Critical Gagliardo-Nirenberg, Trudinger, Brezis-Gallouet-Wainger inequalities on graded groups and ground states

In this paper we investigate critical Gagliardo-Nirenberg, Trudinger-type and Brezis-Gallouet-Wainger inequalities associated with the positive Rockland operators on graded Lie groups, which includes the cases of $\mathbb R^n$, Heisenberg, and general stratified Lie groups. As an application, using the critical Gagliardo-Nirenberg inequality, the existence of least energy solutions of nonlinear Schr\"{o}dinger type equations is obtained. We also express the best constant in the critical Gagliardo-Nirenberg and Trudinger inequalities in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The obtained results are already new in the setting of general stratified Lie groups (homogeneous Carnot groups). Among new technical methods, we also extend Folland's analysis of H\"older spaces from stratified Lie groups to general homogeneous Lie groups.

In [LL13], the authors developed a rearrangement-free argument without using symmetrization to establish the Trudinger-Moser inequalities in the unbounded space R n including Adams type inequalities on the higher order derivatives (and fractional order derivatives). Rearrangement fails on the Heisenberg group or higher order Sobolev spaces. In [LL13] the authors avoid such a rearrangement which is only available in the first order in the Euclidean spaces. We also refer to [Yan14] for a rearrangement free argument on the Heisenberg group, where the author obtained the Trudinger-Moser inequalities when p = Q by gluing local estimates with the cut-off functions. On the Heisenberg group, we also refer to [CL01], [LLT12] for an analogue of inequality (1.1) on domains of finite measure, and to [LL12], [LLT14], [CNLY12] and [Yan12] on the entire Heisenberg group as well as to [LL13], [RY19a] and [RY19b] on stratified (Lie) groups. We also refer to [LLZ18] for the results of concentration-compactness type on the Heisenberg group and beyond using the level set argument.
In this paper, we are interested in obtaining such inequalities on graded Lie groups. We use the strategy developed in [Oza95] and [Oza97] on R n .
A connected simply connected Lie group G is called a graded (Lie) group if its Lie algebra admits a gradation. The graded groups form the subclass of homogeneous nilpotent Lie groups admitting homogeneous hypoelliptic left-invariant differential operators ( [Mil80], [tER97], see also a discussion in [FR16,Section 4.1]). These operators are called Rockland operators from the Rockland conjecture, solved by Helffer and Nourrigat [HN79]. So, we understand by a Rockland operator any leftinvariant homogeneous hypoelliptic differential operator on G.
In this paper, we are interested in obtaining the inequality (1.1) associated with positive Rockland operators on graded groups. We are also interested to obtain critical Gagliardo-Nirenberg and Brezis-Gallouet-Wainger inequalities. Consequently, we give applications of these inequalities to the nonlinear subelliptic equations. As such, this is essentially the most general framework for such inequalities in the setting of nilpotent Lie groups. Indeed, if a nilpotent Lie group has a left-invariant hypoelliptic differential operator, then the group is graded, see Section 2 for definitions and some details.
From now on we let R be a positive Rockland operator, that is, a positive leftinvariant homogeneous hypoelliptic invariant differential operator on G of homogeneous degree ν. Its powers R a for any a > 0 are understood through the functional calculus on the whole of G, extensively analysed in [FR16,FR17]. We denote the Sobolev space by L p a (G) = L p a,R (G), for a > 0, defined by the norm We refer to [FR16,Theorem 4.4.20] for the independence of the spaces L p a (G) of a particular choice of the Rockland operator R.
Thus, in this paper we will show that for a graded group G of homogeneous dimension Q and for a positive Rockland operator R of homogeneous degree ν we have the following results: • (Critical Gagliardo-Nirenberg inequality) Let 1 < p < ∞. Then we have for any q with p ≤ q < ∞ and for any function f from the Sobolev space L p Q/p (G) on graded group G, where the constant C 1 depends only on p and Q. • (Trudinger inequality with remainders) Let 1 < p < ∞. Then there exist positive α and C 2 such that holds for any function f ∈ L p Q/p (G) with R Q νp f L p (G) ≤ 1, where 1/p + 1/p ′ = 1. Furthermore, we show that (1.3) and (1.4) are actually equivalent and give the relation between their best constants. In [RY18] using this result we obtained weighted versions of (1.4) on graded groups. In the case p = Q, for the best constant α in the weighted Trudinger-Moser inequalities we refer to [LL12, Theorem 1.6] on the Heisenberg group and to [LL13,Theorem G] on general stratified groups when for any fixed positive real number τ , and to [LLT14, Theorem 1.1] on the Heisenberg group when ∇ H u L Q (Hn) ≤ 1.
