On the Regularity of Weak Contact p-Harmonic Maps

The study of pseudoharmonic 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 pseudoconvexCRmanifold 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


Introduction
The study of pseudoharmonic 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 where  is a compact strictly pseudoconvex CR manifold of CR dimension , () = ‖() ,  ‖ 2 , V =  ∧ ()  , and  is a contact form on .Also  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, if  and  are two strictly pseudoconvex CR manifolds endowed, respectively, with contact forms  and , then the pseudohermitian analog of the notion of a harmonic morphism (among Riemannian manifolds) is quite obvious: one may consider continuous maps  :  →  such that the pullback V ∘  of any local solution V :   ⊆  → R to Δ   V = 0 in  satisfies Δ  (V ∘ ) = 0 in  =  −1 (  ) in distribution sense.Here Δ  and Δ   are the sublaplacians of (, ) and (, ), respectively.Unlike the situation in [2] (where the target manifold  is Riemannian and  pulls back local harmonic functions on  to distribution solutions of Δ   = 0) such  is not necessarily smooth (since it is unknown whether local coordinate systems (  ,    ) on  such that Δ      = 0 in   might be produced).To give another example, should one look for a pseudohermitian analog to the Fluglede-Ishihara theorem (cf.[3] when  is CR and  is Riemannian), one would face the lack of an Ishihara type lemma (cf.[11]) as it is unknown whether Δ   V = 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 associated to the variational principle  ∫ () /2 V = 0, when  is a Sasakian manifold.Indeed the (2−1)×(2−1) matrix ( 2 )   = −   +     has but rank 2 − 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  = H −1 is the Heisenberg group and  = (  ,  2−1 ) :  ⊆  → H −1 is a solution to (2), then   :  → R 2−2 is subject to yet  2−1 is an arbitrary function (cf.Section 3).For the more appealing case, where  = H  is the Heisenberg group and  =  2−1 is the sphere, (2) may be written as (cf.Proposition 15) which is indeed the form assumed by the Euler-Lagrange equations in [7], yet unlike the situation there  * ⋅  , ̸ = 0 in general (cf.Proposition 16 for the notations).Although  * ⋅  , has a quite explicit form (yielding-for a class of weak solutions  : H  →  2−1 which are close to being horizontal maps-simple estimates on  * ⋅  , ), 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 for some  > 0 and 0 <  < 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 Section 2 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 -harmonic maps.We show that weak contact (2 + 2)-maps  :  ⊂ H  →  2−1 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  = 2 + 2 stems from the facts that ∫  ‖() ,  ‖ 2+2 V is a CR invariant and 2 + 2 is the homogeneous dimension of H  .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.

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 (2 + 1)-dimensional  ∞ differentiable manifold , an almost CR structure is a complex subbundle  1,0 () ⊂ () ⊗ C of the complexified tangent bundle, of complex rank , such that  1,0 ()  ∩  0,1 ()  = (0) for any  ∈ .Here  0,1 () =  1,0 () and overbars indicate complex conjugates.The integer  is the CR dimension of the almost CR manifold (,  1,0 ()).Almost CR structures are a bundle theoretic recast of the tangential Cauchy-Riemann operator   :  ∞ (, C) →  ∞ ( 0,1 () * ) given by (  ) = () for any  ∈  ∞ (, C) and any  ∈  1,0 ().An almost CR structure is ( formally or Frobenius) integrable if [, ] ∈  ∞ (,  1,0 ()) for any ,  ∈  ∞ (,  1,0 ()) and any open set  ⊂ .The tangential C-R operator may be extended to arbitrary (0, )-forms on  and the resulting pseudocomplex   : Ω 0, () → Ω 0,+1 (),  ≥ 0, is a complex (i.e.,  2  = 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  and any real hypersurface (7) is a CR structure on .Here  1,0 () →  is the holomorphic tangent bundle over  (locally the span of {/  : 1 ≤  ≤ } for any local system of complex coordinates (  ) on ).Also  is the complex dimension of , and then the CR dimension of  is  =  − 1. Integrability of (7) follows from the Nijenhuis integrability of the complex structure on .A solution  to    = 0 (the tangential C-R equations) is a CR function on  and, in the context of real hypersurfaces carrying the induced CR structure (7), CR functions appear as traces on  of holomorphic functions defined on a neighborhood of  in .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 .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  into some complex manifold  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 Δ  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  1,0 () 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 (, , , ) on a given contact Riemannian manifold  compensates for the lack of integrability of  1,0 () = { −  :  ∈ Ker()} and specifically providing a generalization of the Tanaka-Webster connection to the nonintegrable context.
Let  be a (2 − 1)-dimensional  ∞ manifold ( ≥ 2).An almost contact structure on  is a synthetic object (, , ) consisting of a (1, 1)-tensor field , a vector field  ∈ X ∞ (), and a 1-form  ∈ Ω 1 () such that with respect to any local coordinate system (  , Associated metrics always exist (cf.[13]).A contact metric structure is an almost contact metric structure (, , , ) such that Ω = , where Ω ∈ Ω 2 () is the 2-form given by Ω  =      .Let  :  →  be a  ∞ map from a strictly pseudoconvex CR manifold  of CR dimension  into a contact Riemannian manifold (, , , , ).Let  be a contact form on  such that the Levi form   is positive definite.Let   = Ker() and let us consider the vector bundle valued form () ,  ∈ Γ ∞ ( * ⊗  −1   ) given by where for any relatively compact domain Ω ⊆ .Contact 2harmonic maps are called contact harmonic maps.

