Sharp Hardy-Littlewood-Sobolev inequalities on compact CR manifold

Assume that $M$ is a CR compact manifold without boundary and CR Yamabe invariant $\mathcal{Y}(M)$ is positive. Here, we devote to study a class of sharp Hardy-Littlewood-Sobolev inequality as follows \begin{equation*} \Bigl| \int_M\int_M [G_\xi^\theta(\eta)]^{\frac{Q-\alpha}{Q-2}} f(\xi) g(\eta) dV_\theta(\xi) dV_\theta(\eta) \Bigr| \leq \mathcal{Y}_\alpha(M) \|f\|_{L^{\frac{2Q}{Q+\alpha}}(M)} \|g\|_{L^{\frac{2Q}{Q+\alpha}}(M)}, \end{equation*} where $G_\xi^\theta(\eta)$ is the Green function of CR conformal Laplacian $\mathcal{L_\theta}=b_n\Delta_b+R$, $\mathcal{Y}_\alpha(M)$ is sharp constant, $\Delta_b$ is Sublaplacian and $R$ is Tanaka-Webster scalar curvature. For the diagonal case $f=g$, we prove that $\mathcal{Y}_\alpha(M)\geq \mathcal{Y}_\alpha(\mathbb{S}^{2n+1})$ (the unit complex sphere of $\mathbb{C}^{n+1}$) and $\mathcal{Y}_\alpha(M)$ can be attained if $\mathcal{Y}_\alpha(M)>\mathcal{Y}_\alpha(\mathbb{S}^{2n+1})$. Particular, if $\alpha=2$, the previous extremal problem is closely related to the CR Yamabe problem. Hence, we can study the CR Yamabe problem by integral equations.

where G θ ξ (η) is the Green function of CR conformal Laplacian L θ = bn∆ b + R, Yα(M ) is sharp constant, ∆ b is Sublaplacian and R is Tanaka-Webster scalar curvature. For the diagonal case f = g, we prove that Yα(M ) ≥ Yα(S 2n+1 ) (the unit complex sphere of C n+1 ) and Yα(M ) can be attained if Yα(M ) > Yα(S 2n+1 ). Particular, if α = 2, the previous extremal problem is closely related to the CR Yamabe problem. Hence, we can study the CR Yamabe problem by integral equations.

