On the Sobolev-Poincare inequality of CR-manifolds
aa r X i v : . [ m a t h . DG ] J a n ON THE SOBOLEV-POINCAR ´E INEQUALITY OFCR-MANIFOLDS
YI WANG AND PAUL YANG
Abstract.
The purpose is to study the CR-manifold with a con-tact structure conformal to the Heisenberg group. In our previouswork [WY17], we have proved that if the Q ′ -curvature is nonneg-ative, and the integral of Q ′ -curvature is below the dimensionalbound c ′ , then we have the isoperimetric inequality. In this pa-per, we manage to drop the condition on the nonnegativity of the Q ′ -curvature. We prove that the volume form e u is a strong A ∞ weight. As a corollary, we prove the Sobolev-Poincar´e inequalityon a class of CR-manifolds with integrable Q ′ -curvature. Introduction
On a four dimensional manifold, the Paneitz operator P and theBranson’s Q -curvature [Bra95] have many analogous properties as theLaplacian operator ∆ g and the Gaussian curvature K g on surfaces. ThePaneitz operator is defined as P g = ∆ + δ ( 23 Rg − Ric ) d, where δ is the divergence, d is the differential, R is the scalar curvatureof g , and Ric is the Ricci curvature tensor. The Q -curvature is definedas Q g = 112 (cid:26) − ∆ R + 14 R − | E | , (cid:27) where E is the traceless part of Ric , and | · | is taken with respect tothe metric g . The most important two properties for the pair ( P g , Q g )are that under the conformal change g w = e w g ,1. P g transforms by P g w ( · ) = e − w P g ( · );2. Q g satisfies the fourth order equation P g w + 2 Q g = 2 Q g w e w . The research of the author is partially supported by NSF grant DMS-1612015.The research of the author is partially supported by NSF grant DMS-1509505.
As proved by Beckner [Bec93] and Chang-Yang [CY13], the pair ( P g , Q g )also apperas in the Moser-Trudinger inequality for higher order opera-tors.On CR-manifold, it is a fundamental problem to study the existenceand analogous properties of CR invariant operator P and curvaturescalar invariant Q . Graham and Lee [GL88] has studied a fourth-orderCR covariant operator with leading term ∆ b + T and Hirachi [Hir93]has identified the Q -curvature which is related to P through a changeof contact form. However, although the integral of the Q -curvature ona compact three-dimensional CR-manifold is a CR invariant, it is al-ways equal to zero. And in many interesting cases when the CR threemanifold is the boundary of a strictly pseudoconvex domains, the Q -curvature vanishes everywhere. As a consequence, it is desirable tosearch for some other invariant operators and curvature invariants ona CR-manifold that are more sensitive in the CR geometry. The workof Branson, Fontana and Morpurgo [BFM13] aims to find such a pair( P ′ , Q ′ ) on a CR sphere. Later, the definition of Q ′ -curvature is gener-alized to all pseudo-Einstein CR-manifolds by the work of Case-Yang[CY13] and that of Hirachi [Hir14]. The construction uses the strategyof analytic continuation in dimension by Branson [Bra95], restricted tothe subspace of the CR pluriharmonic functions. P ′ := lim n → n − P ,n | P . Here P ,n is the fourth-order covariant operator that exists for everycontact form θ by the work of Gover and Graham [GG05]. By [GL88],the space of CR pluriharmonic functions P is always contained in thekernel of P .In this paper, we want explore the geometric meaning of this newlyintroduced conformal invariant Q ′ -curvature.In Riemannian geometry, a classical isoperimetric