Journal of Complex Analysis

Volume 2013 (2013), Article ID 197410, 17 pages

http://dx.doi.org/10.1155/2013/197410

Research Article

On the Regularity of Weak Contact -Harmonic Maps

Sorin Dragomir¹ and Robert Petit²

¹Università degli Studi della Basilicata, Dipartimento di Matematica, Informatica ed Economia, Via Dell'Ateneo Lucano 10, Contrada Macchia Romana, 85100 Potenza, Italy

²Laboratoire de Mathématiques Jean Leray, UMR 6629 CNRS, Université de Nantes, 2, rue de la Houssinière, BP 92208, 44322 Nantes, France

Received 11 September 2012; Accepted 26 January 2013

Academic Editor: Arcadii Grinshpan

Copyright © 2013 Sorin Dragomir and Robert Petit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We prove Caccioppoli type estimates and consequently establish local Hölder continuity for a class of weak contact -harmonic maps from the Heisenberg group into the sphere .

1. 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 ,  , and is a contact form on . Also is a contact Riemannian manifold and in particular an almost CR manifold (of CR codimension ).

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 of any local solution to in satisfies in in distribution sense. Here and 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 ) such is not necessarily smooth (since it is unknown whether local coordinate systems on such that 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 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

when is a Sasakian manifold. Indeed the matrix has but rank 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 is the Heisenberg group and is a solution to (2), then is subject to

yet is an arbitrary function (cf. Section 3). For the more appealing case, where is the Heisenberg group and 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 in general (cf. Proposition 16 for the notations). Although has a quite explicit form (yielding—for a class of weak solutions 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 and , 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 -maps 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 stems from the facts that is a CR invariant and is the homogeneous dimension of . 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 -dimensional differentiable manifold , an almost CR structure is a complex subbundle of the complexified tangent bundle, of complex rank , such that for any . Here and overbars indicate complex conjugates. The integer is the CR dimension of the almost CR manifold . Almost CR structures are a bundle theoretic recast of the tangential Cauchy-Riemann operator given by for any and any . An almost CR structure is (formally or Frobenius) integrable if for any and any open set . The tangential C-R operator may be extended to arbitrary -forms on and the resulting pseudocomplex , , is a complex (i.e., ) 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

is a CR structure on . Here is the holomorphic tangent bundle over (locally the span of for any local system of complex coordinates on ). Also is the complex dimension of , and then the CR dimension of is . Integrability of (7) follows from the Nijenhuis integrability of the complex structure on . A solution to (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 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 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 and specifically providing a generalization of the Tanaka-Webster connection to the nonintegrable context.

Given a CR manifold , let be the Levi, or maximally complex, distribution and , , its complex structure. Let , , be the conormal bundle associated to , a real line bundle over . Since is assumed to be connected and orientable, the conormal bundle is trivial. A globally defined nowhere zero section is a pseudohermitian structure on . For each pseudohermitian structure on the Levi form is

Two pseudohermitian structures are related by for some function . If this is the case, then . A CR manifold is nondegenerate (resp., strictly pseudoconvex) if is nondegenerate (resp., positive definite) for some . If is a nondegenerate CR manifold, of CR dimension , then each pseudohermitian structure is a contact form; that is, is a volume form on . If is nondegenerate and is a contact form on , there is a unique globally defined, nowhere zero, tangent vector field (the Reeb vector field of ()) such that and . The Webster metric is the semi-Riemannian metric on given by

for any . If is strictly pseudoconvex and is chosen such that is positive definite, then is a Riemannian metric on .

Let be a -dimensional manifold (). An almost contact structure on is a synthetic object consisting of a -tensor field , a vector field , and a -form such that

with respect to any local coordinate system on . A Riemannian metric on is associated, or compatible, to the almost contact structure (and is an almost contact metric structure on ) if

Associated metrics always exist (cf. [13]). A contact metric structure is an almost contact metric structure such that , where is the -form given by .

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 and let us consider the vector bundle valued form given by

where is the natural projection associated to the decomposition . Let and let be a local -orthonormal frame of defined on an open neighborhood of . We set

Note that

Definition 1. Let . A map is said to be contact -harmonic if is a critical point of the energy functional

for any relatively compact domain . Contact -harmonic maps are called contact harmonic maps.

