Sharp Global Bounds for the Hessian on Pseudo-Hermitian Manifolds

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1 Sharp Global Bounds for the Hessian on Pseudo-Hermitian anifolds Sagun Chanillo 1 and Juan J. anfredi 1 Department of athematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854, chanillo@math.rutgers.edu Department of athematics, University of Pittsburgh, Pittsburgh PA 1560, manfredi@pitt.edu Dedicated to the memory of our friend and colleague Carlos Segovia. 1 Introduction In PDE theory, Harmonic Analysis enters in a fundamental way through the basic estimate valid for f C 0 R n ), which states, n f x i x j cn, p) f Lp R n ), for 1 < p <. 1) Lp R n ) i,j=1 This estimate is really a statement of the L p boundedness of the Riesz transforms, and thus 1) is a consequence of the multiplier theorems of arcinkiewicz and Hörmander-ikhlin, [15]. ore sophisticated variants of 1) can be proved by relying on the square function [15] and [14]. In particular 1) leads to a-priori W,p estimates for solutions of u = f, for f L p. ) Knowledge of cp, n) allows one to perform a perturbation of ) and study n a ij u x) = f 3) x i x j i,j=1 as was done by Cordes [4], where A = a ij ) is bounded, measurable, elliptic and close to the identity in a sense made precise by Cordes. The availability of the estimates of Alexandrov-Bakelman-Pucci and the Krylov-Safonov theory

2 Chanillo-anfredi [7] allows one to obtain estimates for 3) in full generality without relying on a perturbation argument. See also [1]. Our focus here will be to study the CR analog of 3). Since at this moment in time there is no suitable Alexandrov-Bakelman-Pucci estimate for the CR analog of 3) we will be seeking a perturbation approach based on an analog of 1) on a CR manifold. Our main interest is the case p = in 1). In this case a simple integration by parts suffices to prove 1) in R n. We easily see that for f C0 R n ) we have n f x i x j = f L R n ). 4) i,j=1 L R n ) In the case of 1) on a CR manifold a result has been recently obtained by Domokos-anfredi [6] in the Heisenberg group. The proof in [6] makes uses of the harmonic analysis techniques in the Heisenberg group developed by Strichartz [16] that will not apply to studying such inequalities for the Hessian on a general CR manifold, although other nilpotent groups of step can be treated similarly [5]. Instead we shall proceed by integration by parts and use of the Bochner technique. A Bochner identity on a CR manifold was obtained by Greenleaf [8] and will play an important role in our computations. We now turn to our setup. We consider a smooth orientable manifold n+1. Let V be a vector sub-bundle of the complexified tangent bundle CT. We say that V is a CR bundle if V V = {0}, [V, V] V, and dim C V = n. 5) A manifold equipped with a sub-bundle satisfying 5) will be called a CR manifold. See the book by Trèves [18]. Consider the sub-bundle H = Re V V ). 6) H is a n-dimensional vector sub-bundle of the tangent bundle T. We assume that the real line bundle H T, where T is the cotangent bundle, has a smooth non-vanishing global section. This is a choice of a nonvanishing 1-form θ on and, θ) is said to define a pseudo-hermitian structure. is then called a pseudo-hermitian manifold. Associated to θ we have the Levi form L θ given by L θ V, W ) = i dθv W ), for V, W V. 7) We shall assume that L θ is definite and orient θ by requiring that L θ is positive definite. In this case, we say that is strongly pseudo-convex. We shall always assume that is strongly pseudo-convex. On a manifold that carries a pseudo-hermitian structure, or a pseudohermitian manifold, there is a unique vector field T, transverse to H defined in 6) with the properties

