Sobolev embeddings, rearrangement-invariant spaces and Frostman measures

Cianchi, Andrea; Pick, Luboš; Slavíková, Lenka

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 1, pp. 105-144.

On considère des immersions de Sobolev d'ordre quelconque dans des espaces de fonctions sur des domaines de Rn munis des mesures avec une tendance dans les boules qui est dominée par une puissance d du rayon. Des normes dans les espaces arbitraires invariants par réarrangements sont permises. Nous proposons une approche générale basée sur la réduction des immersions en dimension n à des inégalités du type Hardy en dimension un. On souligne que ces inégalités dépendent de la mesure considérée seulement par le degré de puissance d. Notre résultat permet de détecter l'espace cible optimal dans les immersions de Sobolev, pour une large famille de normes dans des cas où les techniques habituelles ne s'appliquent pas. En particulier on déduit des nouvelles immersions avec espaces-cibles augmentés même dans le cas d'espace de Sobolev standard.

Sobolev embeddings, of arbitrary order, are considered into function spaces on domains of Rn endowed with measures whose decay on balls is dominated by a power d of their radius. Norms in arbitrary rearrangement-invariant spaces are contemplated. A comprehensive approach is proposed based on the reduction of the relevant n-dimensional embeddings to one-dimensional Hardy-type inequalities. Interestingly, the latter inequalities depend on the involved measure only through the power d. Our results allow for the detection of the optimal target space in Sobolev embeddings, for broad families of norms, in situations where customary techniques do not apply. In particular, new embeddings, with augmented target spaces, are deduced even for standard Sobolev spaces.

[1] Adams, D.R. Traces of potentials arising from translation invariant operators, Ann. Sc. Norm. Super. Pisa, Volume 25 (1971), pp. 203–217 | Numdam | MR | Zbl

[2] Adams, D.R. A trace inequality for generalized potentials, Stud. Math., Volume 48 (1973), pp. 99–105 | DOI | MR | Zbl

[3] Alberico, A.; Cianchi, A.; Pick, L.; Slavíková, L. Sharp Sobolev type embeddings on the entire Euclidean space, Commun. Pure Appl. Anal., Volume 17 (2018) no. 5, pp. 2011–2037 | MR

[4] Alvino, A.; Brock, F.; Chiacchio, F.; Mercaldo, A.; Posteraro, M.R. Some isoperimetric inequalities on RN with respect to weights |x|α , J. Math. Anal. Appl., Volume 451 (2017), pp. 280–318 | DOI | MR

[5] Aubin, T. Problèmes isopérimetriques et espaces de Sobolev, J. Differ. Geom., Volume 11 (1976), pp. 573–598 | DOI | MR | Zbl

[6] Bennett, C.; Sharpley, R. Interpolation of Operators, Pure and Applied Mathematics, vol. 129, Academic Press, Boston, 1988 | MR | Zbl

[7] Brézis, H.; Wainger, S. A note on limiting cases of Sobolev embeddings and convolution inequalities, Commun. Partial Differ. Equ., Volume 5 (1980), pp. 773–789 | DOI | MR | Zbl

[8] Carro, M.; Pick, L.; Soria, J.; Stepanov, V. On embeddings between classical Lorentz spaces, Math. Inequal. Appl., Volume 4 (2001), pp. 397–428 | MR | Zbl

[9] Catrina, F.; Wang, Z.-Q. On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Commun. Pure Appl. Math., Volume 54 (2001), pp. 229–258 | DOI | MR | Zbl

[10] Cavaliere, P.; Mihula, Z. Compactness of Sobolev-type trace operators, Nonlinear Anal., Volume 183 (2019), pp. 43–69 | DOI | MR

[11] Cianchi, A. Second-order derivatives and rearrangements, Duke Math. J., Volume 105 (2000), pp. 355–385 | DOI | MR | Zbl

[12] Cianchi, A. Symmetrization and second-order Sobolev inequalities, Ann. Mat. Pura Appl., Volume 183 (2004), pp. 45–77 | DOI | MR | Zbl

[13] Cianchi, A.; Kerman, R.; Pick, L. Boundary trace inequalities and rearrangements, J. Anal. Math., Volume 105 (2008), pp. 241–265 | DOI | MR | Zbl

