# On the Regularity of Weak Contactp-Harmonic Maps

Authors: Sorin Dragomir; Robert Petit
Journal: Journal of Complex Analysis
 (2013)

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## Abstract

We prove Caccioppoli type estimates and consequently establish local Hölder continuity
for a class of weak contact(2n+2)-harmonic maps from the
Heisenberg group ℍn into the sphere S2m−1.


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## Body

## 1. Introduction

The study ofpseudoharmonic maps was started by Barletta et al. [1] (cf. also [2, 3] for successive investigations) as a generalization of the theory of harmonic maps among Riemannian manifolds (cf., e.g., [4]) and by identifying the results of Jost and Xu [5], Zhou [6], Hajłasz and Strzelecki [7], and Wang [8] as local aspects of the theory of pseudoharmonic maps from a strictly pseudoconvex CR manifold into a Riemannian manifold (cf. also [9, pages 225-226]).A similar class of maps, yet with values in another CR manifold, was studied in [10]. These are critical points of the functional
(1)E(ϕ)=12∫M‍Q(ϕ)dv,ϕ∈C∞(M,N),
where M is a compact strictly pseudoconvex CR manifold of CR dimension n,  Q(ϕ)=∥(dϕ)H,H′∥2,dv=θ∧(dθ)n, and θ is a contact form on M. Also N is a contact Riemannian manifold and in particular an almost CR manifold (of CR codimension 1).A moment's thought reveals the augmented difficulties such a theory may present. For instance, ifM and N are two strictly pseudoconvex CR manifolds endowed, respectively, with contact forms θ and η, then the pseudohermitian analog of the notion of a harmonic morphism (among Riemannian manifolds) is quite obvious: one may consider continuous maps ϕ:M→N such that the pullback v∘ϕ of any local solution v:U′⊆N→ℝ to ΔbNv=0 in V satisfies Δb(v∘ϕ)=0 in U=ϕ-1(U′) in distribution sense. Here Δb and ΔbN are the sublaplacians of (M,θ) and (N,η), respectively. Unlike the situation in [2] (where the target manifold N is Riemannian and ϕ pulls back local harmonic functions on N to distribution solutions of Δbu=0) such ϕ is not necessarily smooth (since it is unknown whether local coordinate systems (U′,x′i) on N such that ΔbNx′i=0 in U′ might be produced). To give another example, should one look for a pseudohermitian analog to the Fluglede-Ishihara theorem (cf. [3] when M is CR and N is Riemannian), one would face the lack of an Ishihara type lemma (cf. [11]) as it is unknown whether ΔbNv=0 admits local solutions whose (horizontal) gradient and hessian have prescribed values at a point. Moreover, what would be the appropriate notion of a hessian (cf. [12] for a possible choice)?A third example, discussed at some length in this paper, is that of the “degeneracy” of the Euler-Lagrange equations(2)[(φ2)ji∘ϕ]×{∑a=12n‍div[Q(ϕ)(p-2)/2∇Hϕj]+Q(ϕ)(p-2)/2∑a=12n‍(Γkℓ′j∘ϕ)Xa(ϕk)Xa(ϕℓ)}=0,1≤i≤2m-1,
associated to the variational principle
(3)δ∫Q(ϕ)p/2dv=0,
when N is a Sasakian manifold. Indeed the (2m-1)×(2m-1) matrix (φ2)ji=-δji+ξiηj has but rank 2m-2 at each point (a well-known phenomenon in contact Riemannian geometry, cf., e.g., [13]. See also [14]). Consequently, in general one may not expect regularity of weak solutions to (2). For instance, if N=ℍm-1 is the Heisenberg group and ϕ=(ϕ′,ϕ2m-1):U⊆M→ℍm-1 is a solution to (2), then ϕ′:U→ℝ2m-2 is subject to
(4)∑a=12n‍Xa*(|Xϕ′|p-2Xa(ϕi))=0,1≤i≤2m-2,
yet ϕ2m-1 is an arbitrary function (cf. Section 3). For the more appealing case, where M=ℍn is the Heisenberg group and N=S2m-1 is the sphere, (2) may be written as
(5)X*·VA=Q(ϕ)p/2ϕA,1≤A≤2m,
(cf. Proposition 15) which is indeed the form assumed by the Euler-Lagrange equations in [7], yet unlike the situation there X*·EA,B≠0 in general (cf. Proposition 16 for the notations). Although X*·EA,B has a quite explicit form (yielding—for a class of weak solutions ϕ:ℍn→S2m-1 which are close to being horizontal maps—simple estimates on X*·EA,B), only a weaker form of the duality inequality lemma in [7] may be proved (cf. Lemma 17) leading nevertheless (together with a hole filling argument) to Caccioppoli type estimates
(6)∫BX(x,r)‍|Xϕ|2n+2dv≤Crγ,
for some C>0 and 0<γ<1, which are known (cf., e.g., [7] for a very general argument based on work in [15]) to imply the local Hölder continuity of the given weak solution.The paper is organized as follows. In Section2 we recall a few conventions and basic results obtained in [10]. Sections 3 and 4 are devoted to the study of the local properties of weak contact p-harmonic maps. We show that weak contact (2n+2)-maps ϕ:U⊂ℍn→S2m-1 are locally Hölder continuous (cf. Corollary 21) provided they are close to being horizontal maps; that is, the assumptions (96) are satisfied. The relevance of the number p=2n+2 stems from the facts that ∫M‍∥(dϕ)H,H′∥2n+2dv is a CR invariant and 2n+2 is the homogeneous dimension of ℍn. The authors believe that subelliptic theory should play within CR geometry, as a branch of complex analysis in several complex variables, the strong role played by elliptic theory in Riemannian geometry, and the present paper is a step in this direction.


## 2. Basic Conventions and Results

For all notions of CR and pseudohermitian geometry we adopt the conventions and notations in the monograph [9]. For the approach to contact structures within Riemannian geometry we rely on the presentation in Blair [13], (cf. also Tanno [16]). Given a real (2n+1)-dimensional C∞ differentiable manifold M, an almost CR structure is a complex subbundle T1,0(M)⊂T(M)⊗ℂ of the complexified tangent bundle, of complex rank n, such that T1,0(M)x∩T0,1(M)x=(0) for any x∈M. Here T0,1(M)=T1,0(M)¯ and overbars indicate complex conjugates. The integer n is the CR dimension of the almost CR manifold (M,T1,0(M)). Almost CR structures are a bundle theoretic recast of the tangential Cauchy-Riemann operator∂¯b:C∞(M,ℂ)→C∞(T0,1(M)*) given by (∂¯bf)Z¯=Z¯(f) for any f∈C∞(M,ℂ) and any Z∈T1,0(M). An almost CR structure is (formally or Frobenius) integrable if [Z,W]∈C∞(U,T1,0(M)) for any Z,W∈C∞(U,T1,0(M)) and any open set U⊂M. The tangential C-R operator may be extended to arbitrary (0,q)-forms on M and the resulting pseudocomplex ∂¯b:Ω0,q(M)→Ω0,q+1(M), q≥0, is a complex (i.e., ∂¯b2=0) if and only if the given almost CR structure is integrable (cf. [9]). Integrable almost CR structures are commonly referred to as CR structures and appear mainly on real hypersurfaces of complex manifolds, as induced by the complex structure of the ambient space; that is, for any complex manifold V and any real hypersurface M⊂V(7)T1,0(M)x=[Tx(M)⊗ℝℂ]∩T1,0(V)x,x∈M,
is a CR structure on M. Here T1,0(V)→V is the holomorphic tangent bundle over V (locally the span of {∂/∂zj:1≤j≤N} for any local system of complex coordinates (zj) on V). Also N is the complex dimension of V, and then the CR dimension of M is n=N-1. Integrability of (7) follows from the Nijenhuis integrability of the complex structure on V. A solution f to ∂¯bf=0 (the tangential C-R equations) is a CR function on M and, in the context of real hypersurfaces carrying the induced CR structure (7), CR functions appear as traces on M of holomorphic functions defined on a neighborhood of M in V. Hence to say that the CR structure is given by (7) is to say that the tangential C-R equations are induced by the ordinary Cauchy-Riemann system on V. CR functions which are not traces of holomorphic functions may exist (cf., e.g., [17]). CR structures which are not given by (7), and for which there is not any embedding of M into some complex manifold V yielding (7), do exist as well (cf. again [17, page 172]). An array of geometric objects, such as pseudohermitian structures, the Levi form (cf. [9, 18]) and successively (in the nondegenerate case) contact structures, the Tanaka-Webster connection (cf. [18, 19]), the sublaplacian Δb and the Fefferman metric (cf. [9, 20]), springs from the given CR structure very much the way the complex structure determines the metric structure (up to a conformal invariant) on a Riemann surface and are thought of as geometric tools whose use will ultimately shed light on the properties of solutions, local and global, to the tangential C-R equations. Integrability of T1,0(M) appears as a built-in ingredient of objects such as the Tanaka-Webster connection or the Fefferman metric, yet it is believed to lack the geometric meaning of involutivity of real smooth distributions on manifolds (cf., e.g., [21, page 16]). On the other hand nonintegrable examples of almost CR structures occur frequently, either on real hypersurfaces of almost complex manifolds or on contact Riemannian manifolds (cf. [13, 16]). A remedy was indicated by Tanno [16], showing that the wealth of additional structure (φ,ξ,η,g) on a given contact Riemannian manifold N compensates for the lack of integrability of T1,0(N)={X-iφX:X∈Ker(η)} and specifically providing a generalization of the Tanaka-Webster connection to the nonintegrable context.Given a CR manifold(M,T1,0(M)), let H=Re{T1,0(M)⊕T0,1(M)} be the Levi, or maximally complex, distribution and J(Z+Z¯)=i(Z-Z¯), Z∈T1,0(M), its complex structure. Let Hx⊥={ω∈Tx*(M):Ker(ω)⊇Hx}, x∈M, be the conormal bundle associated to H, a real line bundle over M. Since M is assumed to be connected and orientable, the conormal bundle H⊥→M is trivial. A globally defined nowhere zero section θ∈Γ∞(H⊥) is a pseudohermitian structure on M. For each pseudohermitian structure θ on M the Levi form is
(8)Gθ(X,Y)=(dθ)(X,JY),X,Y∈H.