3 Hessian bounds 3 θt ) = 1 and dθt, ) = 0. 8) T is also called the Reeb vector field. The volume element on is given by dv = θ dθ) n. 9) A complex valued 1-form η is said to be of type 1, 0) if ηw ) = 0 for all W V, and of type 0, 1) if ηw ) = 0 for all W V. An admissible co-frame on an open subset of is a collection of 1, 0) forms {θ 1,..., θ,..., θ n } that locally form a basis for V and such that θ T ) = 0 for 1 n. We set θ = θ. We then have that {θ, θ, θ } locally form a basis of the complex co-vectors, and the dual basis are the complex vector fields {T, Z, Z }. For f C ) we set T f = f 0, Z f = f, Z f = f. 10) We note that in the sequel all our functions f will be real valued. If follows from 5), 7), and 8) that we can express dθ = i h β θ θ β. 11) The hermitian matrix h β ) is called the Levi matrix. On pseudo-hermitian manifolds Webster [19] has defined a connection, with connection forms ω β and torsion forms τ β = A β θ, with structure relations dθ β = θ ω β + θ τ β, ω β + ω β = dh β 1) and Webster defines a curvature form β = dω β ω γ ω β γ, A β = A β. 13) where we have used the Einstein summation convention. Furthermore in [19] it is shown that β = R βρ σ θ ρ θ σ + other terms. Contracting two indices using the Levi matrix h β) we get R β = h ρ σ R βρ σ. 14) The Webster-Ricci tensor RicV, V ) for V V is then defined as RicV, V ) = R βx x β, for V = σ x Z. 15) The torsion tensor is defined for V V as follows

4 4 Chanillo-anfredi TorV, V ) = i Aᾱ βx x β A β x x β). 16) In [19], Prop..), Webster proves that the torsion vanishes if L T preserves H, where L T is the Lie derivative. In particular if is a hypersurface in C n+1 given by the defining function ρ Imz n+1 = ρz, z), z = z 1, z,..., z n ) 17) then Webster s hypothesis is fulfilled and the torsion tensor vanishes on. Thus for the standard CR structure on the sphere S n+1 and on the Heisenberg group the torsion vanishes. Our main focus will be the sub-laplacian b. We define the complex horizontal gradient b and b as follows: b f = f Z, 18) b f = f ᾱ + fᾱ. 19) When n = 1 we will need to frame our results in terms of the CR Paneitz operator. Define the Kohn Laplacian b by Then the CR Paneitz operator P 0 is defined by where See [10] and [9] for further details. b = b + i T. 0) P 0 f = b b + b b ) f Q + Q ) f, 1) Qf = i A 11 f 1 ) 1.

5 Hessian bounds 5 The ain Theorem Theorem 1. Let n+1 be a strictly pseudo-convex pseudo-hermitian manifold. When is non compact assume that f C0 ). When is compact with = we may assume f C ). When f is real valued and n we have f β + f β + Ric + n ) Tor n + ) b f, b f) b f. ) n When n = 1 assume that the CR Paneitz operator P 0 0. For f C0 ) we then have f 11 + f Ric 3 ) Tor b f, b f) 3 b f. 3) Proof. We begin by noting the Bochner identity established by Greenleaf, Lemma 3 in [8]: 1 b b f ) = f β + f β + Re b f, b b f)) 4) + Ric + n ) Tor b, b ) + i f f 0 f fᾱ0 ). where for V, W V we use the notation V, W ) = L θ V, W ) and V = V, V ) 1/. Using the fact that f C0 ) or if =, is compact, integrate 4) over using the volume 9) to get f β + f β + + i Ric + n Tor f f 0 f fᾱ0 ) = ) b f, b f) 5) Re b f, b b f)). Integration by parts in the term on the right yields see 5.4) in [8]) Re b f, b b f)) = 1 b f. 6) Combining 5) and 6) we get f β + f β + + i Ric + n ) Tor b f, b f) 7) f f 0 f fᾱ0 ) = 1 b f.

6 6 Chanillo-anfredi To handle the third integral in the left-hand side, we use Lemmas 4 and 5 of [8] valid for real functions) according to which we have i f f 0 f fᾱ0 ) = f β f β ) ) Ric b f, b f), n 8) and i f f 0 f fᾱ0 ) = 4 f ᾱ 9) n + 1 b f n + Tor b f, b f). Applying the Cauchy-Schwarz inequality to the the first term in the right-hand side of 9) we get i f f 0 f fᾱ0 ) 4 f β 30) + 1 n + b f Tor b f, b f). ultiply 8) by 1 c and 30) by c, 0 < c < 1, and where c will eventually be chosen to be 1/n + 1), and add to get i f f 0 f fᾱ0 ) 1 c) n f β f β ) 31) 1 c) Ric b f, b f) n 4c f β + c n b f + c We now insert 31) into 7) and simplify. We have Tor b f, b f).