[14] Cianchi, A.; Pick, L. Sobolev embeddings into BMO, VMO and L∞ , Ark. Mat., Volume 36 (1998), pp. 317–340 | DOI | MR | Zbl

[15] Cianchi, A.; Pick, L. An optimal endpoint trace embedding, Ann. Inst. Fourier (Grenoble), Volume 60 (2010), pp. 939–951 | DOI | Numdam | MR | Zbl

[16] Cianchi, A.; Pick, L. Optimal Sobolev trace embeddings, Trans. Am. Math. Soc., Volume 368 (2016), pp. 8349–8382 | DOI | MR

[17] Cianchi, A.; Pick, L.; Slavíková, L. Higher-order Sobolev embeddings and isoperimetric inequalities, Adv. Math., Volume 273 (2015), pp. 568–650 | DOI | MR | Zbl

[18] A. Cianchi, L. Pick, L. Slavíková, Sobolev embeddings in Orlicz and Lorentz spaces with measures, preprint. | MR

[19] Cianchi, A.; Randolfi, M. On the modulus of continuity of weakly differentiable functions, Indiana Univ. Math. J., Volume 60 (2011), pp. 1939–1973 | DOI | MR | Zbl

[20] Costea, S.; Maz'ya, V.G. Conductor inequalities and criteria for Sobolev-Lorentz two-weight inequalities (English summary) Sobolev spaces in mathematics. II, Int. Math. Ser. (N. Y.), vol. 9, Springer, New York, 2009, pp. 103–121 | MR | Zbl

[21] Curbera, G.P.; Ricker, W.J. Can optimal rearrangement invariant Sobolev imbeddings be further extended?, Indiana Univ. Math. J., Volume 56 (2007), pp. 1479–1497 | DOI | MR | Zbl

[22] Curbera, G.P.; Ricker, W.J. Optimal domains for the kernel operator associated with Sobolev's inequality, Stud. Math., Volume 158 (2003), pp. 131–152 | DOI | MR | Zbl

[23] De Vore, R.A.; Scherer, K. Interpolation of linear operators on Sobolev spaces, Ann. Math., Volume 109 (1979), pp. 583–599 | MR | Zbl

[24] Dolbeault, J.; Esteban, M.J.; Filippas, S.; Tertikas, S. Rigidity results with applications to best constants and symmetry of Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities, Calc. Var. Partial Differ. Equ., Volume 54 (2015), pp. 2465–2481 | DOI | MR

[25] Dolbeault, J.; Loss, M.J. Esteban M. Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces, Invent. Math., Volume 206 (2016), pp. 397–440 | DOI | MR

[26] Evans, W.D.; Opic, B.; Pick, L. Interpolation of integral operators on scales of generalized Lorentz–Zygmund spaces, Math. Nachr., Volume 182 (1996), pp. 127–181 | DOI | MR | Zbl

[27] Federer, H.; Fleming, W. Normal and integral currents, Ann. Math., Volume 72 (1960), pp. 458–520 | DOI | MR | Zbl

[28] Felli, V.; Schneider, M. Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type, J. Differ. Equ., Volume 191 (2003), pp. 121–142 | DOI | MR | Zbl

[29] Gogatishvili, A.; Moura, S.; Neves, J.; Opic, B. Embeddings of Sobolev-type spaces into generalized Hölder spaces involving $k$-modulus of smoothness, Ann. Mat. Pura Appl., Volume 194 (2015), pp. 425–450 | DOI | MR

[30] Gogatishvili, A.; Neves, J.; Opic, B. Characterization of embeddings of Sobolev-type spaces into generalized Hölder spaces defined by Lp-modulus of smoothness, J. Funct. Anal., Volume 276 (2019), pp. 636–657 | DOI | MR

[31] Holík, M. Reduction theorems for Sobolev embeddings into the spaces of Hölder, Morrey and Campanato type, Math. Nachr., Volume 289 (2016), pp. 1626–1635 | DOI | MR

[32] Holmstedt, T. Interpolation of quasi-normed spaces, Math. Scand., Volume 26 (1970), pp. 177–199 | DOI | MR | Zbl