Introduction
CR geometry, the abstract models of real hypersurfaces in complex manifolds, has attracted much attention in the past decades. Noticing that there is a farreaching analogy between conformal and CR geometry, such as Model space, scalar curvature, Sublaplacian and Yamabe equation etc., many interesting and profound results on CR geometry were obtained, see [2, 5, 6, 8-14, 16, 18-24, 27, 28, 34, 35] and the references therein. Inspired by the idea of [7,16,17,36], we want to study the curvature problem of CR geometry from the point of integral curvature equation. Following, involved notations can be found in the Section 2.
Let (M, J, θ) be a compact psedudohermitian manifold without boundary. Under the transformationθ = φ 4 Q−2 θ with φ ∈ C ∞ (M ) and φ > 0, Tanaka-Webster scalar curvatures R andR, corresponding to θ andθ respectively, satisfy and proved that Y(M ) ≤ Y(S 2n+1 ) and the infimum can be attained if Y(M ) < Y(S 2n+1 ). It is well known that the case of Y(M ) ≤ 0 is easy to deal. While for the positive case, it is complicated. If Y(M ) > 0, then the first eigenvalue λ 1 (L θ ) > 0 and then L θ is invertible. Furthermore, for any ξ ∈ N , there exists a Green function G θ ξ (η) of L θ such that the solution of (1.1) satisfies φ(ξ) = (1.5) As pointed by Zhu in [36], on S n integral curvature equations are equivalent to the classical curvature quation if α is strictly less than dimension; while for the case α strictly greater than dimension, they are not equivalent and integral curvature equation has some advantages. So, it is interesting and valuable to study the integral curvature equation (1.5).
Moreover, if G θ ξ (η) = G θ η (ξ), we note that (1.5) is the Euler-Lagrange equation of the extremal problem which is closely related to a class of Hardy-Littlewood-Sobolev inequalities with .
In this paper, we will mainly devoted to study the extremal problem (1.6) by Hardy-Littlewood-Sobolev inequalities and will prove the following results.
Because of the hypoellipticity of operator L (in fact L satisfies the Hörmander condition [18]), we know that the Green function G θ ξ (η) is C ∞ if ξ = η. Moreover, using CR normal coordinates at ξ and the classical method of parametrix, we can construct the Green function as (without loss of generality, we take the coefficient of singular part as one) where w is the regular part. Particular, if M is locally CR conformal flat, then w satisfies ∆ b w = 0 in some neighbourhood of ξ. Therefore, w is C ∞ in this neighbourhood because of the hypoellipticity of ∆ b . If n = 1, Cheng, Malchiodi and Yang [5] proved that w ∈ C 1,γ (M ) for any γ ∈ (0, 1). In the sequel, we always assume that w(ξ, η) ∈ C 1 (M × M ). Then, we can rewrite G ξ (η) as On the other hand, on a locally CR conformal flat manifold, we note that ρ(ξ, η) = ρ(η, ξ) holds on some neighbourhood of diagonal of M × M . So, we give a sufficient condition for the strict inequality of Theorem 1.1.
The paper is organized as follows. In Section 2, we introduce the definition of CR manifold, some notations and some known results. Section 3 is mainly devoted to the first part of Theorem 1.1, namely, the estimation Y α (M ) ≥ Y α (S 2n+1 ). For the discussion of the second part of Theorem 1.1, we will adopt the blowup analysis. So, we will study the subcritical case of Hardy-Littlewood-Sobolev inequality in the Section 4. Then, in Section 5, we complete the proof of Theorem 1.1 and discuss the condition of strict inequality, namely Proposition 1.2. For completeness, we study the CR comformality of operator (1.3) in the Appendix A.
2.1. CR manifold and CR Yamabe problem. A CR manifold is a real oriented C ∞ manifold M of dimension 2n + 1, n = 1, 2, · · · , together with a subbundle T 1,0 of the complex tangent bundle CT M satisfying: for which there exists a natural volume form dV θ = θ ∧ dθ n . Any such θ is called a pseudo-hermitian structure M .
Associated with each θ, Levi form L θ is defined on H(M ) as By complex linearity, we can extend L θ to CH(M ) and induce a hermitian form on T 1,0 as It is easy to see that Levi form is CR invariant. Namely, if θ is replaced byθ = f θ, L θ changes conformally by Lθ = f L θ . We say M is nondegenerate if the Levi form is nondegenerate at every point, and say M is strictly pseudoconvex if the form is positive definite everywhere. In this paper, we always assume that M is strictly pseudoconvex.
Based on the Levi form L θ , we can take a local unitary frame {T α : α = 1, · · · , n} for T 1,0 (M ). Then, there is a natural seconde order differential operator, namely the Sublaplacian ∆ b , which is defined on the function u as where R is the Tanaka-Webster scalar curvatures and b n = 2Q Q−2 . Take u = φ, we have the prescribed curvature equation (1.1). Furthermore, for given constant curvatureR, the existence of (1.1) is known as CR Yamabe problem, which was introduced by Jerison and Lee, see [19,20].

2.2.
Heisenberg group H n , complex sphere S 2n+1 and the Cayley trans- A class of natural norm function is given by |u| = (|z| 4 + t 2 ) 1/4 and the distance between points u and v is defined by and the corresponding Levi form can be specified as The Sublaplacian are defined with respect to {T j , j = 1, 2, · · · , n} as and the Sublaplacian is On the sphere the distance function is defined as d(ζ, η) 2 = 2|1 − ζ ·η|. Cayley transform C : H n → S 2n+1 \(0, 0, · · · , 0, −1) and its reverse are defined as respectively. The Jacobian of the Cayley transform is Adopting the above notations, we can rewrite the sharp Hardy-Littlewood-Sobolev inequalities on H n and S 2n+1 (see Frank and Lieb's result [12]) as Q+α . Then for any f, g ∈ L p (H n ), where And equality holds if and only if

9)
for some c 1 , c 2 ∈ C, r > 0 and a ∈ H n (unless f ≡ 0 or g ≡ 0). Here H is defined as And equality holds if and only if for some c 1 , c 2 ∈ C and some ξ ∈ C n+1 with |ξ| < 1 (unless f ≡ 0 or g ≡ 0).