Let be the Tanaka-Webster connection of that is the unique linear connection on obeying to (i) is -parallel (i.e., for any and any ), (ii) and , and (iii) the torsion tensor field of is pure (i.e., , for any and , where for any (cf. Theorem 1.3 and Definition 1.25 in [9, pages 25-26]). The vector valued -form is the pseudohermitian torsion of . Let be the generalized Tanaka-Webster connection of given locally by

(cf., e.g., [16]), where are the Christoffel symbols of . Covariant derivatives are meant with respect to the Levi-Civita connection of . For each we consider given by

Let be the connection induced by in the pullback bundle . We set

Let and let be a local -orthonormal frame of defined on an open neighborhood of . We define a section in by setting

where denotes the restriction of to . By a result in [10] the Euler-Lagrange equations associated to the variational principle are

here (cf., e.g., [13]). Also is the pseudohermitian torsion of ; that is, , and for any . are again the local coefficients of with respect to . In particular if is a Sasakian metric, then is contact -harmonic if and only if

3. 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 be a local -orthonormal frame of defined on the open set and the formal adjoint of ; that is,

where and . Also are the local coefficients of the Tanaka-Webster connection of with respect to the local coordinate system on . Clearly for any , where .

Proposition 2. Let be a smooth map and a Sasakian metric on . Then is contact -harmonic if and only if

for any local orthonormal frame of .

Proof. Let us note that , where . Thus (by (22))

on . Then (23) follows from (21).

Example 3 (contact -harmonic maps into the Heisenberg group). Let , , be the Heisenberg group (cf., e.g., [9, pages 11–14]). Let be the Cartesian coordinates on and let

Let be the -tensor field on determined by

where . Next the differential -form given by

is a contact form on ; that is, is a volume form. Let . Finally we shall need the Riemannian metric on given by on , on , and . Then is a Sasakian metric on (and actually is a Sasakian space form of -sectional ; cf., e.g., [13]). A calculation shows that

where and . Let and let be the span of over . Then is a strictly pseudoconvex CR structure on and is a contact form such that the Levi form is positive definite. Let be the Tanaka-Webster connection of . A calculation shows that

where and for simplicity. Hence

and the remaining connection coefficients vanish. The Webster metric of is given by

hence (by a straightforward calculation)

where and . Let us substitute (28)–(32) into (23) so that to obtain

Hence if is a contact -harmonic map, then is subject to (33) while is an arbitrary function. 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.

Lemma 4. A smooth map of a strictly pseudoconvex CR manifold into a Sasakian manifold is contact -harmonic if and only if

for any local orthonormal frame of on and any local coordinate system on such that .

Proof. Let us multiply (23) by a test function and integrate by parts

On the other hand (as both and are parallel with respect to )

where 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 , then are separable Banach spaces with the norms

Also is reflexive provided that . The central concept of this section may be introduced as follows. Let 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, for any and the identities (39) are satisfied for any test function .

Let be a weak contact -harmonic map. By (14)

on , hence

where . Then both integrals in (39) are convergent and the adopted definition is legitimate.

Example 6 (Example 3 continued). A weak solution to (33) is a map such that and

for any . We need to recall the following general result, due to Xu and Zuily [24]. Let be a Hörmander system on an open set , , and a domain such that . Let be a symmetric and positive definite matrix defined in . If for any , then any continuous solution to

in 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.

Proposition 7. For any weak solution to the contact -harmonic map system (33) if , then .

Of course in the particular case any distribution solution is (as the operator is hypoelliptic).

Example 8 (contact -harmonic maps into the sphere). Let and let be the canonical Sasakian metric on . Then a contact -harmonic map is a solution to

for any . Here and with , . Equation (46) follows from (23) by computing the Christoffel symbols of with respect to the local coordinate system

that is

so that

On the other hand (cf. [9])

so that

for any Sasakian metric . When , the identities (49)–(51) lead to

and then to (46) by taking into account that is an -structure on ; that is, . Our next purpose in this example is to prove the following result.

Proposition 9. Let be a horizontal map. Then is contact -harmonic if and only if is subelliptic -harmonic with respect to the canonical Hörmander system on .

According to [7] given a Hörmander system of vector fields defined on an open set , one may adopt the following.

Definition 10. A subelliptic -harmonic map is a solution to the system (the formal adjoint of in [7] is under the conventions adopted in the present paper)

such that .