inequality on acomplete simply connected surface M , called Fiala-Huber’s [Fia41],[Hub57] isoperimetric inequality(1.1) V ol (Ω) ≤ π − R M K + g dv g ) Area ( ∂ Ω) , where K + g is the positive part of the Gaussian curvature K g . Also R M K + g dv g < π is the sharp bound for the isoperimetric inequality tohold.In [Wan15], we generalize the Fiala-Huber’s isoperimetric inequalityto all even dimensions, replacing the role of the Gaussian curvature indimension two by that of the Q -curvature in higher dimensions: N THE SOBOLEV-POINCAR´E INEQUALITY OF CR-MANIFOLDS 3
Let ( M n , g ) = ( R n , g = e u | dx | ) be a complete noncompact evendimensional manifold. Let Q + and Q − denote the positive and negativepart of Q g respectively; and dv g denote the volume form of M . Suppose g = e u | dx | is a “normal” metric, i.e.(1.2) u ( x ) = 1 c n Z R n log | y || x − y | Q g ( y ) dv g ( y ) + C ;for some constant C . If(1.3) α := Z M n Q + dv g < c n where c n = 2 n − ( n − )! π n , and(1.4) β := Z M n Q − dv g < ∞ , then ( M n , g ) satisfies the isoperimetric inequality with isoperimetricconstant depending only on n, α and β . Namely, for any boundeddomain Ω ⊂ M n with smooth boundary,(1.5) | Ω | n − n g ≤ C ( n, α, β ) | ∂ Ω | g . In our previous paper [WY17], we have studied the Q ′ -curvature and P ′ operator, and proved that if ( H , e u θ ) for pluriharmonic function u is a complete CR-manifold with nonnegative Q ′ curvature and nonneg-ative Webster scalar curvature at infinity, if in addition Q ′ curvaturesatisfies(1.6) Z H Q ′ e u θ ∧ dθ < c ′ , then e u is an A weight. Here c ′ is the constant in the fundamentalsolution of P ′ operator (See [WY17]). As a corollary, we have derivedthe isoperimetric inequality on CR-manifold ( H , e u θ ):(1.7) V ol (Ω) ≤ CArea ( ∂ Ω) / . Here the constant C is controlled by c ′ − R H Q ′ e u θ ∧ dθ . To provethis result, we notice that the class of pluriharmonic functions P is therelevant subspace of functions for the conformal factor u .The purpose of the current paper is two-fold. We will first study thecase when Q ′ curvature is negative. Then we will discuss the generalcase when Q ′ curvature does not have a sign. The main results of thepaper are stated in the following. YI WANG AND PAUL YANG
Theorem 1.1.
Let ( H , e u θ ) be a complete CR-manifold, where θ de-notes the contact form on the Heisenberg group H and u is a plurihar-monic funcion on H . If the Q ′ -curvature is negative, and the Websterscalar curvature is nonnegative at infinity. If (1.8) Z H Q ′ e u θ ∧ dθ < ∞ , then e u is a strong A ∞ weight. Note that e u is the volume form of this conformal metric, where 4 isthe homogeneous dimension of H . The descriptions of A weight andstrong A ∞ weight will be in Section 2.We will then discuss the case when the Q ′ -curvature does not havea sign. Theorem 1.2.
Let ( H , e u θ ) be a complete CR-manifold, where θ de-notes the contact form on the Heisenberg group H and u is a plurihar-monic funcion on H . If the Webster scalar curvature is nonnegativeat infinity, If (1.9) α := Z H Q ′ + e u θ ∧ dθ < c ′ , and (1.10) β := Z H Q ′− e u θ ∧ dθ < ∞ , then e u is a strong A ∞ weight. As a corollary of Franchi-Lu-Wheeden [FLW95], we will show that( H , e u θ ) satisfies Sobolev-Poincar´e inequality. We remark that on aCR-manifold ( H , e u θ ), the David-Semme’s [DS90] type of isoperimet-ric inequality is still an open question for strong A ∞ weights. Theorem 1.3.
