Equivalence of sharp Trudinger-Moser-Adams Inequalities

Improved Trudinger-Moser-Adams type inequalities in the spirit of Lions were recently studied in [ 21 ]. The main purpose of this paper is to prove the equivalence of these versions of the Trudinger-Moser-Adams type inequalities and to set up the relations of these Trudinger-Moser-Adams best constants. Moreover, using these identities, we will investigate the existence and nonexistence of the optimizers for some Trudinger-Moser-Adams type ineq


Introduction.
1.1. Trudinger-Moser inequality. Motivated by the applications to the prescribed Gauss curvature problem on two dimensional sphere S 2 , J. Moser proved in [29] an exponential type inequality on S 2 with an optimal constant. In the same paper of Moser [29], he sharpened an inequality on any bounded domain Ω in the Euclidean space R N studied independently by Pohozaev [30], Trudinger [35] and Yudovich [36], namely the embedding W 1,N 0 (Ω) ⊂ L ϕ N (Ω), where L ϕ N (Ω) is the Orlicz space associated with the Young function ϕ N (t) = exp α |t| N/(N −1) − 1 for some α > 0. More precisely, using the Schwarz rearrangement, Moser proved the following inequality in [29] (see also [3,10]): Theorem A (Trudinger-Moser inequality for finite-volume domain). Let Ω be a domain with finite measure in Euclidean N −space R N , N ≥ 2 and 0 ≤ β < N .
Then there exists a constant α N = N ω The constant α N is optimal in the sense that if we replace α N by any number α > α N , then the above supremum is infinite.
Moser used the following symmetrization argument: every function u is associated to a radially symmetric function u * such that the sublevel-sets of u * are balls 974 NGUYEN LAM with the same area as the corresponding sublevel-sets of u. Moreover, u is a positive and non-increasing function defined on B R (0) where |B R (0)| = |Ω|. Hence, by the layer cake representation, we can have that for any function f that is the difference of two monotone functions. In particular, we obtain Moreover, the well-known Pólya-Szegö inequality plays a crucial role in the approach of J. Moser.
As far as the existence of extremal functions of Moser's inequality, the first breakthrough was due to the celebrated work of Carleson and Chang [5] in which they proved that the supremum sup u∈W 1,N 0 (Ω), Ω |∇u| N dx≤1 can be achieved when Ω is an Euclidean ball. This result came as a surprise because it has been known that the Sobolev inequality does not have extremal functions supported on any finite ball. Subsequently, existence of extremal functions has been established on arbitrary domains in [7,8,12,27], and on Riemannian manifolds in [24], etc. We note when the volume of Ω is infinite, the Trudinger-Moser inequality (1) becomes meaningless. Thus, it becomes interesting and nontrivial to extend such inequalities to domains with infinite measure. Here we state the following such results in the Euclidean spaces (see [1,4,9,21,26,31]): Theorem B (Trudinger-Moser inequality for infinite-volume domain). Let 0 ≤ β < N and M > 1. Then we have The constant α N in the above supremums is sharp.
In the recent papers [6,22], it was showed that Theorem C. Let 0 ≤ β < N . Then there exist positive constants c = c (N, β) and C = C (N, β) such that when α is close enough to α N : Also, for a, b > 0, denote Then T M a,b,β (α N ) < ∞ if and only if b ≤ N . The constant α N is sharp. Moreover, we have the following identities: Concerning the existence and nonexistence of the best constants of the Trudinger-Moser type inequalities, the following results were proved in [11,15,16,17,23,26,31]: Then there exists δ ∈ (0, 4π) such that T M 0 (α) can be attained for all δ < α ≤ 4π and is not achieved for all 0 < α < δ. Moreover, in the later case, T M 0 (α) = α for all 0 < α < δ.

