Sharp Global Bounds for the Hessian on Pseudo-Hermitian Manifolds

a r X i v :0704.2833v 1 [m a t h .A P ] 21 A p r 2007∂x i ∂x j L p (R n )≤c (n,p ) ∆f L p (R n ),for 1<p <∞.(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 Marcinkiewicz and H¨o rmander-Mikhlin,[15].More sophisticated variants of (1)can be proved by relying on the square function [15]and [14].In particular (1)leads to a-priori W 2,p estimates for solutions of∆u =f,for f ∈L p .(2)Knowledge of c (p,n )allows one to perform a perturbation of (2)and studyni,j =1a ij (x )∂2u2Chanillo-Manfredi[7]allows one to obtain estimates for(3)in full generality without relying ona perturbation argument.See also[12].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=2in(1).In this case a simple integration by parts suffices to prove(1)in R n.We easily see that for f∈C∞0(R n)we haveni,j=1 ∂2fV={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 Tr`e ves[18].Consider the sub-bundleH=Re V⊕W)=−i dθ(V∧Hessian bounds3θ(T)=1and dθ(T,·)=0.(8) T is also called the Reeb vectorfield.The volume element on M is given bydV=θ∧(dθ)n.(9) A complex valued1-formηis said to be of type(1,0)ifη(α=α} locally form a basis of the complex co-vectors,and the dual basis are the complex vectorfields{T,Zα,Zαf=fβθα∧θβ)is called the Levi matrix.On pseudo-hermitian manifolds Webster[19]has defined a connection, with connection formsωβαand torsion formsτβ=Aβαθα,with structure relationsdθβ=θα∧ωβα+θ∧τβ,ωαβ(12) andAαβ=Aβα.(13) Webster defines a curvature formβα=dωβα−ωγα∧ωβγ,where we have used the Einstein summation convention.Furthermore in[19] it is shown thatβα=Rα¯βρ¯σθρ∧θ¯σ+other terms.Contracting two indices using the Levi matrix(hα¯β)we getRα¯β=hρ¯σRα¯βρ¯σ.(14) The Webster-Ricci tensor Ric(V,V)for V∈V is then defined asRic(V,V)=Rα¯βxα4Chanillo-ManfrediTor(V,V)=i A¯α¯βz),z=(z1,z2,...,z n)(17) then Webster’s hypothesis is fulfilled and the torsion tensor vanishes on M. Thus for the standard CR structure on the sphere S2n+1and on the Heisen-berg group the torsion vanishes.Our main focus will be the sub-Laplacian∆b.We define the horizontal gradient∇b and∆b as follows:∇b f= αfb b+ b Q f,(21) whereQf=2i(A11f1)1.See[10]and[9]for further details.Hessian bounds5 2The Main TheoremTheorem1.Let M2n+1be a strictly pseudo-convex pseudo-hermitian mani-fold.When M is non compact assume that f∈C∞0(M).When M is compact with∂M=∅we may assume f∈C∞(M).When f is real valued and n≥2 we haveα,β M||fαβ||2+||fα¯β||2+ M Ric+n2n M|∆b f|2.(22) When n=1assume that the CR Paneitz operator P0≥0.For f∈C∞0(M) we then haveM||f11||2+||f1¯1||2+ M Ric−32 M|∆b f|2.(23)Here by α,β||fαβ||2we mean the Hilbert-Schmidt norm square of the tensor and similarly for α,β||fα¯β||2.Proof.We begin by noting the Bochner identity established by Greenleaf, Lemma3in[8]:1Tor (∇b,∇b)+i α(f2W)and|V|= (V,V)1/2.We have also abused notation above and represented the Hilbert-Schmidt norm of the tensor fαβin terms of its expression in the local frame which we will continue to do in the rest of the ing the fact that f∈C∞0(M)or if∂M=∅,M is compact,integrate(24)over M using the volume(9)to getM α,β|fαβ|2+|fα¯β|2+ Ric+n−2fα0−fαf¯α0)=− M Re(∇b f,∇b(∆b f)).