Let ( H , e u θ ) satisfy the same assumptions as in Theo-rem 1.2. Let K be a compact subset of Ω . Then there exists r depend-ing on K, Ω , and { X j } such that if B = B ( x, r ) is a ball with x ∈ K and < r < r , and if e u is A p weight for some ≤ p < . Let µ ( x ) := e u dx , ν ( x ) := e (4 − p ) u dx . Then (1.11) ( 1 µ ( B ) Z B | f ( x ) − f B | q dµ ) /q ≤ cr ( 1 ν ( B ) Z B |∇ b f ( x ) | p dν ) /p , for any f ∈ Lip ( ¯ B ) , with f B = µ ( B ) R B f ( x ) dµ . The constant c dependsonly on K, Ω , α, β, p . N THE SOBOLEV-POINCAR´E INEQUALITY OF CR-MANIFOLDS 5 Preliminaries
On a Heisenberg group H n , one can also define the A p weight, inthe same way as on the Euclidean space R n . For a nonnegative localintegrable function ω , we call it an A p weight p >
1, if for all balls B in H n (2.1) 1 | B | Z B ω ( x ) dx · (cid:18) | B | Z B ω ( x ) − p ′ p dx (cid:19) pp ′ ≤ C < ∞ . Here p ′ + p = 1. The constant C is uniform for all B . The definitionof A weight is given by taking the limit process p →
1. Namely, ω iscalled an A weight, if(2.2) M ω ( x ) ≤ Cω ( x ) , for almost all x ∈ B .An important property of A p weight is the reverse H¨older inequality:if ω is an A p weight for some p ≥
1, then there exist an r >
C > B (2.3) (cid:18) | B | Z B ω r dx (cid:19) /r ≤ C | B | Z B ωdx. This would imply that any A p weight ω satisfies the doubling property:there exists a C > Z B ( x , r ) ω ( x ) dx ≤ C Z B ( x ,r ) ω ( x ) dx for all balls B ( x , r ).The notion of strong A ∞ weight was first proposed by David andSemmes in [DS90]. Given a positive continuous weight ω , we define(2.5) δ ω ( x, y ) := Z B x,y ω ( z ) dz ! /n , where B x,y is the ball with diameter | x − y | that contains x and y . Onthe other hand, we can define the geodesic distance with respect to theweight ω to be(2.6) d ω ( x, y ) := inf γ Z γ ω n ( s ) ds. Here γ ⊂ B x,y is a curve connecting x, y such that the tangent vectoris always contact. If ω is an A ∞ weight, then it is easy to prove (seefor example Proposition 3.12 in [Sem93])(2.7) d ω ( x, y ) ≤ Cδ ω ( x, y ) YI WANG AND PAUL YANG for all x, y ∈ H n . If in addition, ω also satisfies the reverse in equality(2.8) δ ω ( x, y ) ≤ Cd ω ( x, y )then we say ω is a strong A ∞ weight.The product of an A weight and an A ∞ weight is an A ∞ weight.This can be proved using the same proof as in the Euclidean space.3. CR-manifold with negative Q ′ -curvature In this section, we will prove Theorem 1.1. It shows that for CR-manifolds with negative Q ′ -curvature, the integral of Q ′ -curvature con-trols the geometry in a very rigid way.We first remark that since Q ′ ( y ) e u ( y ) is integrable, log | y || x − y | Q ′ ( y ) e nu ( y ) is also integrable in y for each fixed x ∈ H .In this section, we consider the analytic property of e u ( x ) . For sim-plicity, we denote it by ω ( x ). We define β := R H | Q ′ | ( y ) e u ( y ) dy < ∞ .Recall that for a nonnegative continuous function ω ( x ), d ω ( x, y ) := ( Z B xy ω ( z ) dz ) n ,δ ω ( x, y ) := inf γ Z γ ω n ( γ ( s )) ds, where B xy is the ball