• (Brezis-Gallouet-Wainger inequality) Let a, p, q ∈ R with 1 < p, q < ∞ and a > Q/q. Then we have for any function f ∈ L p Q/p (G) ∩ L q a (G) with f L p Q/p (G) ≤ 1. • (Existence of ground state solutions) Let 1 < p < q < ∞. Then the Schrödinger type equation (5.1) has a least energy solution φ ∈ L p Q/p (G). Furthermore, we have d = L(φ), for the variational problem (5.4)-(5.7). The nonlinear equation (5.1) mentioned above appears naturally in the analysis of the best constants for the above inequalities: • (Best constants in critical Gagliardo-Nirenberg inequality) Let 1 < p < q < ∞. Let φ be a least energy solution of (5.1) and let C GN,R be the smallest positive constant of C 1 in (1.3). Then we have (1.6) Since (1.3) and (1.4) are equivalent with a relation between constants, this also gives the best constant in Trudinger inequalities.
We note that the above results are already new if G is a stratified group and R is the (positive) sub-Laplacian on G (so that also ν = 2).
The paper is organised as follows. In Section 2 we briefly recall main concepts of graded groups and fix the notation. The critical Gagliardo-Nirenberg inequality and Trudinger-type inequality (1.1) are obtained on graded groups in Section 3, where the constant C is given more explicitly. In Section 4, we prove the Brezis-Gallouet-Wainger inequalities on graded groups. Finally, applications are given to the nonlinear Schrödinger type equations in Section 5.

Preliminaries
Following Folland and Stein [FS82, Chapter 1] and the recent exposition in [FR16, Chapter 3] we recall that G is a graded (Lie) group if its Lie algebra g admits a gradation where the g ℓ , ℓ = 1, 2, ..., are vector subspaces of g, all but finitely many equal to {0}, and satisfying [g ℓ , g ℓ ′ ] ⊂ g ℓ+ℓ ′ ∀ℓ, ℓ ′ ∈ N. It is called stratified if g 1 generates the whole of g through these commutators. We fix a basis {X 1 , . . . , X n } of a Lie algebra g adapted to the gradation. By the exponential mapping exp G : g → G we get points in G: A family of linear mappings of the form is a family of dilations of g. Here A is a diagonalisable linear operator on g with positive eigenvalues. Every D r is a morphism of the Lie algebra g, i.e., D r is a linear mapping from g to itself with the property as usual [X, Y ] := XY − Y X is the Lie bracket. One can extend these dilations through the exponential mapping to the group G by where ν 1 , . . . , ν n are weights of the dilations. The sum of these weights Q := Tr A = ν 1 + · · · + ν n is called the homogeneous dimension of G. We also recall that the standard Lebesgue measure dx on R n is the Haar measure for G (see, e.g. [FR16, Proposition 1.6.6]). A homogeneous quasi-norm on G is a continuous non-negative function with the properties: • |x −1 | = |x| for any x ∈ G, • |λx| = λ|x| for any x ∈ G and λ > 0, • |x| = 0 if and only if x = 0. We will use the following polar decomposition for our analysis: there is a (unique) positive Borel measure σ on the unit sphere such that for any function f ∈ L 1 (G) we have Let G be a unitary dual of G and let H ∞ π be the space of smooth vectors for a representation π ∈ G. A Rockland operator R on G is a left-invariant differential operator which is homogeneous of positive degree and satisfies the condition: (Rockland condition) for each representation π ∈ G, except for the trivial representation, the operator π(R) is injective on H ∞ π , i.e., ∀υ ∈ H ∞ π , π(R)υ = 0 ⇒ υ = 0. Here π(R) := dπ(R) is the infinitesimal representation of the Rockland operator R as of an element of the universal enveloping algebra of G.
Different characterisations of such operators have been obtained by Rockland [Roc78] and Beals [Bea77]. For an extensive presentation about Rockland operators and for the theory of Sobolev spaces on graded groups we refer to [FR17] and [FR16,Chapter 4], and for the Besov spaces on graded groups we refer to [CR17].
