Integral equations on compact CR manifolds

Assume that \begin{document}$ M $\end{document} is a CR compact manifold without boundary and CR Yamabe invariant \begin{document}$ \mathcal{Y}(M) $\end{document} is positive. Here, we devote to study a class of sharp Hardy-Littlewood-Sobolev inequality as follows \begin{document}$ \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*} $\end{document} where \begin{document}$ G_\xi^\theta(\eta) $\end{document} is the Green function of CR conformal Laplacian \begin{document}$ \mathcal{L_\theta} = b_n\Delta_b+R $\end{document} , \begin{document}$ \mathcal{Y}_\alpha(M) $\end{document} is sharp constant, \begin{document}$ \Delta_b $\end{document} is Sublaplacian and \begin{document}$ R $\end{document} is Tanaka-Webster scalar curvature. For the diagonal case \begin{document}$ f = g $\end{document} , we prove that \begin{document}$ \mathcal{Y}_\alpha(M)\geq \mathcal{Y}_\alpha(\mathbb{S}^{2n+1}) $\end{document} (the unit complex sphere of \begin{document}$ \mathbb{C}^{n+1} $\end{document} ) and \begin{document}$ \mathcal{Y}_\alpha(M) $\end{document} can be attained if \begin{document}$ \mathcal{Y}_\alpha(M)> \mathcal{Y}_\alpha(\mathbb{S}^{2n+1}) $\end{document} . So, we got the existence of the Euler-Lagrange equations \begin{document}$ \begin{equation} \varphi^{\frac{Q-\alpha}{Q+\alpha}}(\xi) = \int_M [G_\xi^\theta(\eta)]^{\frac{Q-\alpha}{Q-2}}\varphi(\eta)\ dV_\theta, \quad 0 Moreover, we prove that the solution of (1) is \begin{document}$ \Gamma^\alpha(M) $\end{document} . Particular, if \begin{document}$ \alpha = 2 $\end{document} , 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 ). So, we got the existence of the Euler-Lagrange equations Moreover, we prove that the solution of (1) is Γ α (M ). 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.
1. 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 far-reaching 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 where L θ = b n ∆ b + R is the CR conformal Laplacian related to θ and b n = 2 + 2 n . For a given constant curvatureR, the existence of (1.1) is known as CR Yamabe (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).
which is closely related to a class of Hardy-Littlewood-Sobolev inequalities with Namely, for any f, g ∈ L 2Q/Q+α (M ) with 0 < α < Q, there exists some positive constant C(α, M ) such that .
(1.7) In fact, by the parametrix method, we know by [11] holds by a similar argument with Theorem 15.11 of [11].
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 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 second 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].
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 Theorem 2.1 (Sharp HLS inequality on H n ). For 0 < α < Q and p = 2Q Q+α . Then for any f, g ∈ L p (H n ), where And equality holds if and only if for some c 1 , c 2 ∈ C, r > 0 and a ∈ H n (unless f ≡ 0 or g ≡ 0). Here H is defined as

2.3.
Folland-Stein normal coordinates (see [11,20]). On some open set V ⊂ M , take a set of pseudohermitian frame {W 1 , · · · , W n }. Then, {W i , W i , T, i = 1, · · · , n} forms a local frame, where T is determined by θ(T ) = 1 and dθ(T, X) = 0 for all X ∈ T M . As the Theorem 4.3 and Remark 4.4 of [20], we can summarize the result of Folland-Stein normal coordinates in the following theorem.

Theorem 2.3 (Theorem 4.3 of [20]
). There is a neighbourhood Ω ⊂ M × M of the diagonal and a C ∞ mapping Θ : Ω → H n satisfying: 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 [11], 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
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  [20]). Γ β ⊂ Λ β/2 (loc) for 0 < β < ∞ and there exists some positive constant C such taht f Λ β/2 (U ) ≤ C f Γ β (U ) for any f ∈ C ∞ o (U ). 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 .
3. 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 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 Then,

With (3.3), we have
For I 2 , by HLS inequality (2.7), we have Hence, for small enough , 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 (Θ −1 Sending to 0 and then letting R, δ approach to zero, we obtain the estimate.

Subcritical HLS inequalities and their extremal function.
Proposition 4.1 (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 [11].
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) α−Q .
Define the extremal problem as Obviously, we know that D M,p,q < +∞ because of Proposition 4.2. Moreover, we have 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 (4.7) 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. 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. By Lemma 4.7, 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 .