Optimal limiting embeddings for Δ-reduced Sobolev spaces in L 1
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 2, p. 217-230

We prove sharp embedding inequalities for certain reduced Sobolev spaces that arise naturally in the context of Dirichlet problems with L 1 data. We also find the optimal target spaces for such embeddings, which in dimension 2 could be considered as limiting cases of the Hansson–Brezis–Wainger spaces, for the optimal embeddings of borderline Sobolev spaces W 0 k,n/k .

@article{AIHPC_2014__31_2_217_0,
     author = {Fontana, Luigi and Morpurgo, Carlo},
     title = {Optimal limiting embeddings for $\Delta$-reduced Sobolev spaces in $ {L}^{1}$},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {31},
     number = {2},
     year = {2014},
     pages = {217-230},
     doi = {10.1016/j.anihpc.2013.02.007},
     zbl = {1316.46035},
     mrnumber = {3181666},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2014__31_2_217_0}
}
Fontana, Luigi; Morpurgo, Carlo. Optimal limiting embeddings for Δ-reduced Sobolev spaces in $ {L}^{1}$. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 2, pp. 217-230. doi : 10.1016/j.anihpc.2013.02.007. http://www.numdam.org/item/AIHPC_2014__31_2_217_0/

[1] D.R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. 128 (1988), 385-398 | MR 960950 | Zbl 0672.31008

[2] A. Alberico, A. Cianchi, Optimal summability of solutions to nonlinear elliptic problems, Nonlinear Anal. 67 (2007), 1775-1790 | MR 2326030 | Zbl 05169011

[3] A. Alberico, V. Ferone, Regularity properties of solutions of elliptic equations in 2 in limit cases, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 6 (1995), 237-250 | MR 1382708 | Zbl 0860.35015

[4] A. Alvino, A limit case of the Sobolev inequality in Lorentz spaces, Rend. Accad. Sci. Fis. Mat. Napoli 44 (1977), 105-112 | MR 501652 | Zbl 0412.46024

[5] A. Alvino, V. Ferone, G. Trombetti, Estimates for the gradient of solutions of nonlinear elliptic equations with L 1 data, Ann. Mat. Pura Appl. 178 (2000), 129-142 | MR 1849383 | Zbl 1220.35079

[6] H. Brezis, F. Merle, Uniform estimates and blow-up behavior for solutions of -Δu=V(x)e u in two dimensions, Comm. Partial Differential Equations 16 (1991), 1223-1253 | MR 1132783 | Zbl 0746.35006

[7] C. Bennett, R. Sharpley, Interpolation of Operators, Pure Appl. Math. vol. 129, Academic Press, Inc., Boston, MA (1988) | MR 928802 | Zbl 0647.46057

[8] H. Brezis, W.A. Strauss, Semi-linear second-order elliptic equations in L 1 , J. Math. Soc. Japan 25 (1973), 565-590 | MR 336050 | Zbl 0278.35041

[9] H. Brezis, S. Wainger, A note on limiting cases of Sobolev embeddings, Comm. Partial Differential Equations 5 (1980), 773-789 | MR 579997 | Zbl 0437.35071

[10] A. Cianchi, Higher-order Sobolev and Poincaré inequalities in Orlicz spaces, Forum Math. 18 (2006), 745-767 | MR 2265898 | Zbl 1120.46015

[11] M. Cwikel, E. Pustylnik, Sobolev type embeddings in the limiting case, J. Fourier Anal. Appl. 4 (1998), 433-446 | MR 1658620 | Zbl 0930.46027

[12] D. Cassani, B. Ruf, C. Tarsi, Best constants in a borderline case of second-order Moser type inequalities, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 73-93 | Numdam | MR 2580505 | Zbl 1194.46048

[13] D.E. Edmunds, R. Kerman, L. Pick, Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms, J. Funct. Anal. 170 (2000), 307-355 | MR 1740655 | Zbl 0955.46019

[14] K. Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand. 45 (1979), 77-102 | MR 567435 | Zbl 0437.31009

[15] A. Kaminska, H.J. Lee, M-ideal properties in Marcinkiewicz spaces, Comment. Math. Prace Mat. (2004), 123-144 | MR 2111760 | Zbl 1072.46013

[16] S.G. Krein, J.I. Petunin, E.M. Semenov, Interpolation of Linear Operators, Transl. Math. Monogr. vol. 54, American Mathematical Society, Providence, RI (1982) | MR 649411 | Zbl 0193.09302

[17] D. Ornstein, A non-equality for differential operators in the L 1 norm, Arch. Ration. Mech. Anal. 11 (1962), 40-49 | MR 149331 | Zbl 0106.29602

[18] M. Milman, E. Pustylnik, On sharp higher order Sobolev embeddings, Commun. Contemp. Math. 6 (2004), 495-511 | MR 2068850 | Zbl 1108.46029

[19] G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa 3 (1976), 697-718 | Numdam | MR 601601 | Zbl 0341.35031