7 Hessian bounds 7 ) 1 c) 1 Ric b f, b f) + n ) n ) + c Tor b f, b f) + ) 1 c) 1 + 4c f n β + 3) ) 1 c) 1 1 f β n c ) b f. n Let c = 1/n + 1). Then 3) becomes ) [ n 1 fβ + f n + 1 β ) + Ric + n Tor ) n 1 n + 1 ] b f, b f) 33) ) ) n + b f. n Since n, n 1 > 0 and we can cancel the factor n 1 n+1 from both sides to get ). We now establish 3) using some results by Li-Luk [11] and [9]. When n = 1, identity 7) becomes f f 11 + Ric 1 ) Tor b f, b f) 34) + i f 10 f 1 f 10f 1 ) = 1 b f. By 3.8) in [11] we have i oreover, by 3.6) in [11] we also have f 01 f 1 f 0 1f 1 ) = f0. i f 10 f 1 f 10f 1 ) = i f 01 f 1 f 0 1f 1 ) + Tor b f, b f) and combining the last two identities we get i f 10 f 1 f 10f 1 ) = f0 + Tor b f, b f). 35) Substituting 35) into 34) we obtain f f 11 + Ric + 1 ) Tor b f, b f) f0 36) = 1 b f.

8 8 Chanillo-anfredi Next, we use 3.4) in [9], f0 = b f + Tor b f, b f) 1 Finally, substitute 37) into 36) and simplify to get f f 11 + Ric 3 ) Tor b f, b f) + 1 = 3 P 0 f f. 37) P 0 f f b f. Assuming P 0 0 we obtain 3). We now wish to make some remarks about our theorem: a) It is shown in [6] that on the Heisenberg group the constant n+)/n is sharp. Since the Heisenberg group is a pseudo-hermitian manifold with Ric 0 and Tor 0, we easily conclude our theorem is sharp and contains the result proved in [6]. b) We notice that when we consider manifolds such that Ric+n/)Tor > 0, then for n, in general we have the strict inequality f β + f β < n + b f. n On the Heisenberg group Ric 0, Tor 0 and the constant n + )/n is achieved by a function with fast decay [6]. Thus, the Heisenberg group is, in a sense, extremal for inequality ) in Theorem 1. A similar remark holds for inequality 3). c) The hypothesis on the Paneitz operator in the case n = 1 in our theorem is satisfied on manifolds with zero torsion. A result from [] shows that if the torsion vanishes the Paneitz operator is non-negative. d) We note that Chiu [9] shows how to perturb the standard pseudohermitian structure in S 3 to get a structure with non-zero torsion, for which P 0 > 0 and Ric 3/)Tor > 1. To get such a structure, let θ be the contact form associated to the standard structure on S 3. Fix g a smooth function on S 3. For ɛ > 0 consider θ = e f θ, where f = ɛ 3 sin g ). 38) ɛ Since the sign of the Paneitz operator is a CR invariant and θ has zero torsion we conclude by [] that the CR Paneitz operator P 0 associated to θ satisfies P 0 > 0. Furthermore following the computation in Lemma 4.7) of [9], we easily have for small ɛ that

9 Ric 3 Tor + Oɛ)) e f 1 0. Hessian bounds 9 Thus, the hypothesis of the case n = 1 in our theorem are met, and for such, θ) we have, for f C ) the estimate f 11 + f 1 1 dv 3 b f dv. e) Compact pseudo-hermitian 3-manifolds with negative Webster curvature may be constructed by considering the co-sphere bundle of a compact Riemann surface of genus g, g. Such a construction is given in [3]. 3 Applications to PDE For applications to subelliptic PDE it is helpful to re-state our main result Theorem 1 in its real version. We set X i = ReZ i ) and X i+n = ImZ i ) for i = 1,..., n. The real horizontal gradient of a function is the vector field Its sublaplacian is given by Xf) = X f = n i=1 n i=1 X i f)x i. X i X i f), and the horizontal second derivatives are the n n matrix X f = X i X j f)). For f real we have the following relationships n ) b f = Xf) + i X i f)x i+n X i+n f)x i, i=1 and b f = X f, f β + f β = X i X j f) = X f. i,j

