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

We derive the inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int_\mathbb{R}M(|f'(x)|h(f(x))) dx\leq C(M,h)\int_\mathbb{R}M\left({\sqrt{|f''(x)\tau_h(f(x))|}\cdot h(f(x))}\right)dx$$\end{document}with a constant C(M, h) independent of f, where f belongs locally to the Sobolev space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${W^{2,1}(\mathbb{R})}$$\end{document} and f′ has compact support. Here M is an arbitrary N-function satisfying certain assumptions, h is a given function and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau_h(\cdot)}$$\end{document} is its given transform independent of M. When M(λ) = λp and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h \equiv 1}$$\end{document} we retrieve the well-known inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\int_\mathbb{R}|f'(x)|^{p}dx \leq (\sqrt{p - 1})^{p}\int_\mathbb{R}(\sqrt{|f''(x) f(x)|})^{p}dx}$$\end{document}. We apply our inequality to obtain some generalizations of capacitary estimates and isoperimetric inequalities due to Maz’ya (1985).


Introduction
The following interesting inequality is well known:

272
A. Ka / lamajska and J. Peszek JFPTA another inspiring inequality can be found in the book by Maz'ya (see [10,Section 8]) : (1.2) where f ∈ C ∞ 0 (R) is nonnegative. Motivated by both, one may ask if there is an interplay between (1.1) and (1.2), some general inequality yielding these inequalities as a special case. In [8] we gave an affirmative answer to that question, obtaining the inequality 3) where f belongs locally to the Sobolev space W 2,1 (R) and f has bounded support, h(·) is a given function and T h (·) is its certain transform, independent of p. When h ≡ 1 we retrieve (1.1), while if h(λ) = λ − 1 2 we retrieve (1.2). For example, in the family of those inequalities one finds the following: (1.4) linking (1.1) and (1.2); see [8,Proposition 6.1] for details.
It appears that inequality (1.3) can be applied to the regularity theory in nonlinear boundary value problems; see [8,Section 7].
In this paper, we are inspired by the original motivation of inequality (1.2) due to Maz'ya. Namely, Maz'ya applied (1.2) as a key tool to obtain the capacitary inequality Ω cap + p (N t , Ω)t p−1 dt ≤ C Ω |∇ (2) u(x)| p dx, (1.5) where N t = {x ∈ Ω : u(x) ≥ t}, whenever E is compactly included in Ω. Let μ be a given Borel measure defined on an open set Ω, N the given N -function, N * the Legendre transform of N and L N (Ω, μ) an Orlicz space related to N (see Section 2). It is proven in [10,Theorem 8.3.1] (the original statement is given for R n , but the presented proof applies to the situation described below almost without changes) that the following statements (a) and (b) are equivalent.
(a) The embedding Vol. 13 (2013) On the Gagliardo-Nirenberg inequality 273 holds for every nonnegative u ∈ C ∞ 0 (Ω), with a u-independent finite constant A. (b) The following isoperimetric inequality Moreover, if A and B are the best constants in (1.6) and (1.7), respectively, then B ≤ A ≤ pBC, where C is the same as in (1.5).
One could ask about the validity of a more general embedding: where u ∈ C ∞ 0 (Ω) is nonnegative, with some u-independent constant A, with a (possibly) general convex function M instead of λ p . It appears that, under suitable assumptions on M , (1.8) is equivalent to the isoperimetric inequality The precise statement is given in Section 4, Proposition 4.2. Among these conditions on M , one requires the capacitary inequality holding for all nonnegative u ∈ C ∞ 0 (Ω), with a constantC independent of u. As we show (see Proposition 4.1), inequality (1.10) follows from the following generalization of inequality (1.2): then we apply its special variant (1.11) to prove inequality (1.10) and consequently the equivalence of (1.8) and (1.9). The precise arguments are provided in Section 4. To the best of our knowledge, such Orlicz generalizations was obtained in [7] as a special case of the related inequality in n dimensions. In particular, inequality (1.12), generalizing (1.3), links together inequalities (1.13) and (1.11), which are Orlicz extensions of inequalities (1.1) and (1.2). Apart from the purely theoretical approach, we hope that inequality (1.12) can serve as a tool to derive a priori estimates. For example, it can possibly play a similar role in nonlinear eigenvalue problems as that played by M (λ) = λ p ; see [8].
We also hope to contribute to the investigation of the theory of Sobolev spaces, with particular emphasis on embedding theorems; see, e.g., [1], [2] and [10].