Two pseudohermitian structures θ,θ^∈Γ∞(H⊥) are related by θ^=λθ for some C∞ function λ:M→ℝ∖{0}. If this is the case, then Gθ^=λGθ. A CR manifold M is nondegenerate (resp., strictly pseudoconvex) if Gθ is nondegenerate (resp., positive definite) for some θ. If M is a nondegenerate CR manifold, of CR dimension n, then each pseudohermitian structure θ is a contact form; that is, θ∧(dθ)n is a volume form on M. If M is nondegenerate and θ is a contact form on M, there is a unique globally defined, nowhere zero, tangent vector field T∈𝔛∞(M) (the Reeb vector field of (M,θ)) such that θ(T)=1 and (dθ)(T,·)=0. The Webster metric is the semi-Riemannian metric gθ on M given by
(9)gθ(X,Y)=Gθ(X,Y),gθ(X,T)=0,gθ(T,T)=1,
for any X,Y∈H. If M is strictly pseudoconvex and θ is chosen such that Gθ is positive definite, then gθ is a Riemannian metric on M.LetN be a (2m-1)-dimensional C∞ manifold (m≥2). An almost contact structure on N is a synthetic object (ϕ,ξ,η) consisting of a (1,1)-tensor field φ, a vector field ξ∈𝔛∞(N), and a 1-form η∈Ω1(N) such that
(10)φkiφjk=-δji+ηjξi,ηiφji=0,φjiξj=0,ηiξi=1,
with respect to any local coordinate system (U′,x′i) on N. A Riemannian metric g on N is associated, or compatible, to the almost contact structure (φ,ξ,η) (and (φ,ξ,η,g) is an almost contact metric structure on N) if
(11)gijφkiφℓj=gkℓ-ηkηℓ,gijξj=ηi.
Associated metrics always exist (cf. [13]). A contact metric structure is an almost contact metric structure (φ,ξ,η,g) such that Ω=dη, where Ω∈Ω2(N) is the 2-form given by Ωij=gikφjk.Letϕ:M→N be a C∞ map from a strictly pseudoconvex CR manifold M of CR dimension n into a contact Riemannian manifold (N,φ,ξ,η,g). Let θ be a contact form on M such that the Levi form Gθ is positive definite. Let H′=Ker(η) and let us consider the vector bundle valued form (dϕ)H,H′∈Γ∞(H*⊗ϕ-1H′) given by
(12)((dϕ)H,H′)x=ΠH′,ϕ(x)∘(dxϕ):Hx⟶Hϕ(x)′,x∈M,
where ΠH′:T(N)→H′ is the natural projection associated to the decomposition T(N)=H′⊕ℝξ. Let x∈M and let {Xa:1≤a≤2n} be a local Gθ-orthonormal frame of H defined on an open neighborhood U⊆M of x∈U. We set
(13)Q(ϕ)x=∥(dϕ)H,H′∥x2=∑a=12n‍gϕ(x)(((dϕ)H,H′)xXa,x,((dϕ)H,H′)xXa,x).
Note that
(14)Q(ϕ)=traceGθ{ΠH(ϕ*g)}-∥ΠHϕ*η∥2.Definition 1.
Letp∈(0,+∞). A C∞ map ϕ:M→N is said to be contactp-harmonic if ϕ is a critical point of the energy functional
(15)EΩ,p(ϕ)=∫Ω‍∥(dϕ)H,H′∥pθ∧(dθ)n
for any relatively compact domain Ω⊆M. Contact 2-harmonic maps are called contact harmonic maps.Let∇ be the Tanaka-Webster connection of (M,θ) that is the unique linear connection on M obeying to (i) H is ∇-parallel (i.e., ∇XY∈H for any X∈𝔛∞(M) and any Y∈H), (ii) ∇J=0 and ∇gθ=0, and (iii) the torsion tensor field T∇ of ∇ is pure (i.e., T∇(Z,W)=0, T∇(Z,W¯)=2iGθ(Z,W¯)T for any Z,W∈T1,0(M) and τ∘J+J∘τ=0, where τ(X)=T∇(T,X) for any X∈𝔛∞(M) (cf. Theorem 1.3 and Definition 1.25 in [9, pages 25-26]). The vector valued 1-form τ is the pseudohermitian torsion of ∇. Let ∇′ be the generalized Tanaka-Webster connection of (N,η,g) given locally by
(16)Γjk′i=Γjki+ηjφki-ηk∇jξi+ξi∇jηk,
(cf., e.g., [16]), where Γjki are the Christoffel symbols of gij. Covariant derivatives are meant with respect to the Levi-Civita connection of (M,g). For each X∈𝔛∞(M) we consider ϕ*X∈Γ∞(ϕ-1TN) given by
(17)(ϕ*X)(x)=(dxϕ)Xx∈Tϕ(x)(N)=(ϕ-1TN)x,x∈M.
Let ∇ϕ=ϕ-1∇′ be the connection induced by ∇′ in the pullback bundle ϕ-1TN→M. We set
(18)βϕ(X,Y)=∇Xϕϕ*Y-ϕ*∇XY,X,Y∈𝔛∞(M).
Let x∈M and let {Xa:1≤a≤2n} be a local Gθ-orthonormal frame of H defined on an open neighborhood U of x. We define a C∞ section Γ(ϕ) in ϕ-1TN→M by setting
(19)Γ(ϕ)x=traceGθ{ΠHβϕ}x=∑a=12n‍βϕ(Xa,Xa)x,
where ΠHβϕ denotes the restriction of βϕ to H⊗H. By a result in [10] the Euler-Lagrange equations associated to the variational principle δEΩ,p(ϕ)=0 are
(20)Q(ϕ)-(p-2)/2[(φ2)ji∘ϕ]div(Q(ϕ)(p-2)/2∇Hϕj)=traceGθ{ΠHϕ*(η⊗τN)}-∑a=12n‍[(φ2)ji∘ϕ](Γkℓ′j∘ϕ)Xa(ϕk)Xa(ϕℓ),traceGθ{ΠHϕ*AN}=0,
here φ2=-I+η⊗ξ (cf., e.g., [13]). Also τN is the pseudohermitian torsion of (N,φ,ξ,η,g); that is, τN(X)=T∇′(ξ,X), and AN(X,Y)=g(τNX,Y) for any X,Y∈𝔛∞(N). Γjk′i are again the local coefficients of ∇′ with respect to (U′,x′i). In particular if g is a Sasakian metric, then ϕ:M→N is contact p-harmonic if and only if
(21)[(φ2)ji∘ϕ]×{∑a=12n‍div(Q(ϕ)(p-2)/2∇Hϕj)+Q(ϕ)(p-2)/2∑a=12n‍(Γkℓ′j∘ϕ)Xa(ϕk)Xa(ϕℓ)}=0,1≤i≤2m-1.


## 3. Weak Contact Harmonic Maps

Sections3 and 4 are devoted to the study of local properties of weak critical points of the functional (15). A study of the regularity of weak solutions to subelliptic systems (such as (53)) was started by Wang [8], and Capogna and Garofalo [22], though only for maps from Carnot groups, (cf. also Zhou [23]).LetM be a strictly pseudoconvex CR manifold and θ a contact form on M. Let {Xa:1≤a≤2n} be a local Gθ-orthonormal frame of H defined on the open set U⊆M and Xa* the formal adjoint of Xa; that is,
(22)Xa*u=-Xau-fau,u∈C01(U),
where fa=∂baA/∂xA+baBΓABA and Xa=baA∂/∂xA. Also ΓABC are the local coefficients of the Tanaka-Webster connection of (M,θ) with respect to the local coordinate system (U,xA) on M. Clearly (Xa*u,v)=(u,Xav) for any u∈C01(U), where (u,v)=∫U‍uv¯dv.Proposition 2.
Letϕ:M→N be a smooth map and g a Sasakian metric on N. Then ϕ is contact p-harmonic if and only if
(23)[(φ2)ji∘ϕ]×∑a=12n‍{-Xa*(Q(ϕ)(p-2)/2Xaϕj)×∑a=12n‍+Q(ϕ)(p-2)/2(Γkℓ′j∘ϕ)Xa(ϕk)Xa(ϕℓ)}=0
for any local orthonormal frame {Xa:1≤a≤2n} of H.Proof.
