appl. Anal. 15, No 4 (2012), 536-555; ; · Zbl 1312.35176 [22] B. Dyda, A. Kuznetsov, M. Kwaśnicki, Fractional Laplace operator and Meijer G-function. Constr. Approx., First Online: 20 April 2016, 22 pp.; · Zbl 1365.35204 [23] E. B. Dynkin, Markov processes, Vols. I and II. Springer-Verlag, Berlin-Götingen-Heidelberg (1965). · Zbl 0132.37901 [24] D. W. Fox, J. R. Kuttler, Sloshing frequencies. Z. Angew. Math. Phys. 34 (1983), 668-696. · Zbl 0539.76022 [25] K. O. Friedrichs, H. Lewy, The dock problem. Commun. Pure Appl. Math. 1 (1948), 135-148. · Zbl 0030.37902 [26] M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and SymmetricMarkov Processes. De Gruyter, Berlin-New York (2011). [27] J. E. Gale, P. J. Miana, P. R. Stinga, Extension problem and fractional operators: semigroups and wave equations. J. Evol. Equations13 (2013), 343-368. · Zbl 1336.47049 [28] R. K. Getoor, First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc. 101, No 1 (1961), 75-90. · Zbl 0104.11203 [29] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products. Academic Press (2007). · Zbl 1208.65001 [30] Z. Hao, Y. Jiao, Fractional integral on martingale hardy spaces with variable exponents. Fract. Calc. Appl. Anal. 18, No 5 (2015), 1128-1145; ; · Zbl 1341.42027 [31] R. Hilfer, Experimental implications of Bochner-Lévy-Riesz diffusion. Fract. Calc. Appl. Anal. 18, No 2 (2015), 333-341; ; · Zbl 06418051 [32] 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. [33] N. Jacob, Pseudo Differential Operators and Markov Processes, Vol. 1. Imperial College Press, London (2001). · Zbl 0987.60003 [34] M. Kac, Some remarks on stable processes. Publ. Inst. Statist. Univ. Paris 6 (1957), 303-306. · Zbl 0082.13001 [35] V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral Operators in Non-standard Function Spaces, Vol. I, II. Springer-Birkhäuser (2016). · Zbl 1385.47001 [36] 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. · Zbl 1042.35038 [37] 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. [38] M. Kwaśnicki, Spectral analysis of subordinate Brownian motions on the half-line. Studia Math. 206, No 3 (2011), 211-271. · Zbl 1241.60023 [39] N. S. Landkof, Foundations of Modern Potential Theory. Springer, New York-Heidelberg (1972). · Zbl 0253.31001 [40] W. Luther, Abelian and Tauberian theorems for a class of integral transforms. J. Math. Anal. Appl. 96, No 2 (1983), 365-387. · Zbl 0523.44002 [41] C. Martínez, M. Sanz, The Theory of Fractional Powers of Operators. North-Holland Math. Studies 187, Elsevier, Amsterdam (2001). [42] 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), 827-859; ; · Zbl 1314.35209 [43] S. A. Molchanov, E. Ostrowski, Symmetric stable processes as traces of degenerate diffusion processes. Theor. Prob. Appl. 14, No 1 (1969), 128-131. · Zbl 0281.60091 [44] M. Riesz, Intégrales de Riemann-Liouville et potentiels. Acta Sci. Math. Szeged 9 (1938), 1-42. · JFM 64.0476.03 [45] M. Riesz, Rectification au travail “Intégrales de Riemann-Liouville et potentiels”. Acta Sci. Math. Szeged 9 (1938), 116-118. · JFM 65.1272.03 [46] X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey. Publ. Mat. 60, No 1 (2016), 3-26. · Zbl 1337.47112 [47] B. Rubin, Fractional Integrals and Potentials. Monographs and Surveys in Pure and Applied Mathematics 82, Chapman and Hall/CRC (1996). [48] B. Rubin, On some inversion formulas for Riesz potentials and k-plane transforms. Fract. Calc. Appl. Anal. 15, No 1 (2012), 34-43; ; · Zbl 1285.47052 [49] S. Samko, Hyper singular Integrals and Their Applications. CRC Press, London-New York (2001). [50] 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; ; · Zbl 1312.47045 [51] K. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, Cambridge (1999). [52] R. Schilling, R. Song, Z. Vondraček, Bernstein Functions: Theory and Applications. Studies in Math. 37, De Gruyter, Berlin (2012). [53] 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. · Zbl 0314.44005 [54] F. Spitzer, Some theorems concerning 2-dimensional Brownian motion. Trans. Amer. Math. Soc. 87 (1958), 187-197. · Zbl 0089.13601 [55] E. M. Stein, Singular Integrals And Differentiability Properties Of Functions. Princeton University Press, Princeton (1970). · Zbl 0207.13501 [56] P. R. Stinga, J. L. Torrea, Extension problem and Harnack’s inequality for some fractional operators. Comm. Partial Diff. Equations 35 (2010), 2092-2122. · Zbl 1209.26013
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