Optimal Sobolev trace embeddings

Cianchi, Andrea; Pick, Luboš

doi:10.1090/tran/6606

Optimal Sobolev trace embeddings
HTML articles powered by AMS MathViewer

by Andrea Cianchi and Luboš Pick PDF

Trans. Amer. Math. Soc. 368 (2016), 8349-8382 Request permission

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.

References

David R. Adams, Traces of potentials. II, Indiana Univ. Math. J. 22 (1972/73), 907–918. MR 313783, DOI 10.1512/iumj.1973.22.22075
Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
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, DOI 10.1090/S0002-9947-1990-0989570-0
Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
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, DOI 10.1080/03605308008820154
John E. Brothers and William P. Ziemer, Minimal rearrangements of Sobolev functions, J. Reine Angew. Math. 384 (1988), 153–179. MR 929981
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, DOI 10.1007/978-3-663-11374-4
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, DOI 10.5186/aasfm.2014.3933
Andrea Cianchi, Symmetrization and second-order Sobolev inequalities, Ann. Mat. Pura Appl. (4) 183 (2004), no. 1, 45–77. MR 2044332, DOI 10.1007/s10231-003-0080-6
Andrea Cianchi, Optimal Orlicz-Sobolev embeddings, Rev. Mat. Iberoamericana 20 (2004), no. 2, 427–474. MR 2073127, DOI 10.4171/RMI/396
Andrea Cianchi, Orlicz-Sobolev boundary trace embeddings, Math. Z. 266 (2010), no. 2, 431–449. MR 2678637, DOI 10.1007/s00209-009-0578-8
Andrea Cianchi, David E. Edmunds, and Petr Gurka, On weighted Poincaré inequalities, Math. Nachr. 180 (1996), 15–41. MR 1397667, DOI 10.1002/mana.3211800103
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. MR 2217318, DOI 10.1137/050625011
Andrea Cianchi, Ron Kerman, and Luboš Pick, Boundary trace inequalities and rearrangements, J. Anal. Math. 105 (2008), 241–265. MR 2438426, DOI 10.1007/s11854-008-0036-2
Andrea Cianchi and Luboš Pick, Sobolev embeddings into BMO, VMO, and $L_\infty$, Ark. Mat. 36 (1998), no. 2, 317–340. MR 1650446, DOI 10.1007/BF02384772
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, DOI 10.5802/aif.2543
Andrea Cianchi, Luboš Pick, and Lenka Slavíková, Higher-order Sobolev embeddings and isoperimetric inequalities, Adv. Math. 273 (2015), 568–650. MR 3311772, DOI 10.1016/j.aim.2014.12.027
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, DOI 10.1512/iumj.2011.60.4441
W. D. Evans, B. Opic, and L. Pick, Interpolation of operators on scales of generalized Lorentz-Zygmund spaces, Math. Nachr. 182 (1996), 127–181. MR 1419893, DOI 10.1002/mana.19961820108
Emilio Gagliardo, Proprietà di alcune classi di funzioni in più variabili, Ricerche Mat. 7 (1958), 102–137 (Italian). MR 102740
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, DOI 10.1007/s10231-009-0118-5
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
Kurt Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand. 45 (1979), no. 1, 77–102. MR 567435, DOI 10.7146/math.scand.a-11827
Ron Kerman and Luboš Pick, Optimal Sobolev imbeddings, Forum Math. 18 (2006), no. 4, 535–570. MR 2254384, DOI 10.1515/FORUM.2006.028
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, DOI 10.1007/978-3-642-15564-2
L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115–162. MR 109940
Richard O’Neil, Convolution operators and $L(p,\,q)$ spaces, Duke Math. J. 30 (1963), 129–142. MR 146673
B. Opic and L. Pick, On generalized Lorentz-Zygmund spaces, Math. Inequal. Appl. 2 (1999), no. 3, 391–467. MR 1698383, DOI 10.7153/mia-02-35
Jaak Peetre, Espaces d’interpolation et théorème de Soboleff, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 279–317 (French). MR 221282, DOI 10.5802/aif.232
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
Stanislav I. Pohozaev, On the imbedding Sobolev theorem for $pl=n$, Doklady Conference, Section Math. Moscow Power Inst. (1965), 158–170 (Russian).
E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), no. 2, 145–158. MR 1052631, DOI 10.4064/sm-96-2-145-158
S. L. Sobolev, Applications of functional analysis in mathematical physics, Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963. Translated from the Russian by F. E. Browder. MR 0165337, DOI 10.1090/mmono/007
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. MR 463908, DOI 10.1007/BF02418013
Cristina Tarsi, Adams’ inequality and limiting Sobolev embeddings into Zygmund spaces, Potential Anal. 37 (2012), no. 4, 353–385. MR 2988207, DOI 10.1007/s11118-011-9259-4
Neil S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483. MR 0216286, DOI 10.1512/iumj.1968.17.17028
Viktor I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Soviet Math. Dokl. 2 (1961), 746–749 (Russian).
William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3