10 10 Chanillo-anfredi Theorem. Let n+1 be a strictly pseudo-convex pseudo-hermitian manifold. When is non compact assume that f C0 ). When is compact with = we may assume f C ). When f is real valued and n we have X f + 1 Ric + n ) Tor b f, b f) n + ) X f. 39) n When n = 1 assume that the CR Paneitz operator P 0 0. For f C0 ) we then have X f 1 + Ric 3 ) Tor b f, b f) 3 X f. 40) Let Ax) = a ij x)) a n n matrix. Consider the second order linear operator in non-divergence form Aux) = n i,j=1 a ij x)x i X j ux), 41) where coefficients a ij x) are bounded measurable functions in a domain Ω n+1. Cordes [4] and Talenti [17] identified the optimal condition expressing how far A can be from the identity and still be able to understand 41) as a perturbation of the case Ax) = I n, when the operator is just the sublaplacian. This is the so called Cordes condition that roughly says that all eigenvalues of A must cluster around a single value. Definition 1. [4],[17], [6]) We say that A satisfies the Cordes condition K ε,σ if there exists ε 0, 1] and σ > 0 such that for a. e. x Ω. 0 < 1 σ n i,j=1 a ijx) 1 n 1 + ε n i=1 a ii x)) 4) Let c n = n+) n for n and c 1 = 3 the constants in the right-hand sides of Theorem. We can now adapt the proof of Theorem.1 in [6] to get Theorem 3. Let n+1 be a strictly pseudo-convex pseudo-hermitian manifold such that Ric + n Tor 0 if n and Ric 3 Tor 0 if n = 1. Let 0 < ε 1, σ > 0 such that γ = 1 ε)c n < 1 and A satisfies the Cordes condition K ε,σ. Then for all u C0 Ω) we have the a-priori estimate X u L n 1 γ L Au L, 43) where x) = n Ax), I Ax) = i=1 a iix) n i,j=1 a ij x).

11 Proof. We start from formula.7) in [6] which gives X ux) x)aux) dx 1 ε) Ω Ω Ω Hessian bounds 11 Xu dx. We now apply Theorem to get X ux) x)aux) dx 1 ε)c n X f. The theorem then follows as in [6]. Remark: The hypothesis of Theorem, n, can be weakened to assume only a bound from below to obtain estimates of the type X f n + ) n Ric + n Tor K, with K > 0 X f + K Ω Xf. 44) A similar remark applies to the case n = 1. We finish this paper by indicating how the a priori estimate of Theorem 3 can be used to prove regularity for p-harmonic functions in the Heisenberg group H n when p is close to. We follow [6], where full details can be found. Recall that, for 1 < p <, a p-harmonic function u in a domain Ω H n is a function in the horizontal Sobolev space such that W 1,p X,loc Ω) = {u: Ω R such that u, Xu Lp loc Ω)} n i=1 X i Xu p X i u ) = 0, in Ω 45) in the weak sense. That is, for all φ C0 Ω) we have Xux) p Xux), Xφx) dx = 0. 46) Ω Assume for the moment that u is a smooth solution of 45). We can then differentiate to obtain where n i,j=1 a ij X i X j u = 0, in Ω 47)

12 1 Chanillo-anfredi a ij x) = δ ij + p ) X iux) X j ux) Xux). A calculation shows that this matrix satisfies the Cordes condition 4) precisely when n n ) 4n p + 4n 3 n, n + n 4n + 4n 3 + n n. 48) + n In the case n = 1 this simplifies to 1 5 p, 1 + ) 5. We then deduce a priori estimates for X u from Theorem 3. To apply the Cordes machinery to functions that are only in W 1,p X we need to know that the second derivatives X u exist. This is done in the Euclidean case by a standard difference quotient argument applied to a regularized p-laplacian. In the Heisenberg case this would correspond to proving that solutions to n ) p ) 1 X i m + Xu X i u = 0 49) i=1 are smooth. Contrary to the Euclidean case where solutions to the regularized p-laplacian are C -smooth) in the subelliptic case this is known only for p [, cn)) where cn) = 4 for n = 1,, and lim n cn) = see [13].) The final result will combine the limitations given by 48) and cn). Theorem 4. Theorem 3.1 in [6]) For p < + n + n 4n + 4n 3 n + n we have that p-harmonic functions in the Heisenberg group H n are in W, X,loc Ω). At least in the one-dimensional case H 1 one can also go below p =. See Theorem 3. in [6]. We also note that when p is away from, for example p > 4 nothing is known regarding the regularity of solutions to 45) or its regularized version 49) unless we assume a priori that the length of the gradient is bounded below and above 0 < 1 Xu <. See [1] and [13]. Acknowledgement. S.C. supported in part by NSF Award DS J.J.. supported in part by NSF award DS S.C. wishes to thank Shri S. Devananda and Shri Raghavendra for encouragement in a difficult moment.

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