A horizontal map is a smooth map such that

One may define weak solutions to (54) by requiring that for some 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 is locally Hölder continuous provided that .

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 is contact -harmonic if and only if

for any and any local orthonormal frame of .

By (14) if is a horizontal map, then and one may readily check that (55) is equivalent to (53) for any . Of course the component will satisfy (53) as well (as a consequence of the constraint ). To prove Lemma 11, let us multiply (46) by a test function and integrate over . The left-hand side of the resulting equation is

where . Then (by (37))

hence (46) implies

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 and produce an alternative first variation formula (cf. Theorem 2.22 in [26, page 139]) depending however on the embedding .

A generalization of Nash's embedding theorem to the context of contact Riemannian geometry has been obtained by D'Ambra [27]. Let be the Heisenberg group equipped with the standard Sasakian structure . Let be a contact Riemannian manifold. By a result in [27], if is compact and , there is a -embedding which is both horizontal, that is, , 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 associated to the sub-Riemannian structure . In particular is an isometry among the metric spaces and (cf. Section 7 for the definition of the distance function ). As 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 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 on a strictly pseudoconvex CR manifold (the theory in [29] only deals with second order degenerate elliptic equations on domains in ). Unfortunately the existence of -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 ) is open.

4. Contact Harmonic Maps into Spheres

Let be a bounded open set and a Hörmander system of vector fields such that . We recall (cf., e.g., [9, page 261]) the following.

Definition 12. A number is a homogeneous dimension relative to with respect to if there is a constant such that

for any Carnot-Carathéodory ball of center and radius and any Carnot-Carathéodory ball of center and radius .

The diameter of 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 (i.e., every weak solution to (53) with ) is locally Hölder continuous. Maps with values in a unit sphere 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 which are “close to horizontal” in a sense to be made precise in the sequel.

Let be the Heisenberg group equipped with the standard contact form . Let be a bounded domain. Let be the -orthonormal frame given by and , where as in Example 3. Clearly the coefficients of the 's lie in . We recall that an absolutely continuous curve is admissible if

for some functions such that .

Definition 13. The Carnot-Carathéodory distance among two points is the infimum of all for which there exists an admissible curve such that and . Balls with respect to are denoted by and referred to as Carnot-Carathéodory balls.

We shall characterize horizontal maps in terms of the first order differential operator

defined for .

Proposition 14. Let be a map such that for any . Then is a (weak) horizontal map if and only if for any .

Let be the natural complex coordinates on and set and . The following conventions are adopted as to the range of indices:

Let so that the pointwise restriction of to is a unit normal field on . Let be the complex structure on . Then given by for any is the Reeb vector field on . Here is the inclusion. With respect to the local chart in Example 8 the Reeb vector is given by

Then together with (48) in Example 8 leads to

Finally (66) implies that . Proposition 14 is proved. In particular may be written as

Our next task is to put (55) into a more tractable form.

Proposition 15. Let such that . Let us consider the functions

with . Let for any . Then is a contact -harmonic map if and only if

Here the dot product means . Using and (65) and (66), one obtains

Then substitution into (55) leads to

It remains to be shown that (71) and (72) imply

Let us multiply (71) by , where is an arbitrary test function, and integrate over so that to obtain (after integration by parts)

Similarly let us multiply (72) by so that to obtain

Let us contract the indices and in (74) (resp., and in (75)), add the resulting equations, and use the identities

We get

Let us use (a consequence of (67)). Finally

Now the identity (73) follows from (78) and . Proposition 15 is proved.

The crucial manner of exploiting the constraint is contained in the following.

Proposition 16. Let be a bounded domain and , , a map such that . Then

where one has set . Moreover if is a contact -harmonic map, then

where if , if , and the range of the indices in (80) is meant .

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

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.

Lemma 17. Let be a bounded domain. Let and such that for any . Let with be a Carnot-Carathéodory ball such that and let be a function of compact support. Then for any contact -harmonic map satisfying (96) for some and some

for some constant , where and .

This is similar to Lemma 3.2 (the duality inequality) in [7, page 354] and will be proved later on in this section.