2.3.
Folland-Stein normal coordinates (see [10,20]  . Θ ξ is thus a diffeomorphism of a neighbourhood Ω ξ of ξ onto a neighbourhood of the origin in H n . Denote by u = (z, t) = Θ(ξ, η) the coordinates of H n . Denote by O k , k = 1, 2, · · · , a C ∞ function f of ξ and y such that for each compact set K ⊂⊂ V there is a constant C K , with |f (ξ, y)| ≤ C K |y| k (Heisenberg norm) for ξ ∈ K. Then, in which O k E indicates an operator involving linear combinations of the indicated derivatives with coefficients in O k , and we have used ∂ z to denote any of the derivatives ∂/∂z j , ∂/∂z j . (The uniformity with respect to ξ of bounds on functions in O k is not stated explicitly [10], but follows immediately from the fact that the coefficients are C ∞ .) Theorem 2.4 (Remark 4.4 of [20]). Let T δ (z, t) = (δ −1 z, δ −2 t), K ⊂⊂ V , and let r be fixed. With the notation of Theorem 2.3 and B r = {u ∈ H n : |u| ≤ r}, then T δ • Θ ξ (Ω ξ ) ⊃ B r for sufficiently small δ and all ξ ∈ K. Moreover, for ξ ∈ K and u ∈ B r , .
(Here O k may depeng also on δ, but its derivatives are bounded by multiplies of the frame constants, uniformly as δ → 0. Recall that T δ
Fix the local coordinates of U by u = (z, t) = Θ ξ for some given point ξ ∈ U . Then, for 0 < β < 1, the standard Hölder space Λ β (U ) is While for β ≥ 1, Λ β (U ) can be defined similarly. Furthermore, we have . Now for a compact strictly pseudoconvex pseudohermitian manifold M , choose a finite open covering U 1 , · · · , U m for which each U j has the properties of U above. Choose a C ∞ partition of unity φ i subordinate to this covering, and define Following, for convenience, denote p α = 2Q Q−α and q α = 2Q Q+α .