Adams inequalities.
It is worth noting that symmetrization has been a very useful and efficient (and almost inevitable) method when dealing with the sharp geometric inequalities. Thus, it is very fascinating to investigate such sharp geometric inequalities, in particular, the Trudinger-Moser type inequalities, in the settings where the symmetrization is not available such as on the higher order Sobolev spaces, the Heisenberg groups, Riemannian manifolds, sub-Riemannian manifolds, etc. Indeed, in these settings, an inequality like (2) is not available. In these situations, the first break-through came from the work of D. Adams [2] when he attempted to set up the Trudinger-Moser inequality in the higher order setting in Euclidean spaces. In fact, using a new idea that one can write a smooth function as a convolution of a (Riesz) potential with its derivatives, and then one can use the symmetrization for this convolution, instead of the symmetrization of the higher order derivatives, Adams proved the following inequality with boundary Dirichlet condition [2] which was extended to the Navier boundary condition in [34] when β = 0, and then to the case 0 ≤ β < N in [19]. The following is taken from [2,19]: Furthermore, the constant α(N, m) is optimal in the sense that for any α > α(N , m), the integral can be made as large as possible.
In Theorem E, we use the symbol ∇ m u, m is a positive integer, to denote the m−th order gradient for u ∈ C m , the class of m−th order differentiable functions: where ∇ is the usual gradient operator and is the Laplacian. Also, W (Ω) as a closed subspace. Adams inequalities have been extended to compact Riemannian manifolds in [13]. The Adams inequalities with optimal constants for high order derivatives on domains of infinite volume were recently established by Ruf and Sani in [32] in the case of even order derivatives and by Lam and Lu for all order of derivatives including fractional orders [14,20]. The idea of [32] is to use the comparison principle for polyharmonic equations (thus could deal with the case of even order of derivatives) and thus involves some difficult construction of auxiliary functions. The arguments in [14,20] uses the representation of the (Bessel) potentials and thus avoids dealing with such a comparison principle. In particular, the method developed in [20] adapts the idea of deriving the sharp Moser-Trudinger-Adams inequalities on domains of finite measure to the entire spaces using the level sets of the functions under consideration. Thus, the argument in [20] does not use the symmetrization method and thus also works for the sub-Riemannian setting such as the Heisenberg groups [18]. The following general version is taken from [20].
Theorem F Let 0 < γ < n be an arbitrary real positive number, p = n γ , p = p p−1 , and τ > 0. There holds Furthermore this inequality is sharp in the sense that if β 0 (n, γ) is replaced by any β > β 0 (n, γ), then the supremum is infinite.
Here W γ,p (R n ) is the fractional Sobolev space defined by the Bessel potential (see [33]). It is well-known that when γ ∈ N, W γ,p (R n ) is equivalent to the scale of the regular Sobolev space defined in the distribution sense.
We also have the following versions of the Adams type inequalities in the spirit of Lions (see [21]): Theorem G (Adams type inequalities for infinite-volume domain). Let N ≥ 3, 0 ≤ β < N and M > 1. Then we have Here

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Moreover, the constant α (N, 2) is the best possible. As a consequence, we have that there exists a constant C = C (N, β) > 0 such that The constant α (N, 2) in the above supremums is sharp.
In [22], it was also showed that Theorem H. Let 0 ≤ β < N . Then there exist positive constants c = c (N, β) and C = C (N, β) such that when α is close enough to α (N, 2) : Then A a,b,β (α (N, 2)) < ∞ if and only if b ≤ N 2 . The constant α (N, 2) is sharp. Moreover, we have the following identity: for all 0 < α ≤ α (N, 2) : Very little is known for existence of extremals for Adams inequalities. Existence of extremal functions for the Adams inequality on bounded domains in Euclidean spaces has been established in [28] and compact Riemannian manifolds by [25] only when N = 4 and m = 2 and is still widely open in other cases.
1.3. Our main results. The first main purpose of this article is to study relations of the sharp constants of the Trudinger-Moser type inequalities. We will prove that Theorem 1.1. Let N ≥ 2 and 0 ≤ β < N. Then for 0 < α ≤ α N : Using this result, we will then investigate the existence and nonexistence of optimizers for these Trudinger-Moser best constants. We will prove that As a consequence, from the results in [11,15,16,17,23,26,31] (Theorem D), we get that We also concern the best constants of the Adams type inequalities on W 2, N 2 R N : We will also investigate the attainability/unattainability of the Adams best constants in the singular case with subcritical growth. More precisely, we will prove that 2. Some lemmata. We first recall the following lemma that the proof can be found in [11,15,23]: Lemma 2.2. We have for 0 < α < α N : Lemma 2.3. We have for 0 < α < α N : Proof. Let u ∈ W 1,N R N : Then it is easy to see that Then it is easy to check that Lemma 2.5. N, 2)) . Proof.