Let and as in Lemma 17. Also let and and set and . Let be a test function such that , on , on , and for some constant . Next let us set

Throughout if is a measurable space and a measurable set with , we adopt the notation . Let us take the dot product of (79) with , multiply the resulting equation by , integrate over , and sum over

The first line of (85) may be computed as follows:

and summed over

by the very definition of (cf. Lemma 22) and by (67). Thus (85) becomes

For simplicity let and . Using (88), we may perform the estimates

Lemma 18. Let one set . Then a.e. in and consequently

a.e. in , for any .

The inequalities in Lemma 18 follow easily from and . Using (90), we may write (89) as

In the following estimates denotes some positive constant, not necessarily the same in all formulae. By Hölder's inequality

by the Poincaré inequality

and by . Let us observe that yields

Hence (by (91))

Let us set . Also let us restrict our considerations to maps for which one may control from below. We adopt the following.

Definition 19. A map is said to be close to a horizontal map if there exist constants and such that

If is close to horizontal, then (by (96))

Our main result in this section is the following.

Theorem 20. Let be a bounded domain in the Heisenberg group and , , the Lewy operators. Let and . Let be a map obeying to (96) for some and . If is a weak contact -harmonic map, then there exist constants , and such that

for any and any .

As a consequence of Theorem 20 (by applying a version of the Dirichlet growth theorem due to Macìas and Segovia [15]).

Corollary 21. Let be a bounded domain. Any weak contact -harmonic map satisfying (96) is locally Hölder continuous.

To prove Theorem 20, we use a hole filling technique essentially due to Widman [32], (cf. also Bensoussan et al. [31, page 38–40]). By (95) with and Lemma 17 with , we have

On the other hand, by the very definition of , we may use the Poincaré inequality to estimate

that is,

Using (97) and (101), the inequality (120) yields

By the Vitali absolute continuity of the integral , there is such that for any . As a consequence of (102) we may establish the following.

Lemma 22. There exist and such that

for any .

Proof. The proof is by contradiction. Let us assume that for any and any , there is such that , where is short for . Note that . Then (by (102))

Therefore

The inequality (105) leads to

Indeed, by the contradiction assumption, we may pick a sequence such that as and consider the corresponding radii . By passing to a subsequence, if necessary, one may assume that for some . Let in

and use the absolute continuity of the integral. Then either (yielding (106)) or and then , a contradiction. Finally (106) may be exploited as follows. Let . By the contradiction assumption there is such that (by (106))

and the last integral tends to as , a contradiction. Lemma 22 is proved.

Now we may prove the Caccioppoli type estimate (98). Let so that (103) may be written as

Then (by (109) and induction over )

for any , . Let us consider the family of intervals . It is a cover of , hence for each there is , , such that . Now the inequality implies (by (110))

On the other hand let us set (so that ) and observe that the inequality implies

that is, . One may choose from the very beginning such that for any . Note that (by the very definition of ). Then , hence , where . Theorem 20 is proved.

It remains that we prove Lemma 17. It suffices to prove the inequality (83) for any . Let us consider

where and is the Heisenberg norm of . By a classical result of Folland, [33], is a fundamental solution for the Hörmander operator . In particular for any bounded domain one has the representation formula

for any and any . By a result of Citti et al., [34], we may consider a smooth cut-off function such that on , on , and (the diameter is meant with respect to the Carnot-Carathéodory metric on ). Using (114) for , one may write

where we have set

We wish to prove an estimate on , where for simplicity. As it is well known, for some constant and any and . Here denotes the Lebesgue measure of the set . In particular the Lebesgue measure on has the doubling property. Thus we may apply a result by Macìas and Segovia, [15], to pick a Whitney decomposition of . Precisely let , and given , let us set . Next let us choose among a maximal family of mutually disjoint balls . Then (the Whitney decomposition of ) and there is such that each belongs to at most balls . Moreover, again by a result in [15], we may associate a partition of unity to the Whitney decomposition of ; that is, we may consider a family of smooth functions such that , on , , and . The bounds on the gradients actually follow from the work by Citti et al., [34], quoted above. Then

The presence of term represents of course the main difference with respect to the proof of the so called duality inequality in [7] (there ). Integrating by parts,