αIntegration by parts in the term on the right yields(see(5.4)in[8])− M Re(∇b f,∇b(∆b f))=16Chanillo-ManfrediM α,β|fαβ|2+|fα¯β|2+ M Ric+n−2fα0−fαf¯α0)=1αfα0−fαf¯α0)=2αfα0−fαf¯α0)=−4αn M|∆b f|2+ M Tor(∇b f,∇b f).Applying the Cauchy-Schwarz inequality to the thefirst term in the right-hand side of(29)we geti M α(fn M|∆b f|2+ M Tor(∇b f,∇b f).Multiply(28)by1−c and(30)by c,0<c<1,and where c will eventually be chosen to be1/(n+1),and add to geti M α(f n M α,β |fα¯β|2−|fαβ|2 (31)−2(1−c)β|2c+Hessian bounds7 1−2(1−c)2+c M Tor(∇b f,∇b f)+1+2(1−c)β|2+(32) 1−2(1−c)2−cn+1 M α,β |fαβ|2+|fα¯β|2 + M Ric+nn+1n+2n+1from both sides toget(22).We now establish(23)using some results by Li-Luk[11]and[9].When n=1,identity(27)becomesM|f1¯1|2+|f11|2+ M Ric−12 M|∆b f|2.By(3.8)in[11]we havei M(f01f¯1−f0¯1f1)=− M f20.Moreover,by(3.6)in[11]we also havei(f10f¯1−f¯10f1)=i(f01f¯1−f0¯1f1)+Tor(∇b f,∇b f)and combining the last two identities we geti M(f10f¯1−f¯10f1)=− M f20+ M Tor(∇b f,∇b f).(35)Substituting(35)into(34)we obtainM|f1¯1|2+|f11|2+ M Ric+12 M|∆b f|2.8Chanillo-ManfrediNext,we use(3.4)in[9],M f20= M|∆b f|2+2 M Tor(∇b f,∇b f)−1Tor (∇b f,∇b f)+122 M|∆b f|2. Assuming P0≥0we obtain(23).⊓⊔We now wish to make some remarks about our theorem:(a)It is shown in[6]that on the Heisenberg group the constant(n+2)/2n is sharp.Since the Heisenberg group is a pseudo-hermitian manifold with Ric≡0and 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/2)Tor> 0,then for n≥2,in general we have the strict inequalityα,β M|fαβ|2+|fα¯β|2<n+2).(38)ǫSince the sign of the Paneitz operator is a CR invariant andθhas zero torsion we conclude by[2]that the CR Paneitz operator˜P0associated to˜θsatisfies ˜P>0.Furthermore following the computation in Lemma(4.7)of[9],we 0easily have for smallǫthatHessian bounds9 Ric−32 M|∆b f|2dV.(e)Compact pseudo-hermitian3-manifolds with negative Webster curva-ture may be constructed by considering the co-sphere bundle of a compact Riemann surface of genus g,g≥2.Such a construction is given in[3].3Applications to PDEFor applications to subelliptic PDE it is helpful to re-state our main result Theorem1in its real version.We setX i=Re(Z i)and X i+n=Im(Z i)for i=1,2...,n.The horizontal gradient of a function is the vectorfieldX(f)=2ni=1X i(f)X i.Its sublaplacian is given by∆X f=2ni=1X i X i(f),and the horizontal second derivatives are the2n×2n matrixX2f=(X i X j(f)).For f real we have the following relationships∇b f=X(f)+i n i=1X i(f)X i+n−X i+n(f)X i ,∆b f=2∆X f,and α,β|fαβ|2+|fα¯β|2=2 i,j|X i X j(f)|2=2|X2f|2,where the expression on the extreme right is the Hilbert-Schmidt norm square of the tensor taken by viewing the Levi form as a metric on H.10Chanillo-ManfrediTheorem 2.Let M 2n +1be a strictly pseudo-convex pseudo-hermitian mani-fold.When M is non compact assume that f ∈C ∞0(M ).When M is compactwith ∂M =∅we may assume f ∈C ∞(M ).When f is real valued and n ≥2we haveM|X 2f |2+ M 12Tor (∇b f,∇b f )≤(n +2)2Ric −3σ≤2ni,j =1a 2ij (x )≤1nfor n ≥2and c 1=3the constants in the right-hand sides ofTheorem 2.We can now adapt the proof of Theorem 2.1in [6]to getTheorem 3.Let M 2n +1be a strictly pseudo-convex pseudo-hermitian mani-fold such that Ric +n 2Tor ≥0,P 0≥0if n =1.Let 0<ε≤1,σ>0such that γ=1+21−γα L ∞ A u L 2,(43)whereα(x )=A (x ),I2ni,j =1a 2ij (x ).Hessian bounds11 Proof.We start from formula(2.7)in[6]which givesΩ|∆X u(x)−α(x)A u(x)|2dx≤(1−ε) Ω|X u|2dx.We now apply Theorem2to getΩ|∆X u(x)−α(x)A u(x)|2dx≤(1−ε)c n Ω|∆X f|2.The theorem then follows as in[6].⊓⊔Remark:The hypothesis of Theorem2,n≥2,can be weakened to assume only a bound from belowRic+nn M|∆X f|2+2K M|X f|2.