Let us note thatdiv(Xa)=trace{∂A↦∇∂AXa}=fa, where ∂A=∂/∂xA. Thus (by (22))
(24)div(Q(ϕ)(p-2)/2∇Hϕi)=-∑a=12n‍Xa*(Q(ϕ)(p-2)/2Xaϕi)
on U. Then (23) follows from (21).Example 3 (contactp-harmonic maps into the Heisenberg group).
LetN=ℍm-1, m≥2, be the Heisenberg group (cf., e.g., [9, pages 11–14]). Let (xα,yα,t) be the Cartesian coordinates on ℝ2m-1 and let(25)Xα=∂∂xα+2yα∂∂t,Yα=∂∂yα-2xα∂∂t,aaaaaaaaaaaaaaaaaaaaaaaa1≤α≤m-1.
Let φ be the (1,1)-tensor field on ℍm-1 determined by
(26)φ(Xα)=Yα,φ(Yα)=-Xα,φ(ξ)=0,
where ξ=-∂/∂t. Next the differential 1-form η∈Ω1(ℍm-1) given by
(27)η=2∑α=12m-2‍(yαdxα-xαdyα)-dt
is a contact form on ℍm-1; that is, η∧(dη)m-1 is a volume form. Let H=Ker(η). Finally we shall need the Riemannian metric g on ℍm-1 given by g=-dη(·,φ·) on H⊗H, g(·,ξ)=0 on H, and g(ξ,ξ)=1. Then g is a Sasakian metric on ℍm-1 (and actually (ℍm-1,g) is a Sasakian space form of φ-sectional -3; cf., e.g., [13]). A calculation shows that
(28)φ2:(-δβα000-δβα0-2yβ2xβ0),
where xα=xα and yα=yα. Let Tα=Xα-iYα and let T1,0(ℍm-1)x be the span of {Tα(x):1≤α≤m-1} over ℂ. Then T1,0(ℍm-1) is a strictly pseudoconvex CR structure on ℍm-1 and θ=-η is a contact form such that the Levi form Gθ is positive definite. Let ∇′ be the Tanaka-Webster connection of (ℍm-1,θ). A calculation shows that
(29)∇∂α′∂β=0,∇∂α′∂β+m-1=-2δαβξ,∇∂α+m-1′∂β=2δαβξ,∇∂α+m-1∂β+m-1=0,
where ∂α=∂/∂xα and ∂α+m-1=∂/∂yα for simplicity. Hence
(30)Γα,β+m-1′2m-1=-Γα+m-1,β2m-1=2δαβ
and the remaining connection coefficients vanish. The Webster metric g of (ℍm-1,θ) is given by
(31)g:(2δαβ+4yαyβ-4yαxβ-2yα-4xαyβ2δαβ+4xαxβ2xα-2yβ2xβ1),
hence (by a straightforward calculation)
(32)Q(ϕ)=2∑a=12n‍∑i=12m-2‍|Xa(ϕi)|2=2|Xϕ′|2,
where ϕ=(ϕ′,ϕ2m-1):M→ℍm-1 and ϕ′=(ϕ1,…,ϕ2m-2). Let us substitute (28)–(32) into (23) so that to obtain
(33)∑a=12n‍Xa*(|Xϕ′|p-2Xa(ϕi))=0,1≤i≤2m-2.
Hence if ϕ:M→ℍ2m-1 is a contact p-harmonic map, then ϕ′ is subject to (33) while ϕ2m-1 is an arbitrary function. Therefore, in general one may not expect regularity for a given (weak) contact p-harmonic map.The identity (23) in Proposition 2 leads naturally to the notion of a weak solution to the contact p-harmonic map system. Indeed we may establish the following.Lemma 4.
A smooth mapϕ:M→N of a strictly pseudoconvex CR manifold M into a Sasakian manifold N is contact p-harmonic if and only if
(34)∑a=12n‍{Xa*(Q(ϕ)(p-2)/2[(φ2)ji∘ϕ]Xa(ϕj))-Q(ϕ)(p-2)/2[(φ2)kj∘ϕ](Γjℓ′i∘ϕ)Xa(ϕk)Xa(ϕℓ)}=0
for any local orthonormal frame {Xa:1≤a≤2n} of H on U and any local coordinate system (U′,x′i) on N such that ϕ-1(U′)⊇U.Proof.
Let us multiply (23) by a test function ψ∈C0∞(U) and integrate by parts
(35)∫‍Q(ϕ)(p-2)/2∑a‍Xa(ϕj)Xa((φ2)jiψ)dv=∫‍Q(ϕ)(p-2)/2∑a‍(φ2)jiΓ′kℓjXa(ϕk)Xa(ϕℓ)ψdv.
On the other hand (as both ξ and η are parallel with respect to ∇′)
(36)∂ξi∂x′k=-Γkℓ′iξℓ,∂ηj∂x′k=Γkj'ℓηℓ,(37)∂(φ2)ji∂x′k=ξiηℓΓkj'ℓ-ηjξℓΓkℓ′i,(38)(φ2)jiΓkℓ′i+ηℓξjΓjk′i-ξiηjΓkℓ′j=(φ2)ℓjΓjk′i-Tkℓi,
where Tkℓi are the coefficients of T∇′ with respect to (U′,x′i). Therefore (35) may be written as(39)∫‍Q(ϕ)(p-2)/2×∑a‍{(φ2)jiXa(ϕj)Xa(ψ)-(φ2)kjΓjℓ′iXa(ϕk)Xa(ϕℓ)ψ}dv=0
and Lemma 4 is proved.Let us consider the function spaces(40)WX1,p(U)={u∈Lp(U):Xau∈Lp(U),1≤a≤2n},
where Xau are understood as weak derivatives. If 1≤p<∞, then WX1,p(U) are separable Banach spaces with the norms
(41)∥u∥WX1,p(U)=(∥u∥Lp(U)p+∑a=12n‍∥Xau∥Lp(U)p)1/p.
Also WX1,p(U) is reflexive provided that 1<p<∞. The central concept of this section may be introduced as follows. Let {Xa:1≤a≤2n} be a Gθ-orthonormal frame of H defined on the open set U⊆M. Let U′⊆N be an open set which is relatively compact in a larger coordinate neighborhood in N.Definition 5.
A mapϕ:U→U′ is said to be weak contactp-harmonic if it is a weak solution to (34); that is, ϕj∈WX1,p(U) for any 1≤j≤2m-1 and the identities (39) are satisfied for any test function ψ∈C0∞(U).Letϕ:U→U′ be a weak contact p-harmonic map. By (14)
(42)Q(ϕ)=∑a‍{Xa(ϕi)Xa(ϕj)(gij∘ϕ)-[Xa(ϕi)(ηi∘ϕ)]2}
on U, hence
(43)|Q(ϕ)|≤C|Xϕ|2a.e.inU,|Xϕ|2=∑a=12n‍∑i=12m-1‍|Xa(ϕi)|2,
where C=max{supU′¯|gij|,supU′¯|ηi|:1≤i,j≤2m-1}. Then both integrals in (39) are convergent and the adopted definition is legitimate.Example 6 (Example3 continued).
A weak solution to (33) is a map ϕ=(ϕ′,ϕ2m-1):U→U′⊂⊂ℍm-1 such that ϕ′∈WX1,p(U,ℝ2m-2) and
(44)∑a=12n‍∫U‍|Xϕ′|p-2Xa(ϕi)Xa(ψ)dv=0,1≤i≤2m-2,
for any ψ∈C0∞(U). We need to recall the following general result, due to Xu and Zuily [24]. Let X={X1,…,Xm} be a Hörmander system on an open set U⊆ℝN, N≥2, and Ω⊂ℝN a domain such that U⊃Ω¯. Let aij(x,y) be a symmetric and positive definite matrix defined in Ω×ℝν. If |f(x,y,p)|≤a|p|2+b for any (x,y,p)∈Ω×ℝν×ℝmν, then any continuous solution ϕ=(ϕ1,…,ϕν) to
(45)∑i,j=1m‍Xj*(aij(x,ϕ(x))Xiϕα(x))=fα(x,ϕ(x),Xϕ(x)),aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa1≤α≤ν,
in Ω is actually smooth. Let us assume that U is a domain such that U¯ is contained in a coordinate neighborhood in M. By the result in [24] quoted above.Proposition 7.
For any weak solutionϕ=(ϕ′,ϕ2m):U→U′⊂ℍm-1 to the contact p-harmonic map system (33) if ϕ′∈C0(U,ℝ2m-2), then ϕ′∈C∞(U,ℝ2m-2).Of course in the particular casep=2 any distribution solution ϕ′ is C∞ (as the operator ∑a=12n‍Xa*Xa is hypoelliptic).Example 8 (contactp-harmonic maps into the sphere).