Due to the explicit form of the fundamental solution , one may easily check that

for any . Here it is irrelevant whether differentiation is performed in or . Estimates of the sort in the case of an arbitrary Hörmander system of vector fields have been obtained by Sánchez-Calle [35]. Estimates on itself are available, yet only estimates on the derivatives are needed for the following calculations. Using (119)-(120) and

one has

hence

where . Let . As , the very definition of yields ; hence

and in particular . Thus , where ; hence there is a constant such that

Let us set . Let us apply (123) and (125) and Hölder's inequality to perform the estimates

where we have set for simplicity. By (90) in Lemma 18 and , one has ; hence

At this point we need to apply a version of the Sobolev inequality due to Franchi et al. [36]. Precisely, for any there is a constant such that for any ball with and

By the assumption in Theorem 20 one has ; hence for any . Therefore (by the Sobolev inequality above)

Collecting the information in (127) and (129),

In the sequel we write briefly whenever for some constant . Let . If there is such that , then and (our arguments follow closely those in [7, page 356]). Moreover

hence . Consequently

Also whenever and the estimate (130) may be written as

Next we shall express the estimate on in terms of Riesz potentials and then use the general estimates on norms of Riesz potentials as obtained by Hàjlasz and Koskela [37]. To recall the needed result, let be a metric space endowed with a Borel measure such that for any ball . Let be a bounded open set and let us consider the numbers , , and .

Definition 23. An (abstract) Riesz potential operator is given by

The estimate (133) implies

The needed result in [37] holds for an arbitrary metric space endowed with a Borel measure such that for any ball . Let be a bounded open set such that is doubling on

Let us assume that there are constants and such that

for any and any . Moreover let and . Then (cf. [37])

where and the constant depends only on , , , , , , and the doubling constant. Then (by Hölder's inequality with , resp., with )

with to be determined later on. At this point we need an estimate on . By (80) in Proposition 16 if is a contact -harmonic map obeying to our assumptions (96), then

hence (by (119))

where and . By for any one obtains

that is,

hence

Therefore (by (135) and (144))

that is,

where , respectively,

where

Therefore it must be that

On the other hand

and we may choose such that ; that is, . Consequently

Also

that is, . Summing up (by (139) and (146) and (151)),

which is (83). Lemma 17 is proved.

Acknowledgments

S. Dragomir was a visiting professor of CNRS at Laboratoire de Mathématiques Jean Leray, Université de Nantes, France (June 2008), and expresses his gratitude for the excellent work conditions there. He is also grateful to G. Citti (University of Bologna) for discussions on Example 3 in this paper.

References

E. Barletta, S. Dragomir, and H. Urakawa, “Pseudoharmonic maps from nondegenerate CR manifolds to Riemannian manifolds,” Indiana University Mathematics Journal, vol. 50, no. 2, pp. 719–746, 2001.

E. Barletta, “Hörmander systems and harmonic morphisms,” Annali della Scuola Normale Superiore di Pisa-Scienze Fisiche e Matematiche V, vol. 2, no. 2, pp. 379–394, 2003.

E. Barletta, “Subelliptic F-harmonic maps,” Rivista di Matematica della Università di Parma, vol. 7, no. 2, pp. 33–50, 2003.

F. Hélein, Harmonic Maps, Conservation Laws and Moving Frames, vol. 150 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 2nd edition, 2002.

J. Jost and C. J. Xu, “Subelliptic harmonic maps,” Transactions of the American Mathematical Society, vol. 350, no. 11, pp. 4633–4649, 1998.

Z.-R. Zhou, “Uniqueness of subelliptic harmonic maps,” Annals of Global Analysis and Geometry, vol. 17, no. 6, pp. 581–594, 1999.

P. Hajłasz and P. Strzelecki, “Subelliptic p-harmonic maps into spheres and the ghost of Hardy spaces,” Mathematische Annalen, vol. 312, no. 2, pp. 341–362, 1998.

C. Wang, “Subelliptic harmonic maps from Carnot groups,” Calculus of Variations and Partial Differential Equations, vol. 18, no. 1, pp. 95–115, 2003.

S. Dragomir and G. Tomassini, Differential Geometry and Analysis in CR Manifolds, vol. 246 of Progress in Mathematics, Birkhäuser, Berlin, Germany, 2006.

S. Dragomir and R. Petit, “Contact harmonic maps,” Differential Geometry and Its Applications, vol. 30, pp. 65–84, 2012.

T. Ishihara, “A mapping of Riemannian manifolds which preserves harmonic functions,” Kyoto Journal of Mathematics, vol. 19, no. 2, pp. 215–229, 1979.