[33] Kerman, R.; Pick, L. Optimal Sobolev embeddings, Forum Math., Volume 18 (2006), pp. 535–570 | DOI | MR | Zbl

[34] Kristensen, J.; Korobkov, M.V. The trace theorem, the Luzin $N$ - and Morse-Sard properties for the sharp case of Sobolev-Lorentz mappings, J. Geom. Anal., Volume 28 (2018), pp. 2834–2856 | MR

[35] Malý, J.; Pick, L. An elementary proof of sharp Sobolev embeddings, Proc. Am. Math. Soc., Volume 130 (2002) no. 2, pp. 555–563 | DOI | MR | Zbl

[36] Maligranda, L.; Persson, L.-E. Generalized duality of some Banach function spaces, Nederl. Akad. Wetensch. Indag. Math., Volume 51 (1989), pp. 323–338 | MR | Zbl

[37] Maz'ya, V.G. Classes of regions and imbedding theorems for function spaces, Dokl. Akad. Nauk SSSR, Volume 133 (1960), pp. 527–530 (Russian); English translation: Sov. Math. Dokl., 1, 1960, 882–885 | MR | Zbl

[38] Maz'ya, V.G. On $p$-conductivity and theorems on embedding certain functional spaces into a $C$-space, Dokl. Akad. Nauk SSSR, Volume 140 (1961), pp. 299–302 (Russian); English translation: Sov. Math. Dokl., 3, 1962 | MR | Zbl

[39] Maz'ya, V.G. Certain integral inequalities for functions of many variables, Problems in Mathematical Analysis, Volume vol. 3 (1972), pp. 33–68 (in Russian); English translation: J. Sov. Math., 1, 1973, 205–234 | Zbl

[40] Maz'ya, V.G. Capacity-estimates for “fractional” norms Zap, Nauchn. Semin. Leningr. Otd. Mat. Int. Steklova, Volume 70 (1977), pp. 161–168 (in Russian); English translation: J. Sov. Math., 23, 1983, 1997–2003 | MR | Zbl

[41] Maz'ya, V.G. Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Springer, Berlin, 2011 | DOI | MR | Zbl

[42] Moser, J. A sharp form of an inequality by Trudinger, Indiana Univ. Math. J., Volume 20 (1971), pp. 1077–1092 | DOI | MR | Zbl

[43] O'Neil, R. Convolution operators and L(p,q) spaces, Duke Math. J., Volume 30 (1963), pp. 129–142 | MR | Zbl

[44] Opic, B.; Pick, L. On generalized Lorentz-Zygmund spaces, Math. Inequal. Appl., Volume 2 (1999), pp. 391–467 | MR | Zbl

[45] Peetre, J. Espaces d'interpolation et théorème de Soboleff, Ann. Inst. Fourier, Volume 16 (1966), pp. 279–317 | DOI | Numdam | MR | Zbl

[46] Pick, L.; Kufner, A.; John, O.; Fučík, S. De Gruyter Series in Nonlinear Analysis and Applications, vol. 1, De Gruyter, Berlin (2013), pp. 14 | MR | Zbl

[47] Sawyer, E. A two weight weak type inequality for fractional integrals, Trans. Am. Math. Soc., Volume 281 (1984) no. 1, pp. 339–345 | DOI | MR | Zbl

[48] Sawyer, E. Boundedness of classical operators on classical Lorentz spaces, Stud. Math., Volume 96 (1990), pp. 145–158 | DOI | MR | Zbl

[49] Slavíková, L. Compactness of higher-order Sobolev embeddings, Publ. Mat., Volume 59 (2015), pp. 373–448 | DOI | MR

[50] Stein, E. Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970 | MR | Zbl

[51] Stein, E. Editor's note: the differentiability of functions in Rn , Ann. Math., Volume 113 (1981), pp. 383–385 | MR | Zbl

[52] Talenti, G. Best constant in Sobolev inequality, Ann. Mat. Pura Appl., Volume 110 (1976), pp. 353–372 | DOI | MR | Zbl

[53] H. Turčinová, Basic functional properties of certain scale of rearrangement-invariant spaces, Prague, Preprint, 2019. | MR