General notation
In general we assume that Ω is an open subset of R n , n ≥ 1, and we use the standard notation: C ∞ 0 (Ω), W m,p (Ω) and W m,p loc (Ω) for smooth compactly supported functions and global and local Sobolev functions defined on Ω, respectively. By ∇ (2) u(x) we denote the Hesse matrix of the function u at the point x. We will also be dealing with the special situation when Ω = I ⊆ R is an interval (finite or not). If A ⊆ R and f is defined on A, by fχ A we denote the extension of f by zero outside the set A. More generally, if f is defined on A and g : R → R is zero outside A, by fg we denote the extension of fg by zero outside A. In what follows, M : [0, ∞) → [0, ∞) is continuous and locally absolutely continuous on (0, ∞). By dM we denote the measure M (t)dt defined on [0, ∞).
We will be dealing with integrals of the form R M (|f |c(f )) dx and is a continuous function and it might not be defined at zero. In all such cases we note that on the set A = {x : f (x) = 0} we have f = 0 and f = 0 almost everywhere, so that the functions |f |c(f ) and |f |c(f ) are, by our earlier definition, equal to zero almost everywhere on A. In particular, such integrals are interpreted, respectively, as |f |c(f ) dx.

The special transform
The following definition will be crucial for our considerations.

N -functions
By an N -function we call any function M : [0, ∞) → [0, ∞) which is convex and satisfies the conditions: In what follows, we use the following assumptions.  [3], [11] for definitions and [4], [5], [12] for discussion on those and other indices of Orlicz spaces). (5) If M is a differentiable N -function which satisfies (M) and M (λ) λ 2 is nondecreasing, then the function defined on R, is locally Lipschitz. To verify this we note that when λ > 0, we have A. Ka / lamajska and J. Peszek JFPTA so M 1 is bounded in every neighborhood of zero.
We will use the following lemmas. [9]). Suppose that M is an N -function satisfying (M). Then for every r > 0 and λ > 0,

Orlicz spaces
Let M be an N -function. By L M (Ω, μ) we denote the space of all real, μmeasurable functions, for which where μ is an arbitrary measure. It is known that L M (Ω, μ) is a Banach space with the norm · L M (Ω,μ) , and we have

Capacities
We will be using the following notion of capacity.

Definition 2.2. Let E, F ⊂ Ω and suppose that E is compactly included in F .
Assume further that M is an N -function. We define the capacity of E with respect to F as follows: where G := {u ∈ C ∞ 0 (F ), u ≥ 0 on F, u ≡ 1 in a neighborhood of E}. We recall some well-known properties of the capacities (see [10]).

Useful lemma
In what follows, we will use the following simple observation. [10]

Interpolation inequality
Our goal is to obtain the following result.
Proof. Let f ∈ W 2,1 loc (R) be a given nonnegative function and f := f + .

Denote further
A. Ka / lamajska and J.

Then we have
As d m ≥ 2, the function M 1 (λ) is locally Lipschitz (see Remark 2.1, parts (3), (5) and assumption (M)). Moreover, f h(f ) belongs to W 1,1 (R) and is compactly supported and bounded, in particular f h(f ) is absolutely continuous on R. Therefore, M 1 (|f |h(f )) is absolutely continuous on R and is compactly supported (see Lemma 2.3). By similar arguments, the function H(f ) is locally absolutely continuous on R and we have (H(f )) = f h(f ) (in sense of distributions and almost everywhere on R). This allows us to integrate by parts in the expression above to get From now the proof follows under assumptions (i), (ii) and (iii) considered separately.
(i) Note that where λ f := f h(f ). Consequently, We apply Lemma 2.1 to estimate The first integral equals (D M − 1)δI( ). To estimate the second one, we note that On the Gagliardo-Nirenberg inequality 279 and that according to Lemma 2.2,

This implies that
On the other hand, we have (3.5) Combining estimations (3.4) and (3.5) we obtain Consequently, when (D M − 1)(δ + E) < 1, we have δ < y ≤ 1 and I( ) ≤ C(δ, y)J( ), (3.6) where in our case The minimization of C(δ, y) with respect to δ < y gives the inequality with the constant C(y), achieved at and by Lebesgue's dominated convergence theorem we obtain On the other hand, when <δ, for the sufficiently smallδ and |f (x)| <δ, we can assume that → h(f (x)) either increases when converges to zero or it is bounded by a constant independent of x. Therefore, by Lebesgue's monotonic convergence theorem or by Lebesgue's dominated convergence theorem we obtain Similar arguments applied to the right-hand side in (3.6) give This finishes the proof of part (i).
(ii) In this case, instead of (3.3) we use the precise equation (3.7) Moreover, as (−1)h = |h |, the integrand in the second term above equals under notation (3.2). Therefore, we can equivalently write where A is the same as in This gives This, combined with (3.8) and (3.4), gives Now it suffices to rearrange to get (3.6), where This assumption does not necessarily force the condition δ < 1. Minimization with respect to y > δ > 0 gives inequality (3.6) with constant C(y). Finally, we let converge to zero and complete the proof in the same way as we have finished the proof of part (i).
(iii) Now h ≥ 0 and so we can modify (3.7) to get This implies (3.6) with y = (dM −1)e+1 DM −1 . Minimization of constants, then the final step, letting converge to zero and final conclusion follows by almost the same arguments. Proof. For α > 0 we easily verify condition (iii), while if α < −1, we have condition (ii), as then If −1 < α < 0, the condition e = |α| |α + 1| > 1 d M − 1 must be assumed and then again we can use condition (ii).  Its minimum is achieved at As now T h (f )h(f ) ≡ √ 2, we obtain the inequality