N. Garofalo and S. D. Pauls, The Bernstein Problem in the Heisenberg Group, 2002.

D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, vol. 203 of Progress in Mathematics, Birkhäuser, Berlin, Germany, 2002.

K. Yano, “On a structure defined by a tensor field f of type satisfying ,” North-Holland Mathematics Studies C, vol. 70, pp. 251–261, 1982.

R. A. Macías and C. Segovia, “A decomposition into atoms of distributions on spaces of homogeneous type,” Advances in Mathematics, vol. 33, no. 3, pp. 271–309, 1979.

S. Tanno, “Variational problems on contact Riemannian manifolds,” Transactions of the American Mathematical Society, vol. 314, no. 1, pp. 349–379, 1989.

A. Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex, Studies in Advanced Mathematics, CRC Press, London, UK, 1991.

S. M. Webster, “Pseudohermitian structures on a real hypersurface,” Journal of Differential Geometry, vol. 13, pp. 25–41, 1978.

N. Tanaka, A Differential Geometric Study on Strongly Pseudo-Convex Manifolds, Kinokuniya Book Store, Kyoto, Japan, 1975.

J. M. Lee, “The Feerman metric and pseudohermitian invariants,” Transactions of the American Mathematical Society, vol. 350, no. 11, pp. 4633–4649, 1998.

E. Barletta, S. Dragomir, and K. L. Duggal, Foliations in Cauchy-Riemann Geometry, vol. 140 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 2007.

L. Capogna and N. Garofalo, “Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Hörmander type,” Journal of the European Mathematical Society, vol. 5, no. 1, pp. 1–40, 2003.

Z.-R. Zhou, “A note on boundary regularity of subelliptic harmonic maps,” Kodai Mathematical Journal, vol. 28, no. 3, pp. 525–533, 2005.

C.-J. Xu and C. Zuily, “Higher interior regularity for quasilinear subelliptic systems,” Calculus of Variations and Partial Differential Equations, vol. 5, no. 4, pp. 323–343, 1997.

J. Nash, “The embedding problem for Riemannian manifolds,” Annals of Mathematics, vol. 63, no. 1, pp. 20–63, 1956.

H. Urakawa, Calculus of Variations and Harmonic Maps, vol. 132 of Translations of Math. Monographs, American Mathematical Society, Providence, RI, USA, 1993.

G. D'Ambra, “Nash C1-embedding theorem for Carnot-Caratheodory metrics,” Differential Geometry and its Applications, vol. 5, no. 2, pp. 105–119, 1995.

R. S. Strichartz, “Sub-Riemannian geometry,” Journal of Differential Geometry, vol. 24, pp. 221–263, 1986.

J. M. Bony, “Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés,” Annales de l'institut Fourier, vol. 19, no. 1, pp. 277–304, 1969.

F. Hélein, “Regularite des applications faiblement harmoniques entre une surface et une sphere,” Comptes Rendus de l'Académie des Sciences, vol. 312, pp. 591–596, 1991.

A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, vol. 151 of Applied Mathematical Sciences, Springer, Berlin, Germany, 2002.

K. O. Widman, “Holder continuity of solutions of elliptic equations,” Manuscripta Mathematica, vol. 5, no. 4, pp. 299–308, 1971.

G. B. Folland, “A fundamental solution for a subelliptic operator,” Bulletin of the American Mathematical Society, vol. 79, no. 2, pp. 373–376, 1973.

G. Citti, N. Garofalo, and E. Lanconelli, “Harnak's inequality for sum of squares plus potential,” American Journal of Mathematics, vol. 115, no. 3, pp. 699–734, 1993.

A. Sánchez-Calle, “Fundamental solutions and geometry of the sum of squares of vector fields,” Inventiones Mathematicae, vol. 78, no. 1, pp. 143–160, 1984.

B. Franchi, G. Lu, and R. L. Wheeden, “Representation formulas and weighted Poincaré inequalities for Hörmander vector fields,” Annales de l'institut Fourier, vol. 45, no. 2, pp. 577–604, 1995.

P. Hájlasz and P. Koskela, “Sobolev meets Poincaré,” Comptes Rendus de l'Académie des Sciences, vol. 320, pp. 1211–1215, 1995.