Refined Interpolation Inequality in Besov Spaces With Applications to the Gagliardo

Abstract

We consider the interpolation inequality with respect to the regularity index $s \in R$ in homogeneous Besov spaces ${\dot{B}}_{p, q}^{s} (R^{n})$ . By choosing a general summability index $1 \leq q \leq \infty$ and estimating carefully, we reveal a precise representation of the constant appearing in the interpolation inequality. As an application of the refined interpolation inequality, we show a generalization of the Gagliardo–Nirenberg inequality in homogeneous Besov spaces given by Wadade (2009).

Keywords

interpolation inequality Gagliardo–Nirenberg inequality homogeneous Besov spaces divergence rates

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References

Aubin

(1976). Problèmes isopérimétriques et espaces de Sobolev. Journal of Differential Geometry, 11(4), 573–598. https://doi.org/10.4310/jdg/1214433725

Bahouri

Chemin

J. Y.

Danchin

. (2011). Fourier analysis and nonlinear partial differential equations. Grundlehren der mathematischen wissenschaften, Vol. 343 [Fundamental principles of mathematical sciences, Vol. 343]. Springer.

Bergh

Löfström

. (1976). Interpolation spaces. An introduction. Springer-Verlag. Grundlehren der Mathematischen Wissenschaften, No. 223.

Brezis

Mironescu

(2018). Gagliardo–Nirenberg inequalities and non-inequalities: The full story. Annales de L’Institut Henry Poincaré C Analyse Non Linéaire, 35(5), 1355–1376. https://doi.org/10.1016/j.anihpc.2017.11.007

Brezis

Mironescu

(2019). Where Sobolev interacts with Gagliardo–Nirenberg. Journal of Functional Analysis, 277(8), 2839–2864. https://doi.org/10.1016/j.jfa.2019.02.019

Cianchi

Fusco

Maggi

Pratelli

(2009). The sharp Sobolev inequality in quantitative form. Journal of the European Mathematical Society, 11(5), 1105–1139. https://doi.org/10.4171/JEMS/176

Cohen

Dahmen

Daubechies

DeVore

(2003). Harmonic analysis of the space BV. Revista Matemática Iberoamericana, 19(1), 235–263. https://doi.org/10.4171/RMI/345

Cordero-Erausquin

Nazaret

Villani

(2004). A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities. Advances in Mathematics, 182(2), 307–332. https://doi.org/10.1016/S0001-8708(03)00080-X

Cotsiolis

Tavoularis

N. K.

(2004). Best constants for Sobolev inequalities for higher order fractional derivatives. Journal of Mathematical Analysis and Applications, 295(1), 225–236. https://doi.org/10.1016/j.jmaa.2004.03.034

10.

Dao

N. A.

(2023). Gagliardo–Nirenberg-type inequalities using fractional Sobolev spaces and Besov spaces. Advanced Nonlinear Studies, 23(1), ans-20220080. https://doi.org/10.1515/ans-2022-0080

11.

Dao

N. A.

Lam

(2022). Gagliardo–Nirenberg and Sobolev interpolation inequalities on Besov spaces. Proceedings of the American Mathematical Society, 150(2), 605–616. https://doi.org/10.1090/proc/15567

12.

Del Pino

Dolbeault

(2002). Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions. Journal de Mathématiques Pures et Appliquées, 81(9), 847–875. https://doi.org/10.1016/S0021-7824(02)01266-7

13.

Del Pino

Dolbeault

(2003). The optimal Euclidean

L^{p}

-Sobolev logarithmic inequality. Journal of Functional Analysis, 197(1), 151–161. https://doi.org/10.1016/S0022-1236(02)00070-8

14.

Gagliardo

(1959). Ulteriori proprietà di alcune classi di funzioni in più variabili. Ricerche di Matematica, 8, 24–51.

15.

Gerard

Meyer

Oru

(1997). Inégalités de Sobolev précisées. In Séminaire sur les Équations aux Dérivées Partielles, 1996–1997. École Polytech., Palaiseau. p. Exp. No. IV, 11.

16.