Best constants of the Trudinger-Moser type inequalities.
3.1. Equivalence of Trudinger-Moser type inequalities-Proof of Theorem 1.1.
Proof of Theorem 1.1. We will first show that Indeed, by (5) and (6), we have Hence ∇v n N = 1 and v n N ≤ 1. Also, Now, let (w n ) be the maximizing sequence of LT M β,M (α N ), i.e., w n ∈ W 1,N R N \ {0} : ∇w n N < 1 and Hence ∇z n N = 1 and z n N ≤ 1. Also, Hence, we receive Similarly, we get for 0 < α ≤ α N :
Proof of Theorem 1.2. Let u be the optimizer of T M β (α): ∇u Hence ∇v N = 1 and v N = 1. Also, In other words, w is an optimizer for IT M β,M (α) and LT M β,M (α) . Moreover, the above process can be reversed. Hence, we can conclude that LT M β,M (α) and IT M β,M (α) can be attained if and only if T M β (α) is achieved.

Equivalence of Adams type inequalities-Proof of Theorem 1.6.
Proof of Theorem 1.6. We will show that Indeed, by (7): N, 2)) .
We define a new sequence: Hence, we can conclude that By the similar arguments, we get that for any 0 < α ≤ α (N, 2) :
Proof. We recall that WLOG, we can assume that u n u weakly in W 2, N 2 R N . As a consequence, u n → u a.e. on R N and ∆u N 2 a.e. in {|x| < R} and for some q 1 : by the Egorov theorem, we have that

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As a consequence, Hence, u = 0 and Hence, u is an optimizer for AT A β (α). Next, let ε n ↓ 0 and u n be the optimizer for AT A β (α + ε n ). We note here that by Lemma 2.4, we can assume that ∆u n N 2 = u n N 2 = 1. Again, WLOG, we can assume that u n u weakly in W 2, N 2 R N . As a consequence, u n → u a.e. on R N and ∆u N 2 Also, noting that in {|x| < R}: and there exists q 1 such that we have by the Vitali theorem that the quantity |x| β tends to 0 as n → ∞. Thus AT A β (α + ε n ) − AT A β (α) ↓ 0 as n → ∞.

Proof. We recall that
Also, it is clear that Now, we will also show that Indeed, let s n ↓ 0 and u n : ∆u n N 2 = u n N 2 = 1 be the optimizers for AT A β (s n ). Then By the Sobolev embeddings, Hölder's inequality and the Adams type inequalities, Hence as s n ↓ 0: Now, noting that AT A β (·) is continuous, we get That is, there exists s α ∈ (0, α) such that Let v α : ∆v α N 2 = v α N 2 = 1 be the optimizer for AT A β (s α ). We define

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Hence ∆w α That means that w α is a maximizer for A β (α).
Proof. We choose a smooth function v such that v N 2 = 1, and set v t (x) = t 2/N v t 1/N x , then it is easy to check that Also, for each t 0, there exists a unique c t > 0 such that with u t = c t v t : We note here that c t ↑ 1 as t ↓ 0. Then ! as t ↓ 0. Proof. Indeed, it is enough to show that for α 0 : We first note that by definition: for any γ ∈ (0, α (N, 2)) : Hence it is enough to choose α such that i.e. AT A 0 (γ)

α < γθ
Hence we now can conclude that A 0 (α) cannot be achieved for α 0 if N = 4.