with diameter | x − y | that contains x and y , theinfimum is taken over all contact curves (meaning that the tangentvector on each point of this curve is contact) γ ⊂ B xy connecting x and y , and ds is the arc length.We want to prove ω ( x ) := e u ( x ) is a strong A ∞ weight, i.e. thereexists a constant C = C ( β ) such that(3.1) 1 C ( β ) d ω ( x, y ) ≤ δ ω ( x, y ) ≤ C ( β ) d ω ( x, y ) . Since the Webster scalar curvature is nonnegative at infiniity, by Propo-sition in [WY17], u is normal. Thus(3.2) u ( x ) = − c ′ Z H log | y || x − y | | Q ′ | ( y ) e u ( y ) dy. We first observe that without generality we can assume | x − y | = 2.This is because we can dilate u by a factor λ > u λ ( x ) := u ( λx ) = − c ′ Z H log | y || λx − y | | Q ′ | ( y ) e u ( y ) dy. (3.3) N THE SOBOLEV-POINCAR´E INEQUALITY OF CR-MANIFOLDS 7
By the change of variable, this is equal to − c ′ Z H log | y || x − y | | Q ′ | ( λy ) e u ( λy ) λ dy. Notice | Q ′ | ( λy ) e u ( λy ) λ is still an integrable function on H , with in-tegral equal to β . Thus by choosing λ = | x − y | , the problem reduces toproving inequality (3.1) for u λ and | x − y | = 2.Let us denote the midpoint of x and y by p . And from now on,we adopt the notation λB := B ( p , λ ). Since | x − y | = 2, we have B xy = B ( p ,
1) = B . We also define(3.4) u ( x ) := − c ′ Z B log | y || x − y | | Q ′ | ( y ) e u ( y ) dy, and(3.5) u ( x ) := − c ′ Z H \ B log | y || x − y | | Q ′ | ( y ) e u ( y ) dy. In the following lemma, we prove that when z is close to p , thedifference between u ( z ) and u ( p ) is controlled by β . Lemma 3.1. (3.6) | u ( z ) − u ( p ) | ≤ β c ′ for z ∈ B .Proof. | u ( z ) − u ( p ) | = 1 c ′ (cid:12)(cid:12)(cid:12)(cid:12)Z H \ B − log | y || z − y | | Q ′ | ( y ) e u ( y ) dy + Z H \ B log | y || p − y | | Q ′ | ( y ) e u ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) = 1 c ′ (cid:12)(cid:12)(cid:12)(cid:12)Z H \ B log | z − y || p − y | | Q ′ | ( y ) e u ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ | z − p | c ′ · Z H \ B | (1 − t ∗ )( p − y ) + t ∗ ( z − y ) | | Q ′ | ( y ) e u ( y ) dy, (3.7)for some t ∗ ∈ [0 , y ∈ H \ B and z, p ∈ B ,1 | (1 − t ∗ )( p − y ) + t ∗ ( z − y ) | ≤ , | u ( z ) − u ( p ) | is bounded by | z − p | c ′ · Z H \ B | Q ′ | ( y ) e u ( y ) dy. (3.8) YI WANG AND PAUL YANG
Note that for z ∈ B , | z − p | ≤
2. From this, (3.6) follows. (cid:3)
Now we adopt some techniques used in [BHS04] for potentials to dealwith the ǫ -singular set E ǫ . Lemma 3.2. (Cartan’s lemma) For the Radon measure | Q ′ | ( y ) e u ( y ) dy ,given ǫ > , there exists a set E ǫ ⊆ H , such that H ( E ǫ ) := inf E ǫ ⊆∪ B i { X i diam B i } < ǫ and for all x / ∈ E ǫ and r > , Z B ( x,r ) | Q ′ | ( y ) e u ( y ) dy ≤ rβǫ . The proof of Lemma 1 follows from standard measure theory argu-ment. Thus we omit it here.
Proposition 3.3.
Given ǫ > H (cid:18)(cid:26) x ∈ B : (cid:12)(cid:12)(cid:12)(cid:12) − c ′ Z B log 1 | x − y | | Q ′ | ( y ) e u ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) > C βǫ (cid:27)(cid:19) < ǫ. . Proof.
