On certain generalizations of the Gagliardo–Nirenberg inequality and their applications to capacitary estimates and isoperimetric inequalities

摘要

We derive the inequality$$int_mathbb{R}M(|f'(x)|h(f(x))) dxleq C(M,h)int_mathbb{R}Mleft({sqrt{|f''(x)tau_h(f(x))|}cdot h(f(x))}right)dx$$with a constant C(M, h) independent of f, where f belongs locally to the Sobolev space ({W^{2,1}(mathbb{R})}) and f′ has compact support. Here M is an arbitrary N-function satisfying certain assumptions, h is a given function and ({tau_h(cdot)}) is its given transform independent of M. When M(λ) = λ p and ({h equiv 1}) we retrieve the well-known inequality ({int_mathbb{R}|f'(x)|^{p}dx leq (sqrt{p - 1})^{p}int_mathbb{R}(sqrt{|f''(x) f(x)|})^{p}dx}). We apply our inequality to obtain some generalizations of capacitary estimates and isoperimetric inequalities due to Maz’ya (1985). Mathematics Subject Classification Primary 46E35 Secondary 26D10 Keywords Gagliardo–Nirenberg inequalities interpolation inequalities capacities isoperimetric inequalities References1.B. Bojarski, Geometric properties of the Sobolev mapping. In: Function Spaces, Differential Operators and Nonlinear Analysis (Sodankylä, 1988), Pitman Res. Notes Math. Ser. 211, Longman Sci. Tech., Harlow, 1989, 225–241.2.B. Bojarski, Remarks on Sobolev imbedding inequalities. In: Complex Analysis (Joensuu 1987), Lecture Notes in Math. 1351, Springer, Berlin, 1988, 52–68.3.Boyd D. W.: Indices for the Orlicz spaces. Pacific J. Math. 38, 315–323 (1971)MathSciNetCrossRef4.A. Fiorenza and M. Krbec, Indices of Orlicz spaces and some applications. Comment. Math. Univ. Carolin. 38 (1997), 433–451.5.Gustavsson J., Peetre J.: Interpolation of Orlicz spaces. Studia Math. 60, 33–59 (1977)MathSciNetMATH6.G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities. Reprint of the 1952 edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988.7.K. Pietruska-Pałuba and A. Kałamajska, Interpolation inequalities for derivatives in Orlicz spaces. Indiana Univ. Math. J. 55 (2006), 1767–1790.8.A. Kałamajska and J. Peszek, On some nonlinear extensions of the Gagliardo- Nirenberg inequality with applications to nonlinear eigenvalue problems. Asymptot. Anal. 77 (2012), 169–196.9.M. A. Krasnoseĺskiĭ and Ja. B. Rutickiĭ, Convex Functions and Orlicz Spaces. P. Noordhoff Ltd., Groningen, 1961.10.V. G. Maz’ya, Sobolev Spaces. Springer, Berlin, 1985.11.I. B. Simonenko, Interpolation and extrapolation of linear operators in Orlicz spaces. Mat. Sb. (N.S.) 63 (1964), 536–553 (in Russian).12.I. Yaquiang, Relationship between Matuszewska-Orlicz, Semenov and Simonenko indices of ({varphi}) -functions. Publ. de l’Inst. Math., Nouv. Ser. (73)(87) (2003), 139–147.Copyright information©The Author(s)2013 Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.