Weak Contact Harmonic Maps
Sections 3 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]).
Let  be a strictly pseudoconvex CR manifold and  a contact form on .Let {  : 1 ≤  ≤ 2} be a local  orthonormal frame of  defined on the open set  ⊆  and  *  the formal adjoint of   ; that is, where   = for any local orthonormal frame where   = /  .Thus (by ( 22)) div (() on .Then (23) follows from (21).
Therefore, in general one may not expect regularity for a given (weak) contact -harmonic map.
The identity (23) in Proposition 2 leads naturally to the notion of a weak solution to the contact -harmonic map system.Indeed we may establish the following.
Proof.Let us multiply ( 23) by a test function  ∈  ∞ 0 () and integrate by parts On the other hand (as both  and  are parallel with respect to ∇  ) where   ℓ are the coefficients of  ∇  with respect to (  ,    ).Therefore ( 35) may be written as and Lemma 4 is proved.
Let us consider the function spaces where    are understood as weak derivatives.If 1 ≤  < ∞, then  1,  () are separable Banach spaces with the norms The central concept of this section may be introduced as follows.Let {  : 1 ≤  ≤ 2} be a   -orthonormal frame of  defined on the open set  ⊆ .Let   ⊆  be an open set which is relatively compact in a larger coordinate neighborhood in .Definition 5. A map  :  →   is said to be weak contact harmonic if it is a weak solution to (34); that is,   ∈  Let  :  →   be a weak contact -harmonic map.By ( 14) for any  ∈  ∞ 0 ().We need to recall the following general result, due to Xu and Zuily [24].Let  = { 1 , . . .,   } be a Hörmander system on an open set  ⊆ R  ,  ≥ 2, and Ω ⊂ R  a domain such that  ⊃ Ω.Let   (, ) be a symmetric and positive definite matrix defined in in Ω is actually smooth.Let us assume that  is a domain such that  is contained in a coordinate neighborhood in .
By the result in [24] quoted above.
Example 8 (contact -harmonic maps into the sphere).Let  =  2−1 ⊂ R 2 and let  be the canonical Sasakian metric on for any 1 ≤  ≤ 2 − 1. 46) follows from (23) by computing the Christoffel symbols of  2−1 with respect to the local coordinate system According to [7] given a Hörmander system of vector fields {  } defined on an open set  ⊆ R  , one may adopt the following.Definition 10.A subelliptic -harmonic map is a  ∞ solution  :  → R 2 to the system (the formal adjoint of   in [7] is − *  under the conventions adopted in the present paper) A horizontal map is a smooth map  : One may define weak solutions  : H  →   to (54) by requiring that   ∈  1,  () for some 1 ≤  < ∞ and that (54) holds a.e. in .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 harmonic map  : H  →   is locally Hölder continuous provided that  ≥ 2 + 2.
The proof of Proposition 9 is to write (46) in the form (53).We need the following.Lemma 11.Let  be a strictly pseudoconvex CR manifold.A smooth map  :  →  2−1 is contact -harmonic if and only if for any 1 ≤  ≤ 2 − 1 and any local orthonormal frame {  : 1 ≤  ≤ 2} of .
By (14) if  :  →  2−1 is a horizontal map, then () = || 2 and one may readily check that (55) is equivalent to (53) for any 1 ≤  ≤ 2 − 1.Of course the component  2 will satisfy (53) as well (as a consequence of the constraint ∑ 2 =1  2  = 1).To prove Lemma 11, let us multiply (46) by a test function  ∈  ∞ 0 () and integrate over .The left-hand side of the resulting equation is where  = () (−2)/2 .Then (by ( 37)) which yields (55) because on the sphere Lemma 11 is proved.The notion of a weak contact harmonic map as introduced above is confined to maps  :  →  such that the target contact Riemannian manifold  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  into some Euclidean space R  and produce an alternative first variation formula (cf.Theorem 2.22 in [26, page 139]) depending however on the embedding  → R  .