Hajaiej

Molinet

Ozawa

Wang

(2011). Necessary and sufficient conditions for the fractional Gagliardo–Nirenberg inequalities and applications to Navier–Stokes and generalized boson equations. In Harmonic analysis and nonlinear partial differential equations. RIMS Kôkyûroku Bessatsu. Research Institute for Mathematical Sciences (RIMS) (Vol. B26, pp. 159–175). Kyoto.

17.

Kozono

Ogawa

Taniuchi

(2002). The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations. Mathematische Zeitschrift, 242(2), 251–278. https://doi.org/10.1007/s002090100332

18.

Kozono

Wadade

(2008). Remarks on Gagliardo–Nirenberg type inequality with critical Sobolev space and BMO. Mathematische Zeitschrift, 259(4), 935–950. https://doi.org/10.1007/s00209-007-0258-5

19.

Lunardi

. (2009). Interpolation theory (2nd ed.). Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]. Edizioni della Normale.

20.

Machihara

Ozawa

(2003). Interpolation inequalities in Besov spaces. Proceedings of the American Mathematical Society, 131(5), 1553–1556. https://doi.org/10.1090/S0002-9939-02-06715-1

21.

Nagayasu

Wadade

(2010). Characterization of the critical Sobolev space on the optimal singularity at the origin. Journal of Functional Analysis, 258(11), 3725–3757. https://doi.org/10.1016/j.jfa.2010.02.015

22.

Nakamura

Ozawa

(2000). Small solutions to nonlinear Schrödinger equations in the Sobolev spaces. Journal d’Analyse Mathématique, 81, 305–329. https://doi.org/10.1007/BF02788994

23.

Nirenberg

(1959). On elliptic partial differential equations. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 13(3), 115–162. http://www.numdam.org/item/ASNSP_1959_3_13_2_115_0/

24.

Ogawa

(1990). A proof of Trudinger’s inequality and its application to nonlinear Schrödinger equations. Nonlinear Analysis, 14(9), 765–769. https://doi.org/10.1016/0362-546X(90)90104-O

25.

Ogawa

Ozawa

(1991). Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem. Journal of Mathematical Analysis and Applications, 155(2), 531–540. https://doi.org/10.1016/0022-247X(91)90017-T

26.

Ozawa

(1995). On critical cases of Sobolev’s inequalities. Journal of Functional Analysis, 127(2), 259–269. https://doi.org/10.1006/jfan.1995.1012

27.

Ozawa

Takeuchi

(2023). Refined decay estimate and analyticity of solutions to the linear heat equation in homogeneous Besov spaces. Journal of Fourier Analysis and Applications, 29, 61. https://doi.org/10.1007/s00041-023-10042-2

28.

Sawano

Wadade

(2013). On the Gagliardo–Nirenberg type inequality in the critical Sobolev–Morrey space. Journal of Fourier Analysis and Applications, 19(1), 20–47. https://doi.org/10.1007/s00041-012-9223-8

29.

Talenti

(1976). Best constant in Sobolev inequality. Annali di Matematica Pura ed Applicata, 110(4), 353–372. https://doi.org/10.1007/BF02418013

30.

Van Schaftingen

(2023). Fractional Gagliardo–Nirenberg interpolation inequality and bounded mean oscillation. Comptes Rendus Mathématique: Académie des Sciences, Paris, 361, 1041–1049. https://doi.org/10.5802/crmath.463

31.

Wadade

(2009). Remarks on the Gagliardo–Nirenberg type inequality in the Besov and the Triebel–Lizorkin spaces in the limiting case. Journal of Fourier Analysis and Applications, 15(6), 857–870. https://doi.org/10.1007/s00041-009-9069-x

32.

Wadade

(2010). Remarks on the critical Besov space and its embedding into weighted Besov–Orlicz spaces. Studia Mathematica, 201(3), 227–251. https://doi.org/10.4064/sm201-3-2

33.

Wadade

(2013). Quantitative estimates of embedding constants for Gagliardo–Nirenberg inequalities on critical Sobolev–Besov–Lorentz spaces. Journal of Fourier Analysis and Applications, 19(5), 1029–1059. https://doi.org/10.1007/s00041-013-9287-0

Refined Interpolation Inequality in Besov Spaces With Applications to the Gagliardo–Nirenberg Inequality

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