LetN=S2m-1⊂ℝ2m and let g be the canonical Sasakian metric on S2m-1. Then a C∞ contact p-harmonic map ϕ=(ϕ1,…,ϕ2m):M→S2m-1 is a solution to
(46)[(φ2)ji∘ϕ]∑a=12n‍Xa*(Q(ϕ)(p-2)/2Xaϕj)=Q(ϕ)(p-2)/2×{[(φ2)ji∘ϕ]|Xϕ|2ϕj∑a=12n‍+2∑a=12n‍(ϕ*η)(Xa)(φji∘ϕ)Xa(ϕj)},
for any 1≤i≤2m-1. Here |Xϕ|2=∑β=12m‍∑a=12n‍|Xaϕβ|2 and ∑β=12m‍ϕβ2=1 with ϕβ=ϕβ, 1≤β≤2m. Equation (46) follows from (23) by computing the Christoffel symbols of S2m-1 with respect to the local coordinate system
(47)χ′:U′→ℝ2m-1,χ′(x)=x′,x=(x′,x2m)∈U′,U′=S2m-1∩{x2m>0},x′=(x1,…,x2m-1),
that is
(48)|ijk|=xigjk,gjk=δjk+xjxk1-|x′|2,
so that
(49)∑a=12n‍(|ijk|∘ϕ)Xa(ϕj)Xa(ϕk)=|Xϕ|2ϕi,aaaaaaaaaaaaaaaaaaaaaaa1≤i≤2m-1.
On the other hand (cf. [9])
(50)|ijk|=Γjk′i+ωjkξi+ηjφki+ηkφji
so that
(51)(|ijk|∘ϕ)Xa(ϕj)Xa(ϕk)=(Γjk′i∘ϕ)Xa(ϕj)Xa(ϕk)+2(ϕ*η)(Xa)(φji∘ϕ)Xa(ϕj)
for any Sasakian metric g. When N=S2m-1, the identities (49)–(51) lead to
(52)∑a=12n‍(Γjk′i∘ϕ)Xa(ϕj)Xa(ϕk)=|Xϕ|2ϕi-∑a=12n‍2(ϕ*η)(Xa)(φji∘ϕ)Xa(ϕj)
and then to (46) by taking into account that φ is an f-structure on S2m-1; that is, φ3+φ=0. Our next purpose in this example is to prove the following result.Proposition 9.
Letϕ:ℍn→S2m-1 be a horizontal map. Then ϕ is contact p-harmonic if and only if ϕ is subelliptic p-harmonic with respect to the canonical Hörmander system X={Xγ,Yγ:1≤γ≤n} on ℍn.According to [7] given a Hörmander system of vector fields {Xa} defined on an open set O⊆ℝN, one may adopt the following.Definition 10.
A subellipticp-harmonic map is a C∞ solution ϕ:O→ℝ2m to the system (the formal adjoint of Xa in [7] is -Xa* under the conventions adopted in the present paper)(53)∑a‍Xa*(|Xϕ|p-2Xaϕα)=|Xϕ|pϕα,1≤α≤2m,
such that ∑α=12m‍ϕα2=1.A horizontal map is a smooth mapϕ:ℍn→S2m-1 such that
(54)Xa(ϕi)(ηi∘ϕ)=0,1≤a≤2n.
One may define weak solutions ϕ:ℍn→U′ to (54) by requiring that ϕi∈WX1,p(U) for some 1≤p<∞ and that (54) holds a.e. in U. Then the statement in Proposition 9 holds for weak solutions of the relevant equations as well. In particular, by a result in [7], any weak horizontal contact p-harmonic map ϕ:ℍn→U′ is locally Hölder continuous provided that p≥2n+2.The proof of Proposition9 is to write (46) in the form (53). We need the following.Lemma 11.
LetM be a strictly pseudoconvex CR manifold. A smooth map ϕ:M→S2m-1 is contact p-harmonic if and only if
(55)-∑a=12n‍Xa*(Q(ϕ)(p-2)/2[(φ2)ji∘ϕ]Xa(ϕj))=Q(ϕ)p/2ϕi,
for any 1≤i≤2m-1 and any local orthonormal frame {Xa:1≤a≤2n} of H.By (14) if ϕ:M→S2m-1 is a horizontal map, then Q(ϕ)=|Xϕ|2 and one may readily check that (55) is equivalent to (53) for any 1≤i≤2m-1. Of course the component ϕ2m will satisfy (53) as well (as a consequence of the constraint ∑α=12n‍ϕα2=1). To prove Lemma 11, let us multiply (46) by a test function ψ∈C0∞(U) and integrate over U. The left-hand side of the resulting equation is
(56)∑a‍∫U‍(φ2)jiXa*(ρXa(ϕj))ψdv=∑a‍∫‍ρXa(ϕj)Xa((φ2)jiψ)dv=∑a‍∫‍ρ{Xa(ϕj)(φ2)jiXa(ψ)+ψXa(ϕj)Xa((φ2)ji∘ϕ)}dv=∑a‍∫‍{Xa*(ρ[(φ2)ji∘ϕ]Xa(ϕj))ψ∂(φ2)ji∂x′k+ρψXa(ϕj)Xa(ϕk)∂(φ2)ji∂x′k}dv,
where ρ=Q(ϕ)(p-2)/2. Then (by (37))
(57)∑a‍Xa(ϕj)Xa(ϕk)∂(φ2)ji∂x′k=∑a‍Xa(ϕj)Xa(ϕk)(ξiηℓΓkj'ℓ-ηjξℓΓkℓ′i)=(by(52)and(50))=ξiηℓ(|Xϕ|2ϕℓ-2∑a‍ηjXa(ϕj)φkℓXa(ϕk))-∑a‍Xa(ϕj)Xa(ϕk)ηj(|ikℓ|ξℓ-φki)=ξiηℓϕℓ|Xϕ|2+∑a‍{ηjXa(ϕj)φkiXa(ϕk)-Xa(ϕj)Xa(ϕk)ηjηkϕi},
hence (46) implies
(58)∑a‍Xa*(Q(ϕ)(p-2)/2[(φ2)ji∘ϕ]Xa(ϕj))=Q(ϕ)(p-2)/2(-|Xϕ|2+∑a‍[Xa(ϕj)(ηj∘ϕ)]2)ϕi
which yields (55) because on the sphere
(59)Q(ϕ)=|Xϕ|2-∥ΠHϕ*η∥2.
Lemma 11 is proved.The notion of a weak contact harmonic map as introduced above is confined to mapsϕ:M→N such that the target contact Riemannian manifold N is covered by a single coordinate neighborhood. Another natural approach (customary in the theory of harmonic maps among Riemannian manifolds, cf., e.g., [4, page 38]) is to use Nash's embedding theorem (cf. [25]) in order to embed isometrically the target manifold N into some Euclidean space ℝK and produce an alternative first variation formula (cf. Theorem 2.22 in [26, page 139]) depending however on the embedding N↪ℝK.A generalization of Nash's embedding theorem to the context of contact Riemannian geometry has been obtained by D'Ambra [27]. Let ℍL≈ℂL×ℝ be the Heisenberg group equipped with the standard Sasakian structure (φ0,ξ0,η0,g0). Let (N,(φ,ξ,η,g)) be a contact Riemannian manifold. By a result in [27], if N is compact and L≥dim(N)+1, there is a C1-embedding ι:N→ℍL which is both horizontal, that is, ι*H′⊂ι-1Ker(η0), and isometric in the sense that ι preserves the Levi forms
(60)gp(v,w)=g0,ι(p)((dpι)v,(dpι)w),v,w∈Hp′,p∈N.
Any contact Riemannian manifold N is in particular a sub-Riemannian manifold (in the sense of [28]); hence N carries the Carnot-Carathéodory metric dN:N×N→[0,+∞) associated to the sub-Riemannian structure (H′,g). In particular ι is an isometry among the metric spaces (N,dN) and (ℍL,dX) (cf. Section 7 for the definition of the distance function dX:ℍL×ℍL×[0,+∞)). As ℍL also possesses a linear space structure, the methods in [29] (methods of direct infinitesimal geometry) become available on a contact Riemannian manifold (e.g., one may merely use the balls with respect to dN and the linear structure of the ambient space ℍL to reformulate on N Definition 2.1 in [29, page 280]) and we conjecture that the arguments in [29] may be recovered to study the equation Δbu=0 on a strictly pseudoconvex CR manifold (the theory in [29] only deals with second order degenerate elliptic equations on domains in ℝn). Unfortunately the existence of C1-embeddings of given contact structures is not sufficient for differential geometric purposes, as long as Gauss and Weingarten formulae (which require two derivatives of ι) are involved. The problem of improving D'Ambra's proof (to get a horizontal embedding of class at least C2) is open.


## 4. Contact Harmonic Maps into Spheres

LetΩ⊂ℝN be a bounded open set and X={X1,…,Xm} a Hörmander system of vector fields Xa=baA(x)∂/∂xA∈𝔛(ℝN) such that baA∈C∞(ℝN)∩Lip(ℝN). We recall (cf., e.g., [9, page 261]) the following.Definition 12.