Since we will not be using the representation theoretic interpretation of these operators in this paper, we define Rockland operators as left-invariant homogeneous hypoelliptic differential operators on G. This is equivalent to the Rockland condition as it was shown by Helffer and Nourrigat in [HN79].
The homogeneous and inhomogeneous Sobolev spacesL p a (G) and L p a (G) based on the positive left-invariant hypoelliptic differential operator R have been extensively analysed in [FR17] and [FR16,Section 4.4] to which we refer for the details of their properties. They generalise the Sobolev spaces based on the sub-Laplacian on stratified groups analysed by Folland in [Fol75]. We refer to the above papers for (noncritical) Sobolev inequalities in the setting of graded groups, and to [RTY20] to the determination of the best constants in non-critical Sobolev and Gagliardo-Nirenberg inequalities on graded groups.

Critical Gagliardo-Nirenberg and Trudinger inequalities
We recall the following Gagliardo-Nirenberg inequality (see [RTY20, Theorem 3.2]): Let a ≥ 0, 1 < p < Q a and p ≤ q ≤ pQ Q−ap . Then we have for all functions u from the homogeneous Sobolev spaceL p a (G) on the graded group G: (3.1) In this section, we show this inequality for a = Q/p, which can be viewed as a critical Gagliardo-Nirenberg inequality. Then, we prove Trudinger-type inequality on graded groups, and we show the equivalence of these two inequalities. We note that another version of the Gagliardo-Nirenberg inequalities was also given in [BFKG12].
In order to prove the critical Gagliardo-Nirenberg inequality, we need to recall the following results.
Theorem 3.1 ([FR16, Theorem 4.4.28, Part 6]). Let G be a graded Lie group of homogeneous dimension Q. Let 1 < p < ∞ and a, b, c ∈ R with a < c < b. Then we have for any function and the constant C depends only on a, b and c.
We will also need the following statement where we refer to Definition 4.1 for the precise definition of the operators of type ν.

Now let us state the critical Gagliardo-Nirenberg inequality.
Theorem 3.3. Let G be a graded Lie group of homogeneous dimension Q and let R be a positive Rockland operator of homogeneous degree ν.
for every q with p ≤ q < ∞ and for every function f ∈ L p Q/p (G), where the constant C 1 depends only on p and Q.
Remark 3.4. We note that when G = (R n , +) and R = −△ is the Laplacian, the inequality (3.3) was obtained in [OO91] for p = 2, in [KOS92] for p = n and in [Oza95] for general p as in Theorem 3.3.
Now we state the Trudinger-type inequality with the remainder estimate on graded groups.
Theorem 3.5. Let G be a graded Lie group of homogeneous dimension Q and let R be a positive Rockland operator of homogeneous degree ν. Let 1 < p < ∞. Then there exist positive α and C 2 such that holds for all functions Remark 3.6. The constant C 2 can be expressed in terms of the constant C 1 = C 1 (p, Q) in (3.3) as follows Then, we have (3.12) for all α ∈ (0, (ep ′ C p ′ 1 ) −1 ) and C 2 (α).
Proof of Theorem 3.5. A direct calculation gives Since k ≥ p − 1, we have p ′ k ≥ p, then using Theorem 3.3 for the above integrals in the last line, we calculate which implies (3.12).
Now we show that the obtained critical Gagliardo-Nirenberg inequality (3.3) and Trudinger-type inequality (3.12) are actually equivalent on general graded groups. Note that it is already known in R n , see [Oza97].
Theorem 3.8. The inequalities (3.3) and (3.12) are equivalent. Furthermore, we have . (3.14) Proof of Theorem 3.8. We have already shown that (3.3) implies (3.12) in the proof of Theorem 3.5. By Remark 3.6, (3.14) and taking into account that A ≥ B, we note that α < (ep . (3.15) Then, we see that for any ε with 0 < ε < α there is C ε such that holds for all f ∈ L p Q/p (G). It follows that , that is, for all k ∈ N with k ≥ p−1. Let q > p and p ′ k ≤ q < p ′ (k+1). Then, by interpolating this between L p ′ k (G) and L p ′ (k+1) (G) and taking into account (k + 1)! ≤ Γ(2 + q/p ′ ), we get where Γ is the gamma function. Applying here the Stirling's formula and p ′ k ≥ q −p ′ , we obtain that there is r such that holds for all f ∈ L p Q/p (G) and q with r ≤ q < ∞. Thus, A ≤ (p ′ e( α − ε)) −1/p ′ + δ, then by arbitrariness of ε and δ we obtain This completes the proof of Theorem 3.8.

