no. 5
Gagliardo–Nirenberg inequalities and non-inequalities: The full story
Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 5, pp. 1355-1376.

We investigate the validity of the Gagliardo–Nirenberg type inequality

‖f‖Ws,p(Ω)≲‖f‖Ws1,p1(Ω)θ‖f‖Ws2,p2(Ω)1−θ,
with Ω⊂RN. Here, 0≤s1≤s≤s2 are non negative numbers (not necessarily integers), 1≤p1,p,p2≤∞, and we assume the standard relations
s=θs1+(1−θ)s2,1/p=θ/p1+(1−θ)/p2 for some θ∈(0,1).

By the seminal contributions of E. Gagliardo and L. Nirenberg, (1) holds when s1,s2,s are integers. It turns out that (1) holds for “most” of values of s1,…,p2, but not for all of them. We present an explicit condition on s1,s2,p1,p2 which allows to decide whether (1) holds or fails.

DOI : 10.1016/j.anihpc.2017.11.007
Mots clés : Sobolev spaces, Gagliardo–Nirenberg inequalities, Interpolation inequalities 
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Brezis, Haïm; Mironescu, Petru. Gagliardo–Nirenberg inequalities and non-inequalities: The full story. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 5, pp. 1355-1376. doi : 10.1016/j.anihpc.2017.11.007. http://archive.numdam.org/articles/10.1016/j.anihpc.2017.11.007/

[1] Adams, R.A.; Fournier, J.J.F. Sobolev Spaces, Pure Appl. Math. (Amst.), vol. 140, Elsevier/Academic Press, Amsterdam, 2003 | MR | Zbl

[2] Bourgain, J.; Brezis, H.; Mironescu, P. Lifting in Sobolev spaces, J. Anal. Math., Volume 80 (2000), pp. 37–86 | DOI | MR | Zbl

[3] Brezis, H.; Mironescu, P. Gagliardo–Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ., Volume 1 (2001) no. 4, pp. 387–404 (dedicated to the memory of Tosio Kato) | DOI | MR | Zbl

[4] Chemin, J.-Y. Perfect Incompressible Fluids, Oxf. Lect. Ser. Math. Appl., vol. 14, The Clarendon Press Oxford University Press, New York, 1998 (translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie) | MR | Zbl

[5] Cohen, A. Ondelettes, espaces d'interpolation et applications, Séminaire sur les Équations aux Dérivées Partielles, 1999–2000, École Polytech, Palaiseau, 2000 (pages Exp. No. I, 14 pp.) | Numdam | MR | Zbl

[6] Cohen, A.; Dahmen, W.; Daubechies, I.; DeVore, R. Harmonic analysis of the space BV, Rev. Mat. Iberoam., Volume 19 (2003) no. 1, pp. 235–263 | MR | Zbl

[7] Di Nezza, E.; Palatucci, G.; Valdinoci, E. Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., Volume 136 (2012) no. 5, pp. 521–573 | DOI | MR | Zbl

[8] Gagliardo, E. Ulteriori proprietà di alcune classi di funzioni in più variabili, Ric. Mat., Volume 8 (1959), pp. 24–51 | MR | Zbl

[9] Kašin, B.S. Remarks on the estimation of Lebesgue functions of orthonormal systems, Mat. Sb. (N.S.), Volume 106(148) (1978) no. 3, pp. 380–385 (495) | MR | Zbl

[10] Nirenberg, L. On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa, Volume 3 (1959) no. 13, pp. 115–162 | Numdam | MR | Zbl

[11] F. Oru, Rôle des oscillations dans quelques problèmes d'analyse non-linéaire, PhD thesis, ENS Cachan, 1998, unpublished.

[12] Runst, T. Mapping properties of nonlinear operators in spaces of Triebel–Lizorkin and Besov type, Anal. Math., Volume 12 (1986) no. 4, pp. 313–346 | DOI | MR | Zbl

[13] Runst, T.; Sickel, W. Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter Ser. Nonlinear Anal. Appl., vol. 3, Walter de Gruyter & Co., Berlin, 1996 | MR | Zbl

[14] Stein, E.M. Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser., vol. 30, Princeton University Press, Princeton, NJ, 1970 | MR | Zbl

[15] Triebel, H. Theory of Function Spaces, Monogr. Math., vol. 78, Birkhäuser Verlag, Basel, 1983 | MR | Zbl

[16] Triebel, H. Theory of Function Spaces. II, Monogr. Math., vol. 84, Birkhäuser Verlag, Basel, 1992 | MR | Zbl

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