Estimation of the sharp constant
Proof. Since (G θ ξ (η)) Q−α Q−2 ∼ ρ(ξ, η) α−Q as ρ(ξ, η) → 0, then for any small enough δ > 0, there exists a neighbourhood V of the diagonal of M × M such that Recall that f (u) = H(u) is an extremal function to the sharp HLS inequality in Theorem 2.1, as well as its conformal equivalent class: Thus and f ǫ (u) satisfies integral equation where B is a positive constant. Let Σ R = {u = (z, t) ∈ H n : |z| < R, |t| < R 2 } be a cylindrical set, where R is a fixed constant to be determined later, and take a test function g(u) ∈ L qα (H n ) as With (3.3), we have For I 2 , by HLS inequality (2.7), we have Hence, for small enough ǫ, we have For any given point ξ ∈ M , there exists a neighbourhood V ξ ⊂ V such that Theorem 2.3 hold. So, choose R small enough such that Σ R ⊂ Θ ξ (V ξ ) and Then, sending R to 0 and then letting δ, ǫ approach to zero, we obtain the estimate.  Young's inequality). Let X and Y are measurable spaces, and let the kernel function K : X × Y → R be a measuralble function satisfying where C is some positive constant and r ≥ 1. Then, for any f ∈ L p (Y ) with 1 − 1/r ≤ 1/p ≤ 1, the integral operator Proof. For the case r = 1, the result reduces to the case of Lemma 15.2 of [10]. If r > 1, then 1 ≤ p ≤ r r−1 . When p = 1, by Minkowski's inequality, So, we take (G θ ξ (η)) Q−α Q−2 as the kernel of Proposition 4.1 and have the following subcritical HLS inequalities. where q > 1 and 1 q > 1 p − α Q . Moreover, operator A is compact for any q satisfying q > 1 and 1 q > 1 p − α Q , namely, for any bounded sequence {f j } +∞ j=1 ⊂ L p (M ), there exists a subsequence of {Af j } +∞ j=1 which converges in L q (M ). Proof. Obviously, it is sufficient to prove the compactness of the operator A with kernel K(ξ, η) = ρ(ξ, η) Since the sequence {f j } is bounded in L p (M ), then there exists a subsequence (still denoted by {f j }) and a function f ∈ L p (M ) such that For K(ξ, η), we decompose the integral operator as and K s (ξ, η) = K(ξ, η) − K s (ξ, η), s > 0 will be chosen later. Noting that, for any ξ, Next, we analyze the convergence of M K s (ξ, η)f j (η)dV η by Young's inequality (Proposition 4.1). Take r satisfying 1 r = 1 q − 1 p + 1. Then, r < Q Q−α and M |K s (ξ, η)| r dv η . By the Young's inequality, we have By now, through choosing first s small and then j large, we deduce by (4.3) and (4.4) that So, we complete the proof.
Define the extremal problem as Obviously, we know that D M,p,q < +∞ because of Proposition 4.2. Moreover, we have Af j → Af strongly in L q (M ).
So, f is a maximizer.
where the sharp constant can be attained by some nonnegative function f p ∈ L p (M ) satisfying f p L p (M) = 1 and Remark 4.5. A direct computation deduces that the extremal function f p satisfies the Euler-Lagrange equation Denoted by g(ξ) = f p−1 (ξ). Then (4.6) reduces to where q = p p−1 is the conjugate exponent of p. By a classical routine, we have the following regularity result. Proposition 4.6 (Γ α regularity). If g(ξ) ∈ L p (M ) satisfies (4.7), then g ∈ Γ α (M ).
The proof can be completed by the following two Lemmas.
Proof. Because of the compactness of M , it is sufficient to prove that, for any ξ ∈ M , Lemma 4.7 holds on the neighbourhood V ξ . Hence, without loss of generality, we restrict variable ξ on a neighbourhood V ξ0 for some point ξ 0 ∈ M .
Using the Folland-Stein normal coordinates, we can complete the proof by a similar process of the second part of Lemma 4.3 of [16]. For concise, we omit the details.

Sharp HLS inequalities on compact CR manifold
Following, we will investigate the limitation of the sequence of solutions {f p } ⊂ Γ α (M ) of (4.6), and then complete the proof of Theorem 5.1 by compactness.
First, it is routine to prove Proof. In view of the proof of Proposition 4.6, it is sufficient to prove that {f p } 2Q Q+α <p<2 is uniformly bounded in L ∞ (M ). Following, we will prove it by contradiction. Suppose not. Then f p (ξ p ) → +∞ as p → 2Q Q+α + , where f p (ξ p ) = max ξ∈M f p (ξ).
Let Θ ξp be normal coordinates. We can assume that there is a fixed neighbourhood U = B r (0) of the origin in H n contained in the image of Θ ξp for all p, and for each p we will use Θ ξp to identify U with a neighbourhood of ξ p with coordinates (z, t) = Θ ξp .
For any u = (z, t) = Θ ξp (ξ) ∈ U , we have f p (η)E(η, ξ)dV θ (η) Remark 5.5. In [5], Cheng, Malchiodi and Yang have proved a class of positive mass theorem in three dimensional CR geometry. Namely, under the assumptions of Y(M ) > 0 and the non-negativity of the CR Paneitz operator, they proved that, if M is not CR equivalent to S 3 (endowed with its standard CR structure), then A(ξ) > 0. Therefore, our main result holds for three dimensional compact CR manifold without boundary. While for other cases, it is still an open question as far as we know.