Fix ǫ >
0. By Lemma 3.2, there exists a set E ǫ ⊆ H , s.t. H ( E ǫ ) < ǫ and for x / ∈ E ǫ and r > Z B ( x,r ) | Q ′ | ( y ) e u ( y ) dy ≤ rβǫ . If we can show for some C (3.10)10 B \ E ǫ ⊆ (cid:26) x ∈ B : (cid:12)(cid:12)(cid:12)(cid:12) − c ′ Z B log 1 | x − y | | Q ′ | ( y ) e u ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ǫ β (cid:27) , then H (cid:18)(cid:26) x ∈ B : (cid:12)(cid:12)(cid:12)(cid:12) − c ′ Z B log 1 | x − y | | Q ′ | ( y ) e u ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) > C βǫ (cid:27)(cid:19) ≤ H ( E ǫ ) < ǫ. N THE SOBOLEV-POINCAR´E INEQUALITY OF CR-MANIFOLDS 9
To prove (3.10), we notice for x ∈ B \ E ǫ , r = 2 − j ·
10, (3.9) implies (cid:12)(cid:12)(cid:12)(cid:12) − c ′ Z B log 1 | x − y | | Q ′ | ( y ) e u ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ′ ∞ X j = − (cid:12)(cid:12)(cid:12)(cid:12)Z B ( x, − j · \ B ( x, − ( j +1) · log 1 | x − y | | Q ′ | ( y ) e u ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ′ ∞ X j = − (cid:0) max {| log 2 − j | , | log 2 − ( j +1) |} + log 10 (cid:1) · Z B ( x, − j · \ B ( x, − ( j +1) · | Q ′ | ( y ) e u ( y ) dy ≤ c ′ ∞ X j = − (cid:0) max {| log 2 − j | , | log 2 − ( j +1) |} + log 10 (cid:1) · − j · βǫ ≤ C βǫ , (3.11)where C = 10 P ∞ j = − (cid:0) max {| log 2 − j | , | log 2 − ( j +1) |} + log 10 (cid:1) · − j c ′ < ∞ . This completes the proof of the proposition. (cid:3)
We next estimate the integral of e u ( z ) over 2 B . Proposition 3.4.
Let ¯ c := − c ′ R B log | y || Q ′ | ( y ) e u ( y ) dy . ¯ c < ∞ , since | Q ′ | ( y ) e u ( y ) is continuous thus bounded near the origin. Then(3.12) Z B e u ( z ) dz ≤ C ( β ) e u ( p ) e c , for C depends only on β . Proof.
Recall(3.13) u ( x ) := − c ′ Z B log | y || x − y | | Q ′ | ( y ) e u ( y ) dy, and(3.14) u ( x ) := − c ′ Z H \ B log | y || x − y | | Q ′ | ( y ) e u ( y ) dy. By Lemma 3.1, Z B e u ( z ) dz = Z B e u ( z ) e u ( z ) dz ≤ e βc ′ e u ( p ) Z B e u ( z ) dz. (3.15)To estimate u , by definition β := R B | Q ′ | ( y ) e u ( y ) dy ≤ β < ∞ . If β = 0, then u ( z ) = 0 and ¯ c := − c ′ R B log | y || Q ′ | ( y ) e u ( y ) dy = 0. So(3.12) follows immediately. If β = 0, | Q ′ | ( y ) e u ( y ) β dy is a nonnegativeprobability measure on 10 B . Hence by Jensen’s inequality Z B e u ( z ) dz = e c · Z B e c ′ R B (log | z − y | ) | Q ′ | ( y ) e u ( y ) dy dz ≤ e c · Z B Z B | z − y | β c ′ | Q ′ | ( y ) e u ( y ) β dydz. (3.16)Since z ∈ B and y ∈ B ,(3.17) Z B | z − y | β c ′ dz ≤ C. From this, we get Z B e u ( z ) dz ≤ Ce c Z B | Q ′ | ( y ) e u ( y ) β dy = Ce c . (3.18)Plugging it to (3.15), we finish the proof of the proposition. (cid:3) Now we are ready to prove Theorem 1.1.
