Sharp trace inequalities for fractional Laplacians

Authors:
Amit Einav and Michael Loss

Journal:
Proc. Amer. Math. Soc. **140** (2012), 4209-4216

MSC (2010):
Primary 35A23

DOI:
https://doi.org/10.1090/S0002-9939-2012-11380-2

Published electronically:
April 5, 2012

MathSciNet review:
2957211

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The sharp trace inequality of José Escobar is extended to traces for the fractional Laplacian on , and a complete characterization of cases of equality is discussed. The proof proceeds via Fourier transform and uses Lieb's sharp form of the Hardy-Littlewood-Sobolev inequality.

**1.**Adams, David R. and Hedberg, Lars Inge: Function spaces and potential theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 314. Springer-Verlag, Berlin, 1996. MR**1411441 (97j:46024)****2.**Beckner, William: Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality. Ann. of Math. (2) 138 (1993), no. 1, 213-242. MR**1230930 (94m:58232)****3.**Caffarelli, Luis and Silvestre, Luis: An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245-1260. MR**2354493 (2009k:35096)****4.**Carlen, Eric A. and Loss, Michael: Competing symmetries of some functionals arising in mathematical physics. Stochastic processes, physics and geometry (Ascona and Locarno, 1988), 277-288, World Sci. Publ., Teaneck, NJ, 1990. MR**1124215 (92j:81338)****5.**Cotsiolis, Athanase and Tavoularis, Nikolaos K.: Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295 (2004), no. 1, 225-236. MR**2064421 (2005d:46070)****6.**Escobar, José F.: Sharp constant in a Sobolev trace inequality. Indiana Univ. Math. J. 37 (1988), no. 3, 687-698. MR**962929 (90a:46071)****7.**Lieb, Elliott H.: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. of Math. (2) 118 (1983), no. 2, 349-374. MR**717827 (86i:42010)****8.**Lieb, Elliott H. and Loss, Michael: Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. MR**1817225 (2001i:00001)****9.**Maggi, F. and Villani, Cedric: Balls have the worst best Sobolev inequalities. II. Variants and extensions. Calc. Var. Partial Differential Equations 31 (2008), no. 1, 47-74. MR**2342614 (2009f:46053)****10.**Nazaret, Bruno: Best constant in Sobolev trace inequalities on the half-space. Nonlinear Anal. 65 (2006), no. 10, 1977-1985. MR**2258478 (2007m:46047)****11.**Xiao, Jie: A sharp Sobolev trace inequality for the fractional-order derivatives. Bull. Sci. Math. 130 (2006), no. 1, 87-96. MR**2197182 (2006i:46051)**

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Additional Information

**Amit Einav**

Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160

Email:
aeinav@math.gatech.edu

**Michael Loss**

Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160

Email:
loss@math.gatech.edu

DOI:
https://doi.org/10.1090/S0002-9939-2012-11380-2

Received by editor(s):
May 20, 2011

Published electronically:
April 5, 2012

Additional Notes:
The authors were supported in part by NSF grant DMS-0901304.

Communicated by:
Michael T. Lacey

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.