A numberD is a homogeneous dimension relative to Ω with respect to X if there is a constant C>0 such that
(61)|BX(x,r)||BX(x0,r0)|≥C(rr0)D
for any Carnot-Carathéodory ball B0=BX(x0,r0) of center x0∈Ω and radius 0<r0≤diam(Ω) and any Carnot-Carathéodory ball B=BX(x,r) of center x∈B0 and radius 0<r≤r0.The diameter ofΩ is meant with respect to the Carnot-Carathéodory metric associated to X. Hajłasz and Strzelecki [7] studied local properties of weak solutions to the system (53). Their main finding is that every weak subelliptic D-harmonic map ϕ∈WX1,D(Ω,Sν) (i.e., every weak solution to (53) with p=D) is locally Hölder continuous. Maps ϕ:Ω→Sν with values in a unit sphere Sν⊂ℝν+1 have a special status due to the fact that the subelliptic harmonic map system (here (53)) may be written in a simple form using an approach commonly referred to as the Frédéric Hélein trick (cf. [7, page 353], see also Hélein [30]). The purpose of this section is to start a study of weak solutions to the system (55) following the ideas in [7] though confined to maps ϕ:ℍn→S2m-1 which are “close to horizontal” in a sense to be made precise in the sequel.Letℍn be the Heisenberg group equipped with the standard contact form θ=dt+i∑γ=1n‍(zγdz¯γ-z¯γdzγ). Let U⊆ℍn be a bounded domain. Let {Xa:1≤a≤2n}={Xγ,Yγ:1≤γ≤n} be the Gθ-orthonormal frame given by Xγ=∂/∂xγ+2yγT and Yγ=∂/∂yγ-2xγT, where T=∂/∂t as in Example 3. Clearly the coefficients of the Xa's lie in C∞(ℝ2n+1)∩Lip(ℝ2n+1). We recall that an absolutely continuous curve γ:[0,τ]→ℍn is admissible if
(62)dγdt(t)=∑a=12n‍ua(t)Xa(γ(t))
for some functions ua(t) such that ∑a=12n‍ua(t)2≤1.Definition 13.
TheCarnot-Carathéodory distancedX(x,y) among two points x,y∈ℍn is the infimum of all τ>0 for which there exists an admissible curve γ:[0,τ]→ℍn such that γ(0)=x and γ(τ)=y. Balls with respect to dX:ℍn×ℍn→[0,+∞) are denoted by BX(x,r)={y∈ℍn:dX(x,y)<r} and referred to as Carnot-Carathéodory balls.We shall characterize horizontal maps in terms of the first order differential operator(63)Lau=um+αXa(uα)-uαXa(um+α)
defined for u=(u1,…,u2m)∈WX1,p(U,ℝ2m).Proposition 14.
Letϕ:U→U′=S2m-1∩{x2m>0}⊂ℝ2m be a map such that ϕA∈WX1,p(U) for any 1≤A≤2m. Then ϕ:U→U′ is a (weak) horizontal map if and only if Laϕ=0 for any 1≤a≤2n.Let(z1,…,zm) be the natural complex coordinates on ℂm and set zα=xα+iyα and (x1,…,x2m)=(x1,…,xm,y1,…,ym). The following conventions are adopted as to the range of indices:
(64)1≤A,B,⋯≤2m,1≤i,j,⋯≤2m-1,1≤α,β,⋯≤m,1≤r,s,⋯≤m-1.
Let ν=xα∂/∂xα+yα∂/∂yα∈𝔛∞(ℝ2m) so that the pointwise restriction of ν to S2m-1 is a unit normal field on S2m-1. Let J0 be the complex structure on ℂm. Then ξ∈𝔛∞(S2m-1) given by (dxι)ξx=J0,xνx for any x∈S2m-1 is the Reeb vector field on S2m-1. Here ι:S2m-1→ℝ2m is the inclusion. With respect to the local chart χ′=(x1,…,x2m-1) in Example 8 the Reeb vector ξ is given by
(65)ξα=-yα,ξm+r=xr.
Then ηi=gijξj together with (48) in Example 8 leads to
(66)ηα=-yα-xmymxα,ηm+r=xr-xmymyr.
Finally (66) implies that Xa(ϕi)(ηi∘ϕ)=-Laϕ. Proposition 14 is proved. In particular Q(ϕ) may be written as
(67)Q(ϕ)=|Xϕ|2-∑a=12n‍|Laϕ|2.
Our next task is to put (55) into a more tractable form.Proposition 15.
Letϕ=(ϕ1,…,ϕ2m):U→U′ such that ϕA∈WX1,p(U). Let us consider the functions
(68)Vα,a=Q(ϕ)(p-2)/2{Xa(ϕα)-ϕm+αLaϕ},Vn+α,a=Q(ϕ)(p-2)/2{Xa(ϕm+α)+ϕαLaϕ},
with 1≤α≤m. Let VA=(VA,1,…,VA,2n) for any 1≤A≤2m. Then ϕ:U→U′ is a contact p-harmonic map if and only if
(69)X*·VA=Q(ϕ)p/2ϕA,1≤A≤2m.Here the dot product meansX*·VA=∑a=12n‍Xa*(VA,a). Using φ2=-I+η⊗ξ and (65) and (66), one obtains
(70)[(φ2)ji]1≤i,j≤2m-1=[-δβα+yα(yβ+xmymxβ)-yα(xr-xmymyr)-xs(yβ+xmymxβ)-δrs+xs(xr-xmymyr)].
Then substitution into (55) leads to
(71)∑a=12n‍Xa*[Q(ϕ)(p-2)/2(Xa(ϕα)-ϕm+αLaϕ)]=Q(ϕ)p/2ϕα,(72)∑a=12n‍Xa*[Q(ϕ)(p-2)/2(Xa(ϕm+s)+ϕsLaϕ)]=Q(ϕ)p/2ϕm+s.
It remains to be shown that (71) and (72) imply
(73)∑a=12n‍Xa*[Q(ϕ)(p-2)/2(Xa(ϕ2m)+ϕmLaϕ)]=Q(ϕ)p/2ϕ2m.
Let us multiply (71) by ϕβψ, where ψ∈C0∞(U) is an arbitrary test function, and integrate over U so that to obtain (after integration by parts)
(74)∑a‍Xa*[Q(ϕ)(p-2)/2(Xa(ϕα)-ϕm+αLaϕ)ϕβ]=Q(ϕ)(p-2)/2×{Q(ϕ)ϕαϕβ-∑a‍Xa(ϕα)Xa(ϕβ)+∑a‍ϕm+αXa(ϕβ)Laϕ}.
Similarly let us multiply (72) by ϕm+rψ so that to obtain
(75)∑a‍Xa*[Q(ϕ)(p-2)/2(Xaϕm+s+ϕsLaϕ)ϕm+r]=Q(ϕ)(p-2)/2×{Q(ϕ)ϕm+sϕm+r-∑a‍Xa(ϕm+s)Xa(ϕm+r)-∑a‍ϕsXa(ϕm+r)Laϕ}.
Let us contract the indices α and β in (74) (resp., r and s in (75)), add the resulting equations, and use the identities
(76)Xa(ϕα)ϕα+Xa(ϕm+r)ϕm+r=-Xa(ϕ2m)ϕ2m,-ϕm+αϕα+ϕrϕm+r=-ϕmϕ2m,ϕαϕα+ϕm+rϕm+r=1-ϕ2m2,ϕm+αXa(ϕα)-ϕrXa(ϕm+r)=Laϕ+ϕmXa(ϕ2m).
We get
(77)-∑a‍Xa*[Q(ϕ)(p-2)/2(Xa(ϕ2m)+ϕmLaϕ)ϕ2m]=Q(ϕ)(p-2)/2{Q(ϕ)(1-ϕ2m2)-∑a‍Xa(ϕi)Xa(ϕi)+∑a‍[Laϕ+ϕmXa(ϕ2m)]Laϕ}.
Let us use Q(ϕ)-∑a‍Xa(ϕi)Xa(ϕi)+∑a‍(Laϕ)2=Xa(ϕ2m)2 (a consequence of (67)). Finally
(78)∑a‍Xa*[Q(ϕ)(p-2)/2(Xa(ϕ2m)+ϕmLaϕ)ϕ2m]=Q(ϕ)(p-2)/2×{Q(ϕ)ϕ2m2-∑a‍Xa(ϕ2m)2-∑a‍ϕmXa(ϕ2m)Laϕ}.
Now the identity (73) follows from (78) and Xa*=-Xa-fa. Proposition 15 is proved.The crucial manner of exploiting the constraint∑A=12m‍ϕA2=1 is contained in the following.Proposition 16.
LetU⊂ℍn be a bounded domain and ϕ:U→S2m-1⊂ℝ2m, ϕ=(ϕ1,…,ϕ2m), a map such that ϕA∈WX1,p(U). Then
(79)VA=∑B=12m‍ϕBEA,B,
where one has set EA,B=ϕBVA-ϕAVB. Moreover if ϕ is a contact p-harmonic map, then
(80)X*·EA,B=Q(ϕ)(p-2)/2{σBϕB+mX(ϕA)-σAϕA+mX(ϕB)}Lϕ,
where σA=1 if 1≤A≤m, σA=-1 if m+1≤A≤2m, and the range of the indices in (80) is meant modm.The identity (79) is a consequence of the constraint alone. The identity (80) for A=α and B=β follows from (74) (interchange α and β in (74) and subtract the resulting identity from (74)). In general, for any ψ∈C0∞(U)(81)∫U‍X*·(ϕAVB)ψdv=∫U‍VB·[X(ψϕA)-ψX(ϕA)]dv=∫U‍(X*·VB)ϕAψdv-∫U‍[VB·X(ϕA)]ψdv,
hence (by (69))
(82)∫U‍X*·(ϕAVB)ψdv=∫U‍{Q(ϕ)p/2ϕBϕA-Q(ϕ)(p-2)/2X(ϕB)·X(ϕA)}ψ-∫U‍Q(ϕ)(p-2)/2σBϕB+m[(Lϕ)·X(ϕA)]ψdv.