Proof of Theorem 1.1.
Let us assume ω := e u is an A p weight forsome large p , with bounds depending only on β . The proof of this factfollows that of Proposision 5.1 in [Wan15]. So we omit it here. By thereverse H¨older’s inequality for A p weights, it is easy to prove (see forexample Proposition 3.12 in [Sem93]), δ ω ( x, y ) ≤ C ( β ) d ω ( x, y ) . Hence we only need to prove the other side of the inequality:(3.19) δ ω ( x, y ) ≥ C ( β ) d ω ( x, y ) , for some constant C ( β ). By Proposition 3.3, for a given ǫ >
0, thereexists a Borel set E ǫ ⊆ H , such that(3.20) H ( E ǫ ) ≤ ǫ, and for z ∈ B \ E ǫ , according to (3.10)(3.21) | ˆ u ( z ) | ≤ C ǫ β. N THE SOBOLEV-POINCAR´E INEQUALITY OF CR-MANIFOLDS 11
Here ˆ u ( z ) := − c ′ Z B log 1 | x − y | | Q ′ | ( y ) e u ( y ) dy. With this, we claim the following estimate.
Claim:
Suppose H ( E ǫ ) < ǫ with ǫ ≤ . Then the length of γ \ E ǫ with respect to the metric of Heisenberg group H satisfies(3.22) length ( γ \ E ǫ ) > , where γ ⊂ B xy is a curve connecting x and y . Proof of Claim.
Let P be the projection map from points in B xy tothe contact line segment I xy between x and y . Since the Jacobian ofthe projection map is less or equal to 1,(3.23) length ( γ \ E ǫ ) ≥ length ( P ( γ \ E ǫ )) = m ( P ( γ \ E ǫ )) , where m is the arc length measure on the line segment I xy . Notice P ( γ ) = I xy , and P ( γ ) \ P ( E ǫ ) is a subset of P ( γ \ E ǫ ). Therefore(3.24) m ( P ( γ \ E ǫ )) ≥ m ( P ( γ )) − m ( P ( E ǫ )) = 2 − m ( P ( E ǫ )) . Now by assumption, H ( E ǫ ) < ǫ , so H ( γ ∩ E ǫ ) < ǫ . Hence thereis a covering ∪ i B i of γ ∩ E ǫ , so that X i diam B i < ǫ. This implies that ∪ i P ( B i ) is a covering of the set P ( γ ∩ E ǫ ) and X i diam P ( B i ) = X i diam B i ≤ ǫ. Thus m ( P ( E ǫ )) = H ( P ( E ǫ )) < ǫ < , by choosing ǫ ≤ . Plug itto (3.24), and then to (3.23). This completes the proof of the claim.We now continue the proof of Theorem 1.1. Since γ ⊂ B , then byLemma 3.1, Z γ e u − ( γ ( s )) ds = Z γ e ( u + u )( γ ( s )) ds ≥ e − β c ′ e u ( p ) e ¯ c Z γ e ˆ u ( γ ( s )) ds. (3.25)Here ¯ c is the constant defined in Proposition 3.4. Let ǫ = . By (3.21), | ˆ u ( z ) | ≤ C β for z ∈ B \ E ǫ . Thus(3.26) Z γ e ˆ u ( γ ( s )) ds ≥ e − C β length ( γ \ E ǫ ) . By (3.22), it is bigger than 32 e − C β . Therefore Z γ e u − ( γ ( s )) ds ≥ e − β c ′ e − C β e u ( p ) e ¯ c = C ( β ) e u ( p ) e ¯ c (3.27)for C ( β ) = e − β c ′ e − C β . By inequality (3.27) and Proposition 3.4,we conclude for any curve γ ⊂ B xy connecting x and y , there is a C = C ( β ) such that Z γ e u − ( γ ( s )) ds ≥ C ( β )( Z B xy e u − ( z ) dz ) . (3.28)This implies inequality (3.19) and thus completes the proof of Theorem1.1. 4. Q ′ -curvature without a sign In this section, we consider CR-manifold on which the Q ′ -curvaturedoes not have a sign any more. Suppose ( H , e u θ ) satisfies that(4.1) α := Z H Q ′ + e u θ ∧ dθ < c ′ , (4.2) β := Z H Q ′− e u θ ∧ dθ < ∞ . Suppose also that the Webster scalar curvature is nonnegative at infin-ity.By Theorem 1.1, e u − is a strong A ∞ weight. By Theorem 1.4 in[WY17], e u + is an A weight. Proposition 4.1.