Brezis-Gallouet-Wainger inequalities
In this section we investigate Brezis-Gallouet-Wainger inequalities, which concern the limiting case of the Sobolev estimates (see [Bre82], [BG80] and [BW80]). As part of the proof we extend the analysis of Folland [Fol75] related to Hölder spaces from the setting of stratified to general homogeneous groups. For the background analysis on homogeneous groups we refer to Folland and Stein's fundamental book [FS82] as well as to a more recent treatment in [FR16].
Definition 4.1. Let G be a nilpotent Lie group and let λ be a complex number.
• A measurable function f on G is called homogeneous of degree λ if f •D r = r λ f for all positive r > 0, where D r is the family of dilations on G. • A distribution τ ∈ D ′ is called homogeneous of degree λ if τ, φ • D r = r −Q−λ τ, φ for all φ ∈ D and all positive r > 0. • A distribution which is smooth away from the origin and homogeneous of degree λ − Q is called a kernel of type λ on G.
We also need the following results: Proposition 1.4]). Let G be a nilpotent Lie group and let | · | be a homogeneous quasi-norm on G. Then, there is a positive constant C such that |xy| ≤ C(|x| + |y|), ∀x, y ∈ G. . Let G be a nilpotent Lie group and let | · | be a homogeneous quasi-norm on G. For any f ∈ C 2 (G\{0}) homogeneous of degree λ ∈ R, there are constants C, ε > 0 such that Let us now state the first main result of this section.
Theorem 4.4. Let G be a graded Lie group of homogeneous dimension Q and let R be a positive Rockland operator of homogeneous degree ν. Let a, p, q ∈ R with 1 < p, q < ∞ and a > Q/q. Then there exists C 3 > 0 such that we have for all functions f ∈ L p Q/p (G) ∩ L q a (G) with f L p Q/p (G) ≤ 1. Remark 4.5. In the case G = (R n , +) and R = −△ is the Laplacian, the inequality (4.3) was obtained in [BW80] by employing Fourier transform methods, and in [Eng89] for n/p, m ∈ Z, and in [Oza95] for the general case without using the Fourier transform.
We recall the Lipschitz spaces and obtain an estimate on nilpotent group G, which will be used in the proof of Theorem 4.4. So, let C b (G) be the space of bounded continuous functions on G. Then we define for 0 < α < 1, and when α = 1, we define Note that the Lipschitz space Γ α with 0 < α ≤ 1 is a Banach space with norm f Γα(G) = f L ∞ (G) + |f | α .
Lemma 4.6. Let G be a homogeneous Lie group of homogeneous dimension Q. Let 0 < λ < Q, 0 < α ≤ 1 and 1 < p ≤ ∞ with α = λ − Q/p. Let K be a kernel of type λ. Then we have for the mapping T : f → f * K.
We can also obtain the following estimate using Theorem 3.3: Theorem 4.7. Let G be a graded Lie group of homogeneous dimension Q and let 1 < p < ∞. Then we have for any function f ∈ L p Q/p (G) and for any Lebesgue measurable set Ω with |Ω| < ∞, where the constant C 4 depends only on p and Q, and |Ω| denotes the Lebesgue measure of Ω.

Best constants and nonlinear Schrödinger type equations
In this section using the critical case of the Gagliardo-Nirenberg inequality (3.3) we show the existence of least energy solutions for nonlinear Schrödinger type equations, and we obtain a sharp expression for the smallest positive constant C 1 in (3.3). For non-critical case on nilpotent Lie group, when a = Q/p in the inequality (3.1), similar results were obtained in [CR13] on the Heisenberg group and in [RTY20] on graded groups.
