## Abstract

We derive the inequality

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).

## References

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.B. Bojarski,

*Remarks on Sobolev imbedding inequalities*. In: Complex Analysis (Joensuu 1987), Lecture Notes in Math. 1351, Springer, Berlin, 1988, 52–68.Boyd D. W.: Indices for the Orlicz spaces. Pacific J. Math.

**38**, 315–323 (1971)A. Fiorenza and M. Krbec,

*Indices of Orlicz spaces and some applications*. Comment. Math. Univ. Carolin.**38**(1997), 433–451.Gustavsson J., Peetre J.: Interpolation of Orlicz spaces. Studia Math.

**60**, 33–59 (1977)G. H. Hardy, J. E. Littlewood and G. Pólya,

*Inequalities*. Reprint of the 1952 edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988.K. Pietruska-Pałuba and A. Kałamajska,

*Interpolation inequalities for derivatives in Orlicz spaces*. Indiana Univ. Math. J.**55**(2006), 1767–1790.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.M. A. Krasnoseĺskiĭ and Ja. B. Rutickiĭ,

*Convex Functions and Orlicz Spaces*. P. Noordhoff Ltd., Groningen, 1961.V. G. Maz’ya,

*Sobolev Spaces*. Springer, Berlin, 1985.I. B. Simonenko,

*Interpolation and extrapolation of linear operators in Orlicz spaces*. Mat. Sb. (N.S.)**63**(1964), 536–553 (in Russian).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.

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Kałamajska, A., Peszek, J. On certain generalizations of the Gagliardo–Nirenberg inequality and their applications to capacitary estimates and isoperimetric inequalities.
*J. Fixed Point Theory Appl.* **13**, 271–290 (2013). https://doi.org/10.1007/s11784-013-0106-7

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DOI: https://doi.org/10.1007/s11784-013-0106-7

### Mathematics Subject Classification

- Primary 46E35
- Secondary 26D10

### Keywords

- Gagliardo–Nirenberg inequalities
- interpolation inequalities
- capacities
- isoperimetric inequalities