Assume ω is an A weight, ω is a strong A ∞ weight.If ω r ω for some r ∈ R is an A ∞ weight, then ω r ω a strong A ∞ weight. Remark 4.2.
The proposition for the Euclidean space has been provedin [Sem93]. We prove here the proposition for Heisenberg groups.
Proof.
Let δ ( · , · ) and δ ( · , · ) be the quasidistance associated to ω and ω r ω respectively. Let x , ..., x k ∈ H such that x j ∈ B ( x , | x k − x | )for all j . Notice that it suffices to prove(4.3) δ ( x , x k ) ≤ C k − X j =1 δ ( x j , x j +1 ) . N THE SOBOLEV-POINCAR´E INEQUALITY OF CR-MANIFOLDS 13
Let B = B x ,x k , and B j = B x j ,x j +1 . Since x j ∈ B ( x , | x k − x | ) for all j , B j ⊂ B for all j . By definition δ ,(4.4) δ ( x j , x j +1 ) (cid:3) By Proposition 4.1, in order to prove Theorem 1.2, we only need toshow that e u is an A ∞ weight. In other words, we need to show e u isan A p weight for some p . Proposition 4.3.
Suppose ( H , e u θ ) satisfies the same assumptionsas in Theorem 1.2. Then e u is an A p weight for some p . The A p bounddepends only on the integral of Q ′ curvature. Proof. (4.5) u ( x ) = 1 c ′ Z H log | y || x − y | Q ′ ( y ) e u ( y ) dy with assumptions (1.3) and (1.4), By Theorem 1.4 in [WY17], e u + isan A weight, so there is a uniform constant C = C ( α ), so that for all x ∈ H and r > | B ( x , r ) | Z B ( x ,r ) e u + ( y ) dy ≤ C ( α ) e u + ( x ) . So for all x ∈ B ( x , r )1 | B ( x , r ) | Z B ( x ,r ) e u + ( y ) dy ≤ | B ( x , r ) | Z B ( x, r ) e u + ( y ) dy = 2 | B ( x, r ) | Z B ( x, r ) e u + ( y ) dy ≤ C ( α ) e u + ( x ) . (4.7)Namely, for all ball B in H and x ∈ B ,(4.8) 1 | B | Z B e u + ( y ) dy ≤ C ( α ) e u + ( x ) . We observe that e − ǫu − ( x ) is also an A weight for ǫ = ǫ ( β ) <<
1. Infact,(4.9) e − ǫu − ( x ) = e c ′ R H log | y || x − y | ǫQ − ( y ) e u ( y ) dy .Q − ( y ) e u ( y ) ≥ R H ǫQ − ( y ) e u ( y ) dy < c ′ if ǫ is small enough. Thusby Theorem 1.4 in [WY17], e − ǫu − ( x ) is an A weight. As (4.8), we have(4.10) 1 | B | Z B e − ǫu − ( y ) dy ≤ C ( β ) e − ǫu − ( x ) for all ball B in H and all x ∈ B . Choose 1 < p < ∞ such that ǫ = p ′ /p with p + p ′ = 1. Using e u = e u + · e u − , we get (cid:18)Z B e u ( x ) dx (cid:19) p (cid:18)Z B ( e u ( x ) ) − p ′ p dx (cid:19) p ′ = (cid:18)Z B e u + · ( e − ǫu − ) − ǫ dx (cid:19) p (cid:18)Z B ( e u + ) − p ′ p · e − ǫu − dx (cid:19) p ′ . (4.11)By (4.10), if p is large enough and thus ǫ is small