Fetching the paper…
Reading the bibliography…
This article reviews several definitions of the fractional Laplace operator (-Delta)^{alpha/2} (0 < alpha < 2) in R^d, also known as the Riesz fractional derivative operator, as an operator on Lebesgue spaces L^p, on the space C_0 of continuous functions vanishing at infinity and on the space C_{bu} of bounded uniformly continuous functions.
M. Riesz, Intégrales de Riemann–Liouville et potentiels . Acta Sci. Math. Szeged 9 (1938): 1–42
1938
Earlier work this paper cites.
M. Riesz, Rectification au travail “Intégrales de Riemann–Liouville et potentiels” . Acta Sci. Math. Szeged 9 (1938): 116–118
1938
Earlier work this paper cites.
K. O. Friedrichs, H. Lewy. The dock problem . Commun. Pure Appl. Math. 1 (1948): 135–148
1948
Earlier work this paper cites.
M. Dunford, J. T. Schwartz, Linear Operators. General theory . Interscience Publ., 1953
1953
Earlier work this paper cites.
M. Kac, Some remarks on stable processes . Publ. Inst. Statist. Univ. Paris 6 (1957): 303–306
1957
Earlier work this paper cites.
F. Spitzer, Some theorems concerning 2-dimensional Brownian motion . Trans. Amer. Math. Soc. 87 (1958): 187–197
1958
Earlier work this paper cites.
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
1961
Earlier work this paper cites.
R. K. Getoor, First Passage Times for Symmetric Stable Processes in Space . Trans. Amer. Math. Soc. 101(1) (1961): 75–90
1961
Earlier work this paper cites.
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
1964
Earlier work this paper cites.
E. B. Dynkin, Markov processes, Vols. I and II . Springer-Verlag, Berlin-Götingen-Heidelberg, 1965
1965
Earlier work this paper cites.
S. A. Molchanov, E. Ostrowski, Symmetric stable processes as traces of degenerate diffusion processes . Theor. Prob. Appl. 14(1) (1969): 128–131
1969
Earlier work this paper cites.
E. M. Stein, Singular Integrals And Differentiability Properties Of Functions . Princeton University Press, 1970
1970
Earlier work this paper cites.
N. S. Landkof, Foundations of Modern Potential Theory . Springer, New York–Heidelberg, 1972
1972
Earlier work this paper cites.
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
1975
Earlier work this paper cites.
D. W. Fox, J. R. Kuttler, Sloshing frequencies . Z. Angew. Math. Phys. 34 (1983): 668–696
1983
Earlier work this paper cites.
W. Luther, Abelian and Tauberian theorems for a class of integral transforms . J. Math. Anal. Appl. 96(2) (1983): 365–387
1983
Earlier work this paper cites.
J. Bliedtner, W. Hansen, Potential theory, an analytic and probabilistic approach to balayage . Springer-Verlag, 1986
1986
Cited alongside, same era.
R. D. DeBlassie, The first exit time of a two-dimensional symmetric stable process from a wedge . Ann. Probab. 18 (1990), pp. 1034–1070
1990
Cited alongside, same era.
B. Rubin, Fractional Integrals and Potentials . Monographs and Surveys in Pure and Applied Mathematics 82, Chapman and Hall/CRC, 1996
1996
Cited alongside, same era.
J. Bertoin, Lévy Processes . Cambridge University Press, Melbourne-New York, 1998
1998
Cited alongside, same era.
K. Bogdan, T. Byczkowski, Potential theory for the α \alpha -stable Schrödinger operator on bounded Lipschitz domains . Studia Math. 133(1) (1999): 53–92
1999
Cited alongside, same era.
R. D. DeBlassie, P. J. Méndez-Hernández, α \alpha -continuity properties of the symmetric α \alpha -stable process . Trans. Amer. Math. Soc. 359 (2007), pp. 2343–2359
2007
Later among the works it cites.
I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products . Academic Press, 2007
2007
Later among the works it cites.
K. Bogdan, T. Kulczycki, M. Kwaśnicki, Estimates and structure of α \alpha -harmonic functions . Probab. Theory Related Fields 140(3–4) (2008): 345–381
2008
Later among the works it cites.
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(2) (2008): 425–461
2008
Later among the works it cites.
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, 2009
alphaXiv searches the wider corpus for related work and actual follow-ups.
alphaXiv is searching for related work…
K. Sato, Lévy Processes and Infinitely Divisible Distributions . Cambridge Univ. Press, Cambridge, 1999
1999
Cited alongside, same era.
N. Jacob, Pseudo Differential Operators and Markov Processes. Vol. 1 . Imperial College Press, London, 2001
2001
Cited alongside, same era.
C. Martínez, M. Sanz, The Theory of Fractional Powers of Operators . North-Holland Math. Studies 187, Amsterdam, 2001
2001
Cited alongside, same era.
S. Samko, Hypersingular Integrals and Their Applications . CRC Press, 2001
2001
Cited alongside, same era.
K. Bogdan, K. Burdzy, Z.-Q. Chen, Censored stable processes . Probab. Theory Related Fields 127(1) (2003): 89–152
2003
Cited alongside, same era.
R. Bañuelos, T. Kulczycki, The Cauchy process and the Steklov problem . J. Funct. Anal. 211(2) (2004): 355–423
2004
Cited alongside, same era.
R. D. DeBlassie, Higher order PDE’s and symmetric stable processes . Probab. Theory Related Fields 129 (2004), pp. 495–536
2004
Cited alongside, same era.
2009
Later among the works it cites.
T. Kulczycki, M. Kwaśnicki, J. Małecki, A. Stós, Spectral Properties of the Cauchy Process on Half-line and Interval . Proc. London Math. Soc. 30(2) (2010): 353–368
2010
Later among the works it cites.
M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes . De Gruyter, 2011
2011
Later among the works it cites.
M. Kwaśnicki, Spectral analysis of subordinate Brownian motions on the half-line . Studia Math. 206(3) (2011): 211–271
2011
Later among the works it cites.
B. Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian . Fract. Calc. Appl. Anal. 15(4) (2012): 536–555
2012
Later among the works it cites.
R. Schilling, R. Song, Z. Vondraček, Bernstein Functions: Theory and Applications . De Gruyter, Studies in Math. 37, Berlin, 2012
2012
Later among the works it cites.
J. E. Galé, P. J. Miana, P. R. Stinga, Extension problem and fractional operators: semigroups and wave equations. J. Evol. Equations 13 (2013): 343–368
2013
Later among the works it cites.
K. Bogdan, T. Kumagai, M. Kwaśnicki, Boundary Harnack inequality for Markov processes with jumps . Trans. Amer. Math. Soc. 367(1) (2015): 477–517
2015
Closest in time.
C. Bucur, E. Valdinoci, Non-local diffusion and applications . Preprint, 2015, arXiv:1504.08292
2015
Closest in time.
2015
Closest in time.
P. R. Stinga, J. L. Torrea, Extension Problem and Harnack’s Inequality for Some Fractional Operators. Comm. Partial Diff. Equations 35 (2010): 2092–2122
2092
Closest in time.