(44)A similar remark applies to the case n=1.Wefinish this paper by indicating how the a priori estimate of Theorem 3can be used to prove regularity for p-harmonic functions in the Heisenberg group H n when p is close to2.We follow[6],where full details can be found. Recall that,for1<p<∞,a p-harmonic function u in a domainΩ⊂H n is a function in the horizontal Sobolev spaceW1,pX,loc (Ω)={u:Ω→R such that u,X u∈L ploc(Ω)}such that2ni=1X i |X u|p−2X i u =0,inΩ(45)in the weak sense.That is,for allφ∈C∞0(Ω)we haveΩ|X u(x)|p−2(X u(x),Xφ(x)dx=0.(46)Assume for the moment that u is a smooth solution of(45).We can then differentiate to obtain2ni,j=1a ij X i X j u=0,inΩ(47) where12Chanillo-ManfrediX i u(x)X j u(x)a ij(x)=δij+(p−2)4n2+4n−34n2+4n−355+|X u|2 p−2m4n2+4n−3M≤|X u|≤M<∞.See[1]and[13].Acknowledgement.S.C.supported in part by NSF Award DMS-0600971.J.J.M.sup-ported in part by NSF award DMS-0500983.S.C.wishes to thank Shri S.Devananda and Shri Raghavendra for encouragement in a difficult moment.Hessian bounds13 References1.Capogna,L.,Regularity of quasi-linear equations in the Heisenberg group.Comm.Pure Appl.Math.50(1997),no.9,867–889.2.Chang,S.C.,Cheng,J.H.,Chiu,H.L.,A fourth order Q-curvatureflow on a CR3-manifold,to appear in Indiana Math.J.,/abs/math.DG/0510494.3.Chern,S.S.,Hamilton,R.S.,On Riemannian metrics adapted to three-dimensional contact manifolds.With an appendix by Alan Weinstein.Lecture Notes in Math.,1111,Workshop Bonn1984(Bonn,1984),279–308,Springer, Berlin,1985.4.Cordes,H.O.,Zero order a-priori estimates for solutions of elliptic differentialequations,Proceedings of Symposia in Pure Mathematics IV(1961).5.Domokos,A.,Fanciullo,M.S.,On the best constant for the Friedrichs-Knapp-Stein inequality in free nilpotent Lie groups of step two and applications to subelliptic PDE,The Journal of Geometric Analysis,17(2007),245-252.6.Domokos,A.,Manfredi,J.J.,Subelliptic Cordes estimates.Proc.Amer.Math.Soc.133(2005),no.4,1047–1056.7.Gilbarg,D.,Trudinger,N.S.,Elliptic partial differential equations of second or-der.Reprint of the1998edition.Classics in Mathematics.Springer-Verlag,Berlin, 2001.8.Greenleaf,A.,Thefirst eigenvalue of a sub-Laplacian on a pseudo-Hermitianm.Partial Differential Equations10(1985),no.2,191–217.9.Chiu,H.L.,The sharp lower bound for thefirst positive eigenvalue of the sub-laplacian on a pseudohermitian3-manifold,Ann.Global Anal.Geom.30(2006), no.1,81–96.10.Lee,J.M.,The Fefferman metric and pseudo-Hermitian invariants,Trans.Amer.Math.Soc.296(1986),no.1,411–429.11.Li,S.Y.,Luk,H.S.,The sharp lower bound for thefirst positive eigenvalue ofa sub-Laplacian on a pseudo-Hermitian manifold.Proc.Amer.Math.Soc.132(2004),no.3,789–798.12.Lin,F.H.,Second derivative L p-estimates for elliptic equations of nondivergenttype.Proc.Amer.Math.Soc.96(1986),no.3,447–45113.Manfredi,J.J.,Mingione,G.,Regularity Results for Quasilinear Elliptic Equa-tions in the Heisenberg Group,to appear in Mathematische Annalen,2007.14.Segovia,C.,On the area function of Lusin,Studia Math.331969311–343.15.Stein,E.,Singular integrals and differentiability properties of functions.Prince-ton Mathematical Series,No.30Princeton University Press,Princeton,N.J.1970.16.Strichartz,R.S.,Harmonic analysis and Radon transforms on the Heisenberggroup,J.Funct.Analysis,96(1991),350-406..17.Talenti,G.,Sopra una classe di equazioni ellittiche a coefficienti misurabili.(Italian)Ann.Mat.Pura Appl.(4)69,1965,285–30418.Tr`e ves,F.,Hypo-analytic structures.Local theory,Princeton Mathematical Se-ries,40.Princeton University Press,Princeton,NJ,1992.19.Webster,S.M.,Pseudo-Hermitian structures on a real hypersurface,J.Differ-ential Geom.13(1978),no.1,25–41.。