enough, then( e − ǫu − ) − ǫ ≤ (cid:18) C ( β ) | B | Z B e − ǫu − dx (cid:19) − ǫ . So (cid:18)Z B e u + · ( e − ǫu − ) − ǫ dx (cid:19) p ≤ (cid:18)Z B e u + dx (cid:19) p (cid:18) C ( β ) | B | Z B e − ǫu − dx (cid:19) − ǫp = (cid:18)Z B e u + dx (cid:19) p (cid:18) C ( β ) | B | Z B e − ǫu − dx (cid:19) − p ′ . (4.12)Similarly, by (4.8)( e u + ) − p ′ p ≤ (cid:18) C ( α ) | B | Z B e u + dx (cid:19) − p ′ p . So(4.13) (cid:18)Z B ( e u + ) − p ′ p · e − ǫu − dx (cid:19) p ′ ≤ (cid:18) C ( α ) | B | Z B e u + dx (cid:19) − p (cid:18)Z B e − ǫu − dx (cid:19) p ′ . Applying (4.12) to (4.13) in (4.11), we have(4.14) (cid:18)Z B e u ( x ) dx (cid:19) p (cid:18)Z B ( e u ( x ) ) − p ′ p dx (cid:19) p ′ ≤ ( 1 C | B | ) − p − p ′ = C | B | for p >>
1. This shows that e u ( x ) is an A p weight for p >>
1. Thebound C depends only on α and β . (cid:3) Proof of Theorem 1.3
Theorem 5.1. [FLW95, Theorem 2] Let { X j } be a family of vectorfields that satisfies H¨ormander’s condition. Let K be a compact subsetof Ω. Then there exists r depending on K , Ω and { X j } such that if B = B ( x, r ) is a ball with x ∈ K and 0 < r < r , and if 1 ≤ p < q < ∞ N THE SOBOLEV-POINCAR´E INEQUALITY OF CR-MANIFOLDS 15 and ω , ω are weights satisfying the balance condition (5.2) for B , with ω ∈ A p (Ω , ρ, dx ) and ω doubling, then(5.1)( 1 ω ( B ) Z B | f ( x ) − f B | q ω ( x ) dx ) /q ≤ cr ( 1 ω ( B ) Z B | Xf ( x ) | p ω ( x ) dx ) /p for any f ∈ Lip ( ¯ B ), with f B = ω ( B ) − R B f ( x ) ω ( x ) dx . The con-stant c depends only on K, Ω , { X j } and the constants in the conditionsimposed on ω , and ω .The balance condition is stated as follows: for two weight functions ω , ω on Ω and 1 ≤ p < q < ∞ , a ball B with center in K and r ( B ) < r :(5.2) r ( I ) r ( J ) ( ω ( I ) ω ( J ) ) /q ≤ c ( ω ( I ) ω ( J ) ) /p for all metric balls I, J with I ⊂ J ⊂ B . Proof. of Theorem 1.3. It is obvious that X := ∂∂x + 2 y ∂∂t , X := ∂∂y − x ∂∂t on the Heisenberg group H satisfy the H¨omander’s condition.Let us take ω ( x ) = e ( n − p ) u ( x ) , ω ( x ) = e nu ( x ) , q = npn − p .We only need to check condition (5.2). Namely, we need to show(5.3) ( r ( I ) r ( J ) ) npn − p R I ω dx R J ω dx ≤ c ( R I ω n − pn dx R J ω n − pn dx )This is true because 0 ≤ n − pn < ω = e nu is a strong A ∞ weight,thus it is an A ∞ weight. In fact, for any A ∞ weight w , 0 ≤ s <
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