Geodesic
Embeddings between Grand, Small,
Matematičeskie zametki, Tome 102 (2017) no. 5, pp. 736-748.

Voir la notice de l'article provenant de la source Math-Net.Ru

We give conditions on the exponent function $p(\,\cdot\,)$ that imply the existence of embeddings between the grand, small, and variable Lebesgue spaces. We construct examples to show that our results are close to optimal. Our work extends recent results by the second author, Rakotoson and Sbordone.
Keywords: Banach function spaces, embeddings.
Mots-clés : variable Lebesgue spaces, grand Lebesgue spaces, small Lebesgue spaces 
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D. Cruz-Uribe; A. Fiorenza; O. M. Guzman. Embeddings between Grand, Small,. Matematičeskie zametki, Tome 102 (2017) no. 5, pp. 736-748. http://geodesic.mathdoc.fr/item/MZM_2017_102_5_a7/

[1] T. Iwaniec, C. Sbordone, “On the integrability of the Jacobian under minimal hypotheses”, Arch. Rational Mech. Anal., 119:2 (1992), 129–143 | DOI | MR | Zbl

[2] L. Greco, T. Iwaniec, C. Sbordone, “Inverting the $p$-harmonic operator”, Manuscr. Math., 92:2 (1997), 249–258 | DOI | MR | Zbl

[3] C. Capone, A. Fiorenza, “On small Lebesgue spaces”, J. Funct. Spaces Appl., 3:1 (2005), 73–89 | DOI | MR | Zbl

[4] A. Alberico, T. Alberico, C. Sbordone, “Planar quasilinear elliptic equations with right-hand side in $L(\log L)^\delta$”, Discrete Contin. Dyn. Syst., 31:4 (2011), 1053–1067 | DOI | MR | Zbl

[5] L. D'Onofrio, C. Sbordone, R. Schiattarella, “Grand Sobolev spaces and their applications in geometric function theory and PDEs”, J. Fixed Point Theory Appl., 13:2 (2013), 309–340 | DOI | MR | Zbl

[6] F. Farroni, L. Greco, G. Moscariello, “Stability for $p$-Laplace type equation in a borderline case”, Nonlinear Anal., 116 (2015), 100–111 | DOI | MR | Zbl

[7] T. Radice, G. Zecca, “Existence and uniqueness for nonlinear elliptic equations with unbounded coefficients”, Ric. Mat., 63:2 (2014), 355–367 | DOI | MR | Zbl

[8] A. Fiorenza, “Duality and reflexivity in grand Lebesgue spaces”, Collect. Math., 51:2 (2000), 131–148 | MR | Zbl

[9] G. Di Fratta, A. Fiorenza, “A direct approach to the duality of grand and small Lebesgue spaces”, Nonlinear Anal., 70:7 (2009), 2582–2592 | DOI | MR | Zbl

[10] A. Fiorenza, G. E. Karadzhov, “Grand and small Lebesgue spaces and their analogs”, Z. Anal. Anwendungen, 23:4 (2004), 657–681 | DOI | MR | Zbl

[11] F. Cobos, T. Kühn, “Extrapolation results of Lions–Peetre type”, Calc. Var. Partial Differential Equations, 49:1-2 (2014), 847–860 | DOI | MR | Zbl

[12] D. Cruz-Uribe, A. Fiorenza, Variable Lebesgue Spaces. Foundations and Harmonic Analysis, Birkhäuser, Basel, 2013 | MR | Zbl

[13] A. Fiorenza, J. M. Rakotoson, “Relative rearrangement and Lebesgue spaces $L^{p(\,\cdot\,)}$ with variable exponent”, J. Math. Pures Appl. (9), 88:6 (2007), 506–521 | DOI | MR | Zbl

[14] A. Fiorenza, J. M. Rakotoson, C. Sbordone, “Variable exponents and grand Lebesgue spaces: some optimal results”, Commun. Contemp. Math., 17:6 (2015), 1550023 | MR | Zbl

[15] D. Cruz-Uribe, A. Fiorenza, “$L\log L$ results for the maximal operator in variable $L^p$ spaces”, Trans. Amer. Math. Soc., 361:5 (2009), 2631–2647 | DOI | MR | Zbl

[16] L. Diening, P. Harjulehto, P. Hästö, M. R{ů}žička, Lebesgue and Sobolev spaces with Variable Exponents, Lecture Notes in Math., 2017, Springer, Heidelberg, 2011 | DOI | MR | Zbl

[17] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., 1034, Springer, Berlin, 1983 | DOI | MR | Zbl

[18] C. Bennett, R. Sharpley, Interpolation of Operators, Pure Appl. Math., 129, Acad. Press, Boston, MA, 1988 | MR | Zbl