Now let us interchange A and B in (82) to produce another identity of the sort and subtract it from (82). This yields (80). Proposition 16 is proved.Although regularity of contactp-harmonic maps cannot be expected in general (cf. Example 3), a few fundamental questions may be asked. For instance, what is the the outcome of the ordinary hole filling argument (cf., e.g., [31, pages 38–40]) and of Moser's iteration technique in regularity theory? our finding in this direction is Theorem 20. We shall need the following.Lemma 17.
LetU⊂ℍn be a bounded domain. Let R0>0 and U1⊂⊂U such that BX(x,400R0)⊂U for any x∈U1. Let 𝔹=BX(x0,r) with x0∈U1 be a Carnot-Carathéodory ball such that 0<r≤R0 and let ψ∈WX1,2n+2(𝔹) be a function of compact support. Then for any contact (2n+2)-harmonic map ϕ:ℍn→S2m-1 satisfying (96) for some 0<c<1 and some 0<δ<1(83)|∫𝔹‍[X*·(ϕBEA,B)]ψdv|≤C∥Xψ∥LD(𝔹){∥Xϕ∥LD(100𝔹)D+∥Xϕ∥LD(100𝔹)(1-ϵ)D}
for some constant C=C(U1,n,R0)>0, where ϵ=(1-δ)/D and D=2n+2.This is similar to Lemma 3.2 (the duality inequality) in [7, page 354] and will be proved later on in this section.LetU1⊂⊂U and R0>0 as in Lemma 17. Also let x∈U1 and 0<r<R0 and set 𝔹=BX(x,r) and 2𝔹=BX(x,2r). Let ψ∈C0∞(U) be a test function such that 0≤ψ≤1, ψ=1 on 𝔹, ψ=0 on U∖2𝔹, and |Xψ|≤C/r for some constant C>0. Next let us set
(84)ψA=[ϕA-(ϕA)2𝔹]ψ.
Throughout if (X,μ) is a measurable space and A⊂X a measurable set with μ(A)>0, we adopt the notation uA=(1/μ(A))∫A‍udμ. Let us take the dot product of (79) with X*, multiply the resulting equation by ψA, integrate over 2𝔹, and sum over A(85)∑A=12m‍∫2𝔹‍(X*·VA)ψAdv=∑A,B=12m‍∫2𝔹‍[X*·(ϕBEA,B)]ψAdv.
The first line of (85) may be computed as follows:
(86)∫2𝔹‍(X*·VA)ψAdv=∫2𝔹‍VA·X(ψA)dv=∫2𝔹‍VA·{X(ψ)[ϕA-(ϕA)2𝔹]+ψX(ϕA)}dv
and summed over A(87)∑A‍VA·X(ϕA)=∑a‍{Vα,aXa(ψα)+Vm+α,aXa(ϕm+α)}=Q(ϕ)(p-2)/2{|Xϕ|2-∑a‍(Laϕ)2}=Q(ϕ)p/2
by the very definition of VA (cf. Lemma 22) and by (67). Thus (85) becomes
(88)∫2𝔹‍ψQ(ϕ)p/2dv+∑A‍∫2𝔹‍[ϕA-(ϕA)2𝔹]VA·X(ψ)dv=∑A,B‍∫2𝔹‍[X*·(ϕBEA,B)]ψAdv.
For simplicity let IA,B=∫2𝔹‍[X*·(ϕBEA,B)]ψAdv and C0=∑A,B‍|IA,B|. Using (88), we may perform the estimates
(89)∫𝔹‍Q(ϕ)p/2dv≤∫2𝔹‍ψQ(ϕ)p/2dv≤C0+∑A‍∫2𝔹‍|ϕA-(ϕA)2𝔹||VA||Xψ|dv.Lemma 18.
Let one set|Lϕ|2=∑a=12n‍|Laϕ|2. Then |LX|≤2|Xϕ| a.e. in U and consequently
(90)|VA|≤6Q(ϕ)(p-2)/2|Xϕ|
a.e. in U, for any 1≤A≤2m.The inequalities in Lemma18 follow easily from |ϕA|≤1 and |XϕA|≤|Xϕ|. Using (90), we may write (89) as
(91)∫𝔹‍Q(ϕ)p/2dv≤C0+6∑A‍∫2𝔹‍Q(ϕ)(p-2)/p|ϕA-(ϕA)2𝔹||Xϕ||Xψ|dv.
In the following estimates C denotes some positive constant, not necessarily the same in all formulae. By Hölder's inequality
(92)∫2𝔹‍Q(ϕ)(p-2)/2|ϕA-(ϕA)2𝔹||Xϕ||Xψ|dv≤(∫2𝔹‍|ϕA-(ϕA)2𝔹|pdv)1/p×(∫2𝔹∖𝔹‍(Q(ϕ)(p-2)/2|Xϕ||Xψ|)p/(p-1)dv)(p-1)/p≤C(∫2𝔹‍|XϕA|pdv)1/p×(∫2𝔹∖𝔹‍Q(ϕ)p(p-2)/2(p-1)|Xϕ|p/(p-1))(p-1)/p,
by the Poincaré inequality
(93)(∫2𝔹‍|ϕA-(ϕA)2𝔹|pdv)1/p≤Cr(∫2𝔹‍|XϕA|pdv)1/pand by |Xψ|≤C/r. Let us observe that Q(ϕ)≤|Xϕ|2 yields
(94)(∫2𝔹∖𝔹‍Q(ϕ)p(p-2)/2(p-1)|Xϕ|p/(p-1))(p-1)/p≤(∫2𝔹∖𝔹‍|Xϕ|pdv)(p-1)/p.
Hence (by (91))
(95)∫𝔹‍Q(ϕ)p/2dv≤C0+C(∫2𝔹‍|Xϕ|pdv)1/p×(∫2𝔹∖𝔹‍|Xϕ|pdv)(p-1)/p.
Let us set Ip(r)=∫BX(x,r)‍|Xϕ|pdv. Also let us restrict our considerations to maps ϕ:ℍn→S2m-1 for which one may control Q(ϕ) from below. We adopt the following.Definition 19.
A mapϕ:ℍn→S2m-1 is said to be close to a horizontal map if there exist constants 0<c<1 and 0<δ<1 such that
(96)|Lϕ|≤c|Xϕ|δa.e.in{x∈ℍn:|Xϕ|(x)≥1},|Lϕ|≤c|Xϕ|a.e.in{x∈ℍn:|Xϕ|(x)<1}.Ifϕ:ℍn→S2m-1 is close to horizontal, then (by (96))
(97)Q(ϕ)≥a|Xϕ|2,a=1-c2>0.
Our main result in this section is the following.Theorem 20.
LetU⊂ℍn be a bounded domain in the Heisenberg group and Z¯α=∂/∂z¯α-izα∂/∂t, 1≤α≤n, the Lewy operators. Let X={Zα+Z¯α,i(Zα-Z¯α):1≤α≤n} and U1⊂⊂U. Let ϕ∈WX1,2n+2(U,S2m-1) be a map obeying to (96) for some 0<c<1 and 0<δ<1. If ϕ:U→S2m-1 is a weak contact (2n+2)-harmonic map, then there exist constants r0>0, C>0 and 0<γ<1 such that
(98)∫BX(x,r)‍|Xϕ|2n+2dv≤Crγ
for any x∈U1 and any 0<r≤r0.As a consequence of Theorem20 (by applying a version of the Dirichlet growth theorem due to Macìas and Segovia [15]).Corollary 21.
LetU⊂ℍn be a bounded domain. Any weak contact (2n+2)-harmonic map ϕ:U→S2m-1 satisfying (96) is locally Hölder continuous.To prove Theorem20, we use a hole filling technique essentially due to Widman [32], (cf. also Bensoussan et al. [31, page 38–40]). By (95) with p=D=2n+2 and Lemma 17 with ψ=ψA, we have
(99)∫𝔹‍Q(ϕ)D/2dv≤C{∑A=12m‍ID(2r)1/D(ID(2r)-ID(r))(D-1)/D+[ID(200r)+ID/(D+1)(200r)1/(D+1)]×∑A=12m‍∥XψA∥LD(2𝔹)}.
On the other hand, by the very definition of ψA, we may use the Poincaré inequality to estimate
(100)∑A=12m‍∥XψA∥LD(2𝔹)≤∑a‍{∥(Xψ)[ϕA-(ϕA)2𝔹]∥LD(2𝔹)+∥ψXϕA∥LD(2𝔹)}=∑A‍(∫2𝔹‍|Xψ|D|ϕA-(ϕA)2𝔹|Ddv)1/D+∑A‍(∫2𝔹‍|ψ|D|XϕA|Ddv)1/D≤Cr∑A‍(∫2𝔹‍|ϕA-(ϕA)2𝔹|Ddv)1/D+∑A‍(∫2𝔹‍|XϕA|Ddv)1/D,
that is,
(101)∑A=12m‍∥XψA∥LD(2𝔹)≤CID(2r)1/D.
Using (97) and (101), the inequality (120) yields
(102)aID(r)≤C{ID(2r)1/D(ID(2r)-ID(r))(D-1)/D+[ID(200r)+ID(200r)1-ϵ]ID(2r)1/D}.
