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Optimal Sobolev trace embeddings


Authors: Andrea Cianchi and Luboš Pick
Journal: Trans. Amer. Math. Soc. 368 (2016), 8349-8382
MSC (2010): Primary 46E35, 46E30
DOI: https://doi.org/10.1090/tran/6606
Published electronically: January 19, 2016
MathSciNet review: 3551574
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Abstract: Optimal target spaces are exhibited in arbitrary-order Sobolev type embeddings for traces of $ n$-dimensional functions on lower dimensional subspaces. Sobolev spaces built upon any rearrangement-invariant norm are allowed. A key step in our approach consists of showing that any trace embedding can be reduced to a one-dimensional inequality for a Hardy type operator depending only on $ n$ and on the dimension of the relevant subspace. This can be regarded as an analogue for trace embeddings of a well-known symmetrization principle for first-order Sobolev embeddings for compactly supported functions. The stability of the optimal target space under iterations of Sobolev trace embeddings is also established and is part of the proof of our reduction principle. As a consequence, we derive new trace embeddings, with improved (optimal) target spaces, for classical Sobolev, Lorentz-Sobolev and Orlicz-Sobolev spaces.


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  • [1] David R. Adams, Traces of potentials. II, Indiana Univ. Math. J. 22 (1972/73), 907-918. MR 0313783 (47 #2337)
  • [2] Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957 (56 #9247)
  • [3] Miguel A. Ariño and Benjamin Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for nonincreasing functions, Trans. Amer. Math. Soc. 320 (1990), no. 2, 727-735. MR 989570 (90k:42034), https://doi.org/10.2307/2001699
  • [4] Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802 (89e:46001)
  • [5] Haïm Brézis and Stephen Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations 5 (1980), no. 7, 773-789. MR 579997 (81k:46028), https://doi.org/10.1080/03605308008820154
  • [6] John E. Brothers and William P. Ziemer, Minimal rearrangements of Sobolev functions, J. Reine Angew. Math. 384 (1988), 153-179. MR 929981 (89g:26013)
  • [7] Victor I. Burenkov, Sobolev spaces on domains, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 137, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998. MR 1622690 (99g:46040)
  • [8] Paola Cavaliere and Andrea Cianchi, Classical and approximate Taylor expansions of weakly differentiable functions, Ann. Acad. Sci. Fenn. Math. 39 (2014), no. 2, 527-544. MR 3237035, https://doi.org/10.5186/aasfm.2014.3933
  • [9] Andrea Cianchi, Symmetrization and second-order Sobolev inequalities, Ann. Mat. Pura Appl. (4) 183 (2004), no. 1, 45-77. MR 2044332 (2005b:46067), https://doi.org/10.1007/s10231-003-0080-6
  • [10] Andrea Cianchi, Optimal Orlicz-Sobolev embeddings, Rev. Mat. Iberoamericana 20 (2004), no. 2, 427-474. MR 2073127 (2005d:46069), https://doi.org/10.4171/RMI/396
  • [11] Andrea Cianchi, Orlicz-Sobolev boundary trace embeddings, Math. Z. 266 (2010), no. 2, 431-449. MR 2678637 (2011g:46062), https://doi.org/10.1007/s00209-009-0578-8
  • [12] Andrea Cianchi, David E. Edmunds, and Petr Gurka, On weighted Poincaré inequalities, Math. Nachr. 180 (1996), 15-41. MR 1397667 (97e:26015)
  • [13] Andrea Cianchi and Adele Ferone, On symmetric functionals of the gradient having symmetric equidistributed minimizers, SIAM J. Math. Anal. 38 (2006), no. 1, 279-308 (electronic). MR 2217318 (2007f:49018), https://doi.org/10.1137/050625011
  • [14] Andrea Cianchi, Ron Kerman, and Luboš Pick, Boundary trace inequalities and rearrangements, J. Anal. Math. 105 (2008), 241-265. MR 2438426 (2009m:46048), https://doi.org/10.1007/s11854-008-0036-2
  • [15] Andrea Cianchi and Luboš Pick, Sobolev embeddings into BMO, VMO, and $ L_\infty $, Ark. Mat. 36 (1998), no. 2, 317-340. MR 1650446 (99k:46052), https://doi.org/10.1007/BF02384772
  • [16] Andrea Cianchi and Luboš Pick, An optimal endpoint trace embedding, Ann. Inst. Fourier (Grenoble) 60 (2010), no. 3, 939-951 (English, with English and French summaries). MR 2680820 (2011m:46046)
  • [17] Andrea Cianchi, Luboš Pick, and Lenka Slavíková, Higher-order Sobolev embeddings and isoperimetric inequalities, Adv. Math. 273 (2015), 568-650. MR 3311772, https://doi.org/10.1016/j.aim.2014.12.027
  • [18] Andrea Cianchi and Monia Randolfi, On the modulus of continuity of weakly differentiable functions, Indiana Univ. Math. J. 60 (2011), no. 6, 1939-1973. MR 3008258, https://doi.org/10.1512/iumj.2011.60.4441
