PARTIAL SCHAUDER ESTIMATES FOR A SUB-ELLIPTIC EQUATION?

Na WEIYongsheng JIANGYonghong WU

1.School of Statistic and Mathematics，Zhongnan University of Economics and Law，Wuhan 430073，China2.Department of Mathematics and Statistics，Curtin University of Technology，Perth，WA 6845，AustraliaE-mail:weina@znufe.edu.cn;jiangys@znufe.edu.cn;Y.Wu@curtin.edu.au

PARTIAL SCHAUDER ESTIMATES FOR A SUB-ELLIPTIC EQUATION?

1.School of Statistic and Mathematics，Zhongnan University of Economics and Law，Wuhan 430073，China
2.Department of Mathematics and Statistics，Curtin University of Technology，Perth，WA 6845，Australia
E-mail:weina@znufe.edu.cn;jiangys@znufe.edu.cn;Y.Wu@curtin.edu.au

In this paper，we establish the partial Schauder estimates for the Kohn Laplace equation in the Heisenberg group based on the mean value theorem，the Taylor formula and a priori estimates for the derivatives of the Newton potential.

partial Schauder estimates;Kohn Laplace equation;Heisenberg group

2010 MR Subject Classification35R05;35B45;35H20

1　Introduction

In the Euclidean space，Schauder estimates for elliptic and parabolic equations were well studied in［5，6，19，22］，etc.，which play an important role in the theory of partial differential equations.In brief，if u∈C2is a solution of Δu=f，then one can have the estimates for the modulus of D2u when f is H¨older continuous.The partial Schauder estimates for the solutions of elliptic equations in Euclidean spaces can be derived under incomplete H¨older continuity assumptions，see［9，11，20］.Here，the partial Schauder estimates means that the partial derivatives of the solution in some directions are H¨older continuous but fail in others.One of the motivations of this paper is to study the phenomenon about the partial Schauder estimates for sub-elliptic equations.

Some research was done for the Schauder estimates of the operators structured on the nonabelian vector fields.Capogna and Han［7］showed the pointwise Schauder estimates for the operatorin the Carnot group，wherespan the first layer of the Lie algebra of a Carnot group in RN.Then，Guti′errez and Lanconelli［18］considered the Schauder estimates for a class of more general sub-elliptic equationsX0by using the Taylor formula.Bramanti and Brandolini［3］gave the Schauder estimates forthe operators aijXiXj-?twith Xisatisfying H¨ormander’s rank condition，by making use of the properties of the fundamental solution of the frozen operator.Capogna showed the Cαregularity of Quasi-linear equations in the Heisenberg