By the Vitali absolute continuity of the integral ID(200r), there is r0′>0 such that ID(200r)<1 for any 0<r≤r0′. As a consequence of (102) we may establish the following.Lemma 22.
There exist0<r0≤r0′ and 1/2≤λ<1 such that
(103)ID(r)≤λID(200r)1-ϵ
for any 0<r≤r0.Proof.
The proof is by contradiction. Let us assume that for any0<r0≤r0′ and any 1/2≤λ<1, there is 0<r≤r0 such that I(r)>λI(200r)1-ϵ, where I is short for ID. Note that I(200r)≤I(200r)1-ϵ. Then (by (102))
(104)λI(200r)1-ϵ<I(r)≤C{I(200r)(1-λ)(D-1)/D≤C+[I(200r)+I(200r)1-ϵ]I(2r)1/D}≤CI(200r)1-ϵ{(1-λ)(D-1)/D+I(2r)1/D}.
Therefore
(105)12≤λ<C{(1-λ)(D-1)/D+I(2r)1/D}.
The inequality (105) leads to
(106)(12C)D≤∫2𝔹‍|Xϕ|Ddv.
Indeed, by the contradiction assumption, we may pick a sequence λj∈[1/2,1) such that λj→1 as j→∞ and consider the corresponding radii 0<rj≤r0. By passing to a subsequence, if necessary, one may assume that limj→∞rj=r∞ for some r∞∈[0,r0]. Let j→∞ in
(107)12≤λj<C{(1-λj)(D-1)/D+I(2rj)1/D}
and use the absolute continuity of the integral. Then either r∞>0 (yielding (106)) or r∞=0 and then 1/2≤0, a contradiction. Finally (106) may be exploited as follows. Let r0=1/k. By the contradiction assumption there is 0<r≤1/k such that (by (106))
(108)(12C)D≤I(2r)≤∫BX(x,2/k)‍|Xϕ|Ddv
and the last integral tends to 0 as k→∞, a contradiction. Lemma 22 is proved.
Now we may prove the Caccioppoli type estimate (98). Let τ=1/200 so that (103) may be written as(109)ID(τr)≤λID(r)1-ϵ.
Then (by (109) and induction over m)
(110)ID(τmr)≤λ[1-(1-ϵ)m]/ϵID(r)(1-ϵ)m
for any m∈ℤ, m≥1. Let us consider the family of intervals {(τm,τm-1]:m∈ℤ,m≥1}. It is a cover of (0,1], hence for each 0<r≤r0 there is m∈ℤ, m≥1, such that τm<r/r0≤τm-1. Now the inequality r≤τm-1r0 implies (by (110))
(111)ID(r)≤ID(τm-1r0)≤λ[1-(1-ϵ)m-1]/ϵID(r0)(1-ϵ)m-1.
On the other hand let us set γ=(logλ)/(logτ) (so that 0<γ<1) and observe that the inequality r/r0≥τm implies
(112)(rr0)γ>τmγ=τ(logλm)/(logτ)=λm,
that is, λm<(r/r0)γ. One may choose r0>0 from the very beginning such that ID(r0)<λ for any x∈U1. Note that 0<ϵ≤1/2 (by the very definition of ϵ). Then [1-(1-ϵ)m]/ϵ=∑j=0m-1‍(1-ϵ)j≥1+(m-1)(1/2)=(m+1)/2, hence λ[1-(1-ϵ)m]/ϵ≤Crγ/2, where C=r0-γ/2λ. Theorem 20 is proved.It remains that we prove Lemma17. It suffices to prove the inequality (83) for any ψ∈C0∞(𝔹). Let us consider
(113)w(x)=(14a0)|x|-2n,x∈ℍn,
where a0=(22-2nπn+1/Γ(n/2))2 and |x|=(|z|4+t2)1/4 is the Heisenberg norm of x=(z,t). By a classical result of Folland, [33], G(x,y)=w(xy-1) is a fundamental solution for the Hörmander operator ∑a=12n‍Xa2. In particular for any bounded domain U⊂ℍn one has the representation formula
(114)u(x)=∫U‍XyG(y,x)·Xu(y)dv(y)
for any u∈C0∞(U) and any x∈U. By a result of Citti et al., [34], we may consider a smooth cut-off function 0≤ψ0≤1 such that ψ0=1 on 2𝔹, ψ0=0 on U∖4𝔹, and |Xψ0|≤C/diam(𝔹) (the diameter is meant with respect to the Carnot-Carathéodory metric on ℍn). Using (114) for u=ψ, one may write
(115)∫𝔹‍[X*·(ϕBEA,B)]ψdv=∫𝔹‍X*·(ϕBEA,B)(x)ψ(x)ψ0(x)dv(x)=∫𝔹‍dv(x)[X*·(ϕBEA,B)](x)ψ0(x)×∫𝔹‍XyG(y,x)·Xψ(y)dv(y)=∫𝔹‍𝒜A,B·(Xψ)dv,
where we have set
(116)𝒜A,B(y)=∫𝔹‍[X*·(ϕBEA,B)](x)ψ0(x)XyG(y,x)dv(x).
We wish to prove an estimate on |𝒜(y)|, where 𝒜=𝒜A,B for simplicity. As it is well known, |BX(x,r)|=Cr2n+2 for some constant C>0 and any x∈ℍn and r>0. Here |A| denotes the Lebesgue measure of the set A. In particular the Lebesgue measure on (ℍn,dX) has the doubling property. Thus we may apply a result by Macìas and Segovia, [15], to pick a Whitney decomposition of Uy=U∖{y}. Precisely let y∈𝔹, and given x∈Uy, let us set rx=dX(x,ℍn∖Uy)/1000. Next let us choose among {BX(x,rx)}x∈Uy a maximal family of mutually disjoint balls {BX(xα,rα)}α∈I. Then Uy=⋃α∈I‍BX(xα,3rα) (the Whitney decomposition of Uy) and there is N≥1 such that each x∈U belongs to at most N balls BX(xα,6rα). Moreover, again by a result in [15], we may associate a partition of unity to the Whitney decomposition of Uy; that is, we may consider a family of smooth functions {θα}α∈I such that 0≤θα≤1, ∑α∈I‍θα=1 on Uy, Supp(θα)⊂𝔹α=BX(xα,6rα), and |Xθα|≤C/rα. The bounds on the gradients actually follow from the work by Citti et al., [34], quoted above. Then
(117)𝒜a(y)=∑α∈I‍∫𝔹α‍[X*·(ϕBEA,B)](x)ψ0(x)×θα(x)Xa,yG(y,x)dv(x)=∑α∈I‍∫𝔹α‍[X*·(ϕB-(ϕB)𝔹α)EA,B](x)ψ0(x)×θα(x)Xa,yG(y,x)dv(x)+∑α∈I‍(ϕB)𝔹α∫𝔹α‍(X*·EA,B)(x)ψ0(x)×θα(x)Xa,yG(y,x)dv(x)=𝒜a′(y)+𝒜a′′(y).
The presence of term 𝒜a′′(y) represents of course the main difference with respect to the proof of the so called duality inequality in [7] (there X*·EA,B=0). Integrating by parts,(118)𝒜a′(y)=∑α∈I‍∫𝔹α‍(ϕB(x)-(ϕB)𝔹α)=∑α∈I‍∫𝔹α‍×EA,B(x)·Xx[ψ0(x)θα(x)Xa,yG(y,x)]dv(x).Due to the explicit form of the fundamental solution G(x,y), one may easily check that
(119)|XaG(x,y)|≤CdX(x,y)-2n-1,(120)|XaXbG(x,y)|≤CdX(x,y)-2n-2,
for any x,y∈U. Here it is irrelevant whether differentiation is performed in x or y. Estimates of the sort in the case of an arbitrary Hörmander system of vector fields have been obtained by Sánchez-Calle [35]. Estimates on G(x,y) itself are available, yet only estimates on the derivatives are needed for the following calculations. Using (119)-(120) and
(121)|Xψ0(x)|≤CdX(x,y)-1,|θα(x)|≤CdX(x,y)-1,α∈I,
one has
(122)|Xb,x[ψ0(x)θα(x)Xa,yG(y,x)]|≤CdX(x,y)-2n-2,
hence
(123)|𝒜α′(y)|≤C∑α∈I‍∫Γα‍|ϕB(x)-(ϕB)𝔹α||EA,B(x)|dX(x,y)2n+2dv(x),
where Γα=Supp(θα). Let x∈𝔹α=BX(xα,6rα). As y∈ℍn∖Uy, the very definition of rα yields dX(y,xα)≥1000rα; hence
(124)1000rα≤dX(y,xα)≤dX(y,x)+dX(x,xα)≤dX(x,y)+6rα
and in particular 6rα≤dX(x,y). Thus |𝔹α|=Crα2n+2≤C′dX(x,y)2n+2, where C′=C6-2n-2; hence there is a constant C>0 such that
(125)dX(x,y)2n+2≥C|𝔹α|,x∈𝔹α.
