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September 2018, 17(5): 2011-2037. doi: 10.3934/cpaa.2018096

## Sharp Sobolev type embeddings on the entire Euclidean space

 1 Istituto per le Applicazioni del Calcolo "M. Picone", Consiglio Nazionale delle Ricerche, Via Pietro Castellino 111, 80131 Napoli, Italy 2 Dipartimento di Matematica e Informatica "U. Dini", Università di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy 3 Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic 4 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

Received  September 2017 Revised  January 2018 Published  April 2018

A comprehensive approach to Sobolev type embeddings, involving arbitrary rearrangement-invariant norms on the entire Euclidean space ${\mathbb R^n}$, is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in our analysis is a new reduction principle for the relevant embeddings, showing their equivalence to a couple of considerably simpler one-dimensional inequalities. Applications to the classes of the Orlicz-Sobolev and the Lorentz-Sobolev spaces are also presented. These contributions fill in a gap in the existing literature, where sharp results in such a general setting are only available for domains of finite measure.

Citation: Angela Alberico, Andrea Cianchi, Luboš Pick, Lenka Slavíková. Sharp Sobolev type embeddings on the entire Euclidean space. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2011-2037. doi: 10.3934/cpaa.2018096
##### References:
 [1] E. Acerbi and R. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 164 (2002), 213-259. [2] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [3] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63 (1976/77), 337-403. [4] C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics Vol. 129, Academic Press, Boston, 1988. [5] P. Baroni, Riesz potential estimates for a general class of quasilinear equations, Calc. Var. Partial Differential Equations, 53 (2015), 803-846. [6] D. Breit and O. D. Schirra, Korn-type inequalities in Orlicz-Sobolev spaces involving the trace-free part of the symmetric gradient and applications to regularity theory, Z. Anal. Anwend., 31 (2012), 335-356. [7] D. Breit, B. Stroffolini and A. Verde, A general regularity theorem for functionals with φ-growth, J. Math. Anal. Appl., 383 (2011), 226-233. [8] M. Bulíček, L. Diening and S. Schwarzacher, Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems, Anal. PDE, 9 (2016), 1115-1151. [9] M. Bulíček, M. Majdoub and J. Málek, Unsteady flows of fluids with pressure dependent viscosity in unbounded domains, Nonlinear Anal. Real World Appl., 11 (2010), 3968-3983. [10] M. Carro, A. García del Amo and J. Soria, Weak-type weights and normable Lorentz spaces, Proc. Amer. Math. Soc., 124 (1996), 849-857. [11] M. Carro, L. Pick, J. Soria and V. Stepanov, On embeddings between classical Lorentz spaces, Math. Inequal. Appl., 4 (2001), 397-428. [12] A. Cianchi, A sharp embedding theorem for Orlicz-Sobolev spaces, Indiana Univ. Math. J., 45 (1996), 39-65. [13] A. Cianchi, Boundedness of solutions to variational problems under general growth conditions, Comm. Partial Differential Equations, 22 (1997), 1629-1646. [14] A. Cianchi, Optimal Orlicz-Sobolev embeddings, Rev. Mat. Iberoamericana, 20 (2004), 427-474. [15] A. Cianchi, Higher-order Sobolev and Poincar´e inequalities in Orlicz spaces, Forum Math., 18 (2006), 745-767. [16] A. Cianchi and L. Pick, Sobolev embeddings into BMO, VMO and L∞, Ark. Mat., 36 (1998), 317-340. [17] A. Cianchi and L. Pick, Optimal Sobolev trace embeddings, Trans. Amer. Math. Soc., 368 (2016), 8349-8382. [18] A. Cianchi, L. Pick and L. Slavíková, Higher-order Sobolev embeddings and isoperimetric inequalities, Adv. Math., 273 (2015), 568-650. [19] A. Cianchi and M. Randolfi, On the modulus of continuity of weakly differentiable functions, Indiana Univ. Math. J., 60 (2011), 1939-1973. [20] D. E. Edmunds, R. Kerman and L. Pick, Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms, J. Funct. Anal., 170 (2000), 307-355. [21] H. J. Eyring, Viscosity, plasticity, and diffusion as example of absolute reaction rates, J. Chemical Physics, 4 (1936), 283-291. [22] R. Kerman and L. Pick, Optimal Sobolev imbeddings, Forum Math., 18 (2006), 535-570. [23] A. G. Korolev, On the boundedness of generalized solutions of elliptic differential equations with nonpower nonlinearities, Mat. Sb., 180 (1989), 78-100. [24] G. M. Lieberman, The natural generalization of the natural conditions of Ladyzenskaya and Ural'tseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361. [25] G. G. Lorentz, On the theory of spaces Λ, Pacific J. Math., 1 (1951), 411-429. [26] P. Marcellini, Regularity for elliptic equations with general growth conditions, J. Differential Equations, 105 (1993), 296-333. [27] L. Pick, A. Kufner, O. John and S. Fučík, Function Spaces, Vol. 1. Second revised and extended edition, Math. Inequal. Appl., de Gruyter & Co., Berlin, 2013. [28] S. I. Pohozaev, On the imbedding theorem by S.L.Sobolev in the case pl = n, Dokl. Conf., Sect. Math. Moscow Power Inst., (1965), 158-170. [29] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math., 96 (1990), 145-158. [30] J. Soria, Lorentz spaces of weak type, Quart. J. Math. Oxford Ser. (2), 49 (1998), 93-103. [31] R. S. Strichartz, A note on Trudinger' s extension of Sobolev' s inequality, Indiana Univ. Math. J., 21 (1972), 841-842. [32] G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl., 120 (1979), 159-184. [33] G. Talenti, Boundedness of minimizers, Hokkaido Math. J., 19 (1990), 259-279. [34] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. [35] V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, in Russian, Soviet Math. Dokl., 2 (1961), 746-749. [36] J. Vybíral, Optimal Sobolev imbeddings on ${\mathbb{R}^n}$, Publ. Mat., 51 (2007), 17-44. [37] A. Wrióblewska, Steady flow of non-Newtonian fluids–monotonicity methods in generalized Orlicz spaces, Nonlinear Anal., 72 (2010), 4136-4147.