Let R be a positive Rockland operator of homogeneous degree ν, on a graded group G of homogeneous dimension Q. Let 1 < p < q < ∞. We consider the nonlinear equation with the power nonlinearity Such an equation appears naturally in the analysis of best constants of the established critical inequalities. We prove the existence of least energy solutions to (5.1) and their relation to the best constants in the Gagliardo-Nirenberg inequalities and hence, in view of Theorem 3.8, also to the best constants in the Trudinger inequalities. For example, for p = 2, if G is a stratified Lie group of homogeneous dimension Q and R = L is the positive sub-Laplacian, equation (5.1) becomes and if p = 2, equation (5.1) can be regarded as the p-sub-Laplacian version of (5.2). In the Euclidean case, when Q = 2, q = 4 and L = −∆, one can note that if u(x) is a solution of (5.2) then the function w(x) = u(x)e it/2 solves the following nonlinear Schrödinger equation Such equations arise in modeling the propagation of a thin electromagnetic beam through a medium with an index of refraction dependent on the field intensity (see for example [CGT64] and [Wei83, Section V]). Now, let us give some notations and definitions for this section.
Definition 5.1. A function u ∈ L p Q/p (G) is said to be a solution of (5.1) if and only if for any function ψ ∈ L p Q/p (G) the equality holds true.
We define the functionals L : L p Q/p (G) → R and I : L p Q/p (G) → R acting on L p Q/p (G) ∩ L q (G) as follows: Theorem 5.3. Let G be a graded Lie group of homogeneous dimension Q, let 1 < p < q < ∞. Then the Schrödinger type equation (5.1) has a least energy solution φ ∈ L p Q/p (G). Furthermore, we have d = L(φ).
In the sequel, we assume 1 < p < q < ∞. Let us state and prove the following lemmas, which will be used in the proof of Theorem 5.3.
Proof of Lemma 5.4. For any u ∈ L p Q/p (G) \ {0} and we note that µ u u ∈ N . It is easy to see that this µ u is unique. Then, by (5.8) one gets 0 < µ u < 1 provided that u p L p Q/p (G) < u q L q (G) .
Lemma 5.5. For all functions u ∈ N , we have inf u u L p Q/p (G) > 0. Proof of Lemma 5.5. Using (3.3) we calculate for all u ∈ N . From this we obtain u q−p L p Q/p (G) ≥ C −1 , which implies u L p Q/p (G) ≥ κ for any function u ∈ N after setting κ = C − 1 q−p > 0.
Let us show the following Rellich-Kondrachev type lemma. On the Heisenberg group a similar result was obtained by Garofalo and Lanconelli [GL92].
Lemma 5.6. Let 1 < p < q < ∞. Then, we have the compact embedding L p Q/p (D) ֒→ L q (D) for any smooth bounded domain D ⊂ G, where Proof of Lemma 5.6. Because of the density argument, it is enough to proveL p Q/p (D) ֒→ L q (D), whereL p Q/p (D) denotes the closure of C ∞ 0 (D) with respect to the norm (1.2) with a = Q/p.
where f ∈ L 1 loc (G) and φ ε (x) := ε −Q φ(ε −1 x) for every ε > 0. We will also use the following lemma, which is an analogue of [GL92, Lemma 3.1] for graded groups. (1) Z is bounded; ( Proof of Lemma 5.7. Let us briefly sketch the proof of the lemma. To show the necessity, we extend functions in L q (D) with zero outside D. We can take f 1 , . . . , f n from Z and r > 0, so that the balls in L q (D) centred at f k with radius r cover Z.
For a given f , let us take f k such that f k − f q < r. Then we have Taking into account K ε f q ≤ f q and K ε f → f in L q (D) as ε → 0 also letting r → 0, we get (2). We now show sufficiency. Let f n be a bounded sequence in Z. By the Banach-Alouglu theorem and the reflexivity of L q , 1 < q < ∞, we know that there exists a subsequence, still denoted by f n weakly convergent in L q , that is, there exists f ∈ L q such that (5.10) Let us now show that it actually converges strongly. For this, we write By the assumption (2), we note that the first and third summands vanish when ε → 0. For the second summand, (5.10) implies that for all x and for all ε we have Thus, from (5.11) we can conclude that Z is relatively compact in L q (D).
We now come back to the proof of Lemma 5.6. Setting f ≡ 0 in G\D for f ∈ L p Q/p (D), we get a function in L p Q/p (G). Let us now use Lemma 5.7. Let Z be a bounded set inL p Q/p (D). Then, using |B(x, r)| = r Q |B(0, 1)| for the Haar measure of any open quasi-ball (see e.g. [FR16,Page 140]) and the critical Gagliardo-Nirenberg inequality (3.3), we note that Z is bounded in L q (D).