A generalization of Nash's embedding theorem to the context of contact Riemannian geometry has been obtained by D' Ambra [27].Let H  ≈ C  × R be the Heisenberg group equipped with the standard Sasakian structure ( 0 ,  0 ,  0 ,  0 ).Let (, (, , , )) be a contact Riemannian manifold.By a result in [27], if  is compact and  ≥ dim() + 1, there is a  1 -embedding  :  → H  which is both horizontal, that is,  *   ⊂  −1 Ker( 0 ), and isometric in the sense that  preserves the Levi forms Any contact Riemannian manifold  is in particular a sub-Riemannian manifold (in the sense of [28]); hence  carries the Carnot-Carathéodory metric   :  ×  → [0, +∞) associated to the sub-Riemannian structure (  , ).
In particular  is an isometry among the metric spaces (,   ) and (H  ,   ) (cf.Section 7 for the definition of the distance function   : As H  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   and the linear structure of the ambient space H  to reformulate on  Definition 2.1 in [29, page 280]) and we conjecture that the arguments in [29] may be recovered to study the equation Δ   = 0 on a strictly pseudoconvex CR manifold (the theory in [29] only deals with second order degenerate elliptic equations on domains in R  ).Unfortunately the existence of  1 -embeddings of given contact structures is not sufficient for differential geometric purposes, as long as Gauss and Weingarten formulae (which require two derivatives of ) are involved.The problem of improving D' Ambra's proof (to get a horizontal embedding of class at least  2 ) is open.
The diameter of Ω is meant with respect to the Carnot-Carathéodory metric associated to .Hajłasz and Strzelecki [7] studied local properties of weak solutions to the system (53).Their main finding is that every weak subelliptic harmonic map  ∈  1,  (Ω,  ] ) (i.e., every weak solution to (53) with  = ) is locally Hölder continuous.Maps  : Ω →  ] with values in a unit sphere  ] ⊂ R ]+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  : H  →  2−1 which are "close to horizontal" in a sense to be made precise in the sequel.
The identity ( 79) is a consequence of the constraint alone.The identity (80) for  =  and  =  follows from (74) (interchange  and  in (74) and subtract the resulting identity from (74)).In general, for any  ∈  ∞ 0 () hence (by (69)) Now let us interchange  and  in (82) to produce another identity of the sort and subtract it from (82).This yields (80).Proposition 16 is proved.
Although regularity of contact -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.

Next let us set
Throughout if (, ) is a measurable space and  ⊂  a measurable set with () > 0, we adopt the notation   = (1/()) ∫  .Let us take the dot product of (79) with  * , multiply the resulting equation by   , integrate over 2B, and sum over  ( The first line of (85) may be computed as follows: In the following estimates  denotes some positive constant, not necessarily the same in all formulae.By Hölder's inequality V) by the Poincaré inequality and by || ≤ /.Let us observe that () ≤ || 2 yields (∫ 2B\B () (−2)/2(−1)         /(−1) ) Hence (by (91)) Let us set   () = ∫   (,) ||  V.Also let us restrict our considerations to maps  : H  →  2−1 for which one may control () from below.We adopt the following.
If  : H  →  2−1 is close to horizontal, then (by (96)) Our main result in this section is the following.
As a consequence of Theorem 20 (by applying a version of the Dirichlet growth theorem due to Macìas and Segovia [15]).