  • [19] W. Des Evans, Bohumír Opic, and Luboš Pick, Interpolation of operators on scales of generalized Lorentz-Zygmund spaces, Math. Nachr. 182 (1996), 127-181. MR 1419893 (97m:46041), https://doi.org/10.1002/mana.19961820108
  • [20] Emilio Gagliardo, Proprietà di alcune classi di funzioni in più variabili, Ricerche Mat. 7 (1958), 102-137 (Italian). MR 0102740 (21 #1526)
  • [21] Filippo Gazzola, Hans-Christoph Grunau, and Guido Sweers, Optimal Sobolev and Hardy-Rellich constants under Navier boundary conditions, Ann. Mat. Pura Appl. (4) 189 (2010), no. 3, 475-486. MR 2657420 (2011j:46052), https://doi.org/10.1007/s10231-009-0118-5
  • [22] Amiran Gogatishvili, Bohumír Opic, and Luboš Pick, Weighted inequalities for Hardy-type operators involving suprema, Collect. Math. 57 (2006), no. 3, 227-255. MR 2264321 (2007g:26019)
  • [23] Kurt Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand. 45 (1979), no. 1, 77-102. MR 567435 (81j:31007)
  • [24] Ron Kerman and Luboš Pick, Optimal Sobolev imbeddings, Forum Math. 18 (2006), no. 4, 535-570. MR 2254384 (2007g:46052), https://doi.org/10.1515/FORUM.2006.028
  • [25] Vladimir Maz'ya, Sobolev spaces with applications to elliptic partial differential equations, Second, revised and augmented edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, Springer, Heidelberg, 2011. MR 2777530 (2012a:46056)
  • [26] Louis Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115-162. MR 0109940 (22 #823)
  • [27] Richard O'Neil, Convolution operators and $ L(p,\,q)$ spaces, Duke Math. J. 30 (1963), 129-142. MR 0146673 (26 #4193)
  • [28] Bohumír Opic and Luboš Pick, On generalized Lorentz-Zygmund spaces, Math. Inequal. Appl. 2 (1999), no. 3, 391-467. MR 1698383 (2000m:46067), https://doi.org/10.7153/mia-02-35
  • [29] Jaak Peetre, Espaces d'interpolation et théorème de Soboleff, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 279-317 (French). MR 0221282 (36 #4334)
  • [30] Luboš Pick, Alois Kufner, Oldřich John, and Svatopluk Fučík, Function spaces. Vol. 1, Second revised and extended edition, De Gruyter Series in Nonlinear Analysis and Applications, vol. 14, Walter de Gruyter & Co., Berlin, 2013. MR 3024912
  • [31] Stanislav I. Pohozaev, On the imbedding Sobolev theorem for $ pl=n$, Doklady Conference, Section Math. Moscow Power Inst. (1965), 158-170 (Russian).
  • [32] Eric T. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), no. 2, 145-158. MR 1052631 (91d:26026)
  • [33] Sergei L. Sobolev, Applications of functional analysis in mathematical physics, translated from the Russian by F. E. Browder. Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963. MR 0165337 (29 #2624)
  • [34] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 (44 #7280)
  • [35] Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353-372. MR 0463908 (57 #3846)
  • [36] Cristina Tarsi, Adams' inequality and limiting Sobolev embeddings into Zygmund spaces, Potential Anal. 37 (2012), no. 4, 353-385. MR 2988207, https://doi.org/10.1007/s11118-011-9259-4
  • [37] Neil S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473-483. MR 0216286 (35 #7121)
  • [38] Viktor I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Soviet Math. Dokl. 2 (1961), 746-749 (Russian).
  • [39] William P. Ziemer, Weakly differentiable functions, Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. MR 1014685 (91e:46046)

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Additional Information

Andrea Cianchi
Affiliation: Dipartimento di Matematica e Informatica \lq\lq U. Dini", Università di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy
Email: cianchi@unifi.it

Luboš Pick
Affiliation: Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
Email: pick@karlin.mff.cuni.cz

DOI: https://doi.org/10.1090/tran/6606
Keywords: Sobolev spaces, trace embeddings, optimal target, rearrangement-invariant spaces, Orlicz spaces, Lorentz spaces, supremum operators
Received by editor(s): April 24, 2014
Received by editor(s) in revised form: September 27, 2014
Published electronically: January 19, 2016
Additional Notes: This research was partly supported by the research project Prin 2008 “Geometric aspects of partial differential equations and related topics” of MIUR (Italian Ministry of University), by GNAMPA of the Italian INdAM (National Institute of High Mathematics), and by the grants 201/08/0383 and P201/13/14743S of the Grant Agency of the Czech Republic.
Article copyright: © Copyright 2016 American Mathematical Society

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