To complete the proof, it remains to verify the second part of Lemma 5.7. For f ∈ Z by denoting ψ ε := K ε f − f and using (3.3), we obtain Therefore, it is enough to show that Therefore, by Lemma 5.7 we can conclude that Z is relatively compact in L q (D).
We also note the following property of least energy solutions.
Lemma 5.8. If v ∈ N and L(v) = d then v must be a least energy solution of the Schrödinger type equation (5.1).
Proof of Lemma 5.8. By the Lagrange multiplier rule there is θ ∈ R such that for any ψ ∈ L p Q/p (G) we have due to the assumption on v, where ·, · G is a dual product between L p Q/p (G) and its dual space.
Taking into account q > p, one gets On the other hand, we have Combining (5.14) and (5.15), we obtain θ = 0 from (5.13). It implies that L ′ (v) = 0. By Definition 5.2, we obtain that v is a least energy solution of the nonlinear equation (5.1).
Now we are ready to prove Theorem 5.3.
Proof of Theorem 5.3. We choose (v k ) k ⊂ N as a minimising sequence. By Ekeland variational principle we obtain a sequence (u k ) k ⊂ N satisfying L(u k ) → d and L ′ (u k ) → 0. Applying the Sobolev inequality and Lemma 5.5 we see that there exist two positive constants A 1 and A 2 with the properties From this and the equality we obtain the existence of a positive constant A 3 so that lim sup Taking into account the bi-invariance of the Haar measure and the left invariance of the operator R one has for allx k ∈ G L(u k (xx)) = L(u k (x)) and I(u k (xx)) = I(u k (x)).
Let us denote ω k (x) := u k (xx). Then we have L(ω k ) = L(u k ) and I(ω k ) = I(u k ). Moreover, it gives the bounded sequence (ω k ) k from L p Q/p (G) with There exists a subsequence, denoted by ω k that weakly converges to φ in L p Q/p (G). Then from Lemma 5.6 we see that ω k strongly converges to φ in L q loc (G). By this and (5.20), we have φ = 0. Now let us show that ω k converges strongly to φ in L p Q/p (G). First we show that I(φ) = 0. We proceed by contradiction. Suppose that I(φ) < 0. Lemma 5.4 gives that there is a positive number µ φ < 1 with µ φ φ ∈ N for I(φ) < 0. Since I(ω k ) = 0, applying the Fatou lemma we calculate which implies d > L(µ φ φ). Since µ φ φ ∈ N , we arrive at a contradiction. Suppose now that I(φ) > 0. We need the following lemma: Applying Lemma 5.4, there exists a sequence µ k := µ ψ k with µ k ψ k ∈ N and lim sup k→∞ µ k ∈ (0, 1). Indeed, assume that lim sup k→∞ µ k = 1. Then we have a subsequence (µ k j ) j∈N with the property lim j→∞ µ k j = 1. Since µ k j ψ k j ∈ N we get that I(ψ k j ) = I(µ k j ψ k j ) + o(1) = o(1), which is a contradiction because of (5.22). So, we have that lim sup k→∞ µ k ∈ (0, 1).
A direct calculation gives that (5.23) Therefore, the fact lim sup k→∞ µ k ∈ (0, 1) gives that d > L(µ k ψ k ). It contradicts µ k φ k ∈ N . Thus, we must have I(φ) = 0. Now we prove that ψ k = ω k − φ → 0 in the space L p Q/p (G). Indeed, suppose that ψ k L p Q/p (G) does not vanish as k → ∞. Then we consider the following cases. The first one is when G |ψ k (x)| q dx does not converge to 0 as k → ∞. Then the following identity In order to show (5.27), using (5.4) and (5.26), we calculate Then, it follows that which is (5.27).
Proof of Lemma 5.12. From the definition of T ρ 0 ,p,q , one has φ p L p Q/p (G) ≥ T ρ 0 ,p,q . We claim that T ρ 0 ,p,q ≥ φ p L p Q/p (G) . Indeed, using Lemma 5.4 for any u ∈ L p Q/p (G) satisfying G |u(x)| q dx = G |φ(x)| q dx there is a unique We estimate the sharp expression C GN,R by studying the following minimisation problem C −1 GN,R = inf{J(u) : u ∈ L p Q/p (G) \{0}}.