show all references

##### References:
 [1] E. Acerbi and R. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 164 (2002), 213-259. [2] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [3] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63 (1976/77), 337-403. [4] C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics Vol. 129, Academic Press, Boston, 1988. [5] P. Baroni, Riesz potential estimates for a general class of quasilinear equations, Calc. Var. Partial Differential Equations, 53 (2015), 803-846. [6] D. Breit and O. D. Schirra, Korn-type inequalities in Orlicz-Sobolev spaces involving the trace-free part of the symmetric gradient and applications to regularity theory, Z. Anal. Anwend., 31 (2012), 335-356. [7] D. Breit, B. Stroffolini and A. Verde, A general regularity theorem for functionals with φ-growth, J. Math. Anal. Appl., 383 (2011), 226-233. [8] M. Bulíček, L. Diening and S. Schwarzacher, Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems, Anal. PDE, 9 (2016), 1115-1151. [9] M. Bulíček, M. Majdoub and J. Málek, Unsteady flows of fluids with pressure dependent viscosity in unbounded domains, Nonlinear Anal. Real World Appl., 11 (2010), 3968-3983. [10] M. Carro, A. García del Amo and J. Soria, Weak-type weights and normable Lorentz spaces, Proc. Amer. Math. Soc., 124 (1996), 849-857. [11] M. Carro, L. Pick, J. Soria and V. Stepanov, On embeddings between classical Lorentz spaces, Math. Inequal. Appl., 4 (2001), 397-428. [12] A. Cianchi, A sharp embedding theorem for Orlicz-Sobolev spaces, Indiana Univ. Math. J., 45 (1996), 39-65. [13] A. Cianchi, Boundedness of solutions to variational problems under general growth conditions, Comm. Partial Differential Equations, 22 (1997), 1629-1646. [14] A. Cianchi, Optimal Orlicz-Sobolev embeddings, Rev. Mat. Iberoamericana, 20 (2004), 427-474. [15] A. Cianchi, Higher-order Sobolev and Poincar´e inequalities in Orlicz spaces, Forum Math., 18 (2006), 745-767. [16] A. Cianchi and L. Pick, Sobolev embeddings into BMO, VMO and L∞, Ark. Mat., 36 (1998), 317-340. [17] A. Cianchi and L. Pick, Optimal Sobolev trace embeddings, Trans. Amer. Math. Soc., 368 (2016), 8349-8382. [18] A. Cianchi, L. Pick and L. Slavíková, Higher-order Sobolev embeddings and isoperimetric inequalities, Adv. Math., 273 (2015), 568-650. [19] A. Cianchi and M. Randolfi, On the modulus of continuity of weakly differentiable functions, Indiana Univ. Math. J., 60 (2011), 1939-1973. [20] D. E. Edmunds, R. Kerman and L. Pick, Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms, J. Funct. Anal., 170 (2000), 307-355. [21] H. J. Eyring, Viscosity, plasticity, and diffusion as example of absolute reaction rates, J. Chemical Physics, 4 (1936), 283-291. [22] R. Kerman and L. Pick, Optimal Sobolev imbeddings, Forum Math., 18 (2006), 535-570. [23] A. G. Korolev, On the boundedness of generalized solutions of elliptic differential equations with nonpower nonlinearities, Mat. Sb., 180 (1989), 78-100. [24] G. M. Lieberman, The natural generalization of the natural conditions of Ladyzenskaya and Ural'tseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361. [25] G. G. Lorentz, On the theory of spaces Λ, Pacific J. Math., 1 (1951), 411-429. [26] P. Marcellini, Regularity for elliptic equations with general growth conditions, J. Differential Equations, 105 (1993), 296-333. [27] L. Pick, A. Kufner, O. John and S. Fučík, Function Spaces, Vol. 1. Second revised and extended edition, Math. Inequal. Appl., de Gruyter & Co., Berlin, 2013. [28] S. I. Pohozaev, On the imbedding theorem by S.L.Sobolev in the case pl = n, Dokl. Conf., Sect. Math. Moscow Power Inst., (1965), 158-170. [29] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math., 96 (1990), 145-158. [30] J. Soria, Lorentz spaces of weak type, Quart. J. Math. Oxford Ser. (2), 49 (1998), 93-103. [31] R. S. Strichartz, A note on Trudinger' s extension of Sobolev' s inequality, Indiana Univ. Math. J., 21 (1972), 841-842. [32] G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl., 120 (1979), 159-184. [33] G. Talenti, Boundedness of minimizers, Hokkaido Math. J., 19 (1990), 259-279. [34] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. [35] V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, in Russian, Soviet Math. Dokl., 2 (1961), 746-749. [36] J. Vybíral, Optimal Sobolev imbeddings on ${\mathbb{R}^n}$, Publ. Mat., 51 (2007), 17-44. [37] A. Wrióblewska, Steady flow of non-Newtonian fluids–monotonicity methods in generalized Orlicz spaces, Nonlinear Anal., 72 (2010), 4136-4147.
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