Let us set J={α∈I:Γα∩4𝔹≠∅}. Let us apply (123) and (125) and Hölder's inequality to perform the estimates
(126)|𝒜a′(y)|≤C∑α∈J‍1|𝔹α|∫𝔹α‍|ϕB(x)-(ϕB)𝔹α||EA,B(x)|dv(x)≤C∑α∈J‍(1|𝔹α|∫𝔹α‍|ϕB-(ϕB)𝔹α|D2dv)1/D2×(1|𝔹α|∫𝔹α‍|EA,B|D2/(D2-1)dv)(D2-1)/D2,
where we have set D=2n+2 for simplicity. By (90) in Lemma 18 and Q≤|Xϕ|2, one has |EA,B|≤26|Xϕ|D-1; hence
(127)(1|𝔹α|∫𝔹α‍|EA,B|D2/(D2-1)dv)(D2-1)/D2≤C(1|𝔹α||Xϕ|D2/(D+1)dv)(D2-1)/D2.
At this point we need to apply a version of the Sobolev inequality due to Franchi et al. [36]. Precisely, for any 1≤p<2n+2 there is a constant C>0 such that for any ball BX(x,r) with x∈U and 0<r≤diam(U)(128)(1|BX(x,r)|∫BX(x,r)‍|u-uBX(x,r)|p*dv)1/p*≤Cr(1|BX(x,r)|∫BX(x,r)‍|Xu|pdv)1/p,p*=2(n+1)p2n+2-p.
By the assumption in Theorem 20 one has XaϕB∈L2n+2(U); hence XaϕB∈Lν(U) for any 0<ν≤2n+2. Therefore (by the Sobolev inequality above)
(129)(1|𝔹α||ϕB-(ϕB)𝔹α|D2dv)1/D2≤Crα(1|𝔹α|∫𝔹α‍|Xϕ|d2/(D+1)dv)(D+1)/D2.
Collecting the information in (127) and (129),
(130)|𝒜a′(y)|≤C∑α∈J‍rα(1|𝔹α|∫𝔹α‍|Xϕ|D2/(D+1)dv)(D+1)/D.
In the sequel we write briefly a≈b whenever a/C≤b≤Ca for some constant C≥1. Let α∈J. If there is k∈ℤ such that xα∈BX(y,2k-1)∖BX(y,2k-2), then rα≈2k and 𝔹α⊂BX(y,2k) (our arguments follow closely those in [7, page 356]). Moreover
(131)|𝔹α||BX(y,2k)|=(6rα2k)2n+2,
hence |𝔹α|≈|BX(y,2k)|. Consequently
(132)rα(1|𝔹α|∫𝔹α‍|Xϕ|D2/(D+1)dv)(D+1)/D≤C2k(1|BX(y,2k)|∫BX(y,2k)‍|Xϕ|D2/(D+1)dv)(D+1)/D.
Also {α∈J:xα∈BX(y,2k-1)∖BX(xα,2k-2)}=∅ whenever 2k-2≥diam(8𝔹) and the estimate (130) may be written as(133)|𝒜a′(y)|≤C×∑2k≤4diam(8𝔹)‍2k(1|BX(y,2k)|∫BX(y,2k)‍|Xϕ|D2/(D+1)dv)(D+1)/D.Next we shall express the estimate on |𝒜a′(y)| in terms of Riesz potentials and then use the general estimates on Lp norms of Riesz potentials as obtained by Hàjlasz and Koskela [37]. To recall the needed result, let (X,ρ) be a metric space endowed with a Borel measure μ such that μ(B)>0 for any ball B⊂X. Let A⊂X be a bounded open set and let us consider the numbers q>0, σ≥1, and h>0.Definition 23.
An (abstract) Riesz potential operator Jh,qσ,A is given by
(134)(Jh,qσ,Ag)(x)=∑2k≤2σdiam(A)‍2kh(1|B(x,2k)|∫B(x,2k)‍|g(z)|qdμ(z))1/q.The estimate (133) implies
(135)|𝒜a′(y)|≤C(J1,q2,8𝔹|Xϕ|)(y),q=DD+1.
The needed result in [37] holds for an arbitrary metric space (X,ρ) endowed with a Borel measure μ such that μ(B)>0 for any ball B⊂X. Let A⊂X be a bounded open set such that μ is doubling on
(136)V={x∈X:dist(x,A)<2σdiam(A)}.
Let us assume that there are constants b>0 and D>0 such that
(137)μ(B(x,R))≥b(Rdiam(A))Dμ(A)
for any x∈A and any 0<R≤2σdiam(A). Moreover let h>0 and 0<q≤s<D/h. Then (cf. [37])
(138)∥Jh,qσ,Ag∥Ls*(A,μ)≤C(diam(A)μ(A)1/D)h∥g∥Ls(V,μ),
where s*=sD/(D-hs) and the constant C>0 depends only on h, σ, q, s, b, D, and the doubling constant. Then (by Hölder's inequality with 1/(2n+2)+1/D′=1, resp., with 1/μ+1/μ′=1)
(139)|∫𝔹‍[X*·(ϕBEA,B)]ψdv|≤∑a=12n‍∥Xψ∥L2n+2(𝔹)×(∫𝔹‍|𝒜a′(y)|2(n+1)/(2n+1)dv(y))(2n+1)/[2(n+1)]+∑a‍∥Xψ∥Lμ(𝔹)(∫𝔹‍|𝒜a′′(y)|μ′dv(y))1/μ′
with 1<μ<D to be determined later on. At this point we need an estimate on |𝒜a′′(y)|. By (80) in Proposition 16 if ϕ:ℍn→S2m-1 is a contact (2n+2)-harmonic map obeying to our assumptions (96), then
(140)|X*·EA,B|≤2Q(ϕ)(D-2)/2|Xϕ||Lϕ|≤|2cXϕ|D-1+δ,
hence (by (119))
(141)|𝒜a′′(y)|≤∑α∈I‍∫𝔹α‍|(X*·EA,B)(x)||ψ0(x)|×|θα(x)||Xa,yG(y,x)|dv(x)≤C∑α∈I‍∫Γα‍|Xϕ|νdX(x,y)2n+1dv(x),
where ν=D-1+δ and 0<ν<D. By dX(x,y)2n+1≥C|𝔹α|/rα for any x∈𝔹α one obtains
(142)|𝒜a′′(y)|≤C∑α∈J‍rα1|𝔹α|∫𝔹α‍|Xϕ|νdv≤C∑α∈J‍rα|𝔹α|(∫𝔹α‍|Xϕ|D)ν/D|𝔹α|(D-ν)/D,
that is,
(143)|𝒜a′′(y)|≤C∑2k≤4diam(8𝔹)‍2k(1|BX(y,2k)|∫BX(y,2k)‍|Xϕ|Ddv)ν/D,
hence
(144)|𝒜a′′(y)|≤C(J1,D/ν2,8𝔹|Xϕ|ν)(y).
Therefore (by (135) and (144))
(145)(∫𝔹‍|𝒜a′|D/(D-1)dv)(D-1)/D≤C∥J1,D/(D+1)2,8𝔹|Xϕ|D∥LD/(D-1)(8𝔹)≤Cdiam(8𝔹)|8𝔹|1/D∥|Xϕ|D∥L1(V),
that is,
(146)(∫𝔹‍|𝒜a′|D/(D-1)dv)(D-1)/D≤C∥Xϕ∥LD(100𝔹)D,
where V={x∈ℍn:dist(x,8𝔹)≤4diam(8𝔹)}, respectively,
(147)(∫𝔹‍|𝒜a′′|μ/(μ-1)dv)(μ-1)/μ≤C∥J1,D/ν2,8𝔹|Xϕ|ν∥Lμ/(μ-1)(8𝔹)≤Cdiam(8𝔹)|8𝔹|1/D∥|Xϕ|ν∥Ls(V)≤C∥|Xϕ|ν∥Ls(100𝔹),
where
(148)0<Dν≤s<D,μμ-1=s*=sDD-s.
Therefore it must be that
(149)1<μ≤D2-δ,s=μD(D+1)μ-D.
On the other hand
(150)∥|Xϕ|ν∥Ls(100𝔹)=(∫100𝔹‍(|Xϕ|D)νμ/[(D+1)μ-D]dv)[(D+1)μ-D]/(μD)
and we may choose μ such that νμ/[(D+1)μ-D]=1; that is, μ=D/(2-δ). Consequently
(151)(D+1)μ-DμD=1-ϵ,ϵ=1-δD,∥|Xϕ|ν∥Ls(100𝔹)=(∫100𝔹‍|Xϕ|Ddv)1-ϵ,(∫𝔹‍|𝒜a′′|μ/(μ-1)dv)(μ-1)/μ≤C(∫100𝔹‍|Xϕ|Ddv)1-ϵ.
Also
(152)∥Xψ∥Lμ(𝔹)μ=∫𝔹‍|Xψ|μdv≤(∫𝔹‍|Xψ|μ(2-δ)dv)1/(2-δ)|𝔹|(1-δ)/(2-δ)≤C(∫𝔹‍|Xψ|Ddv)1/(2-δ),
that is, ∥Xψ∥Lμ(𝔹)≤C∥Xψ∥LD(𝔹). Summing up (by (139) and (146) and (151)),
(153)|∫𝔹‍[X*·(ϕBEA,B)]ψdv|≤∥Xψ∥LD(𝔹){∥Xϕ∥LD(100𝔹)D+∥Xϕ∥LD(100𝔹)(1-ϵ)D}
which is (83). Lemma 17 is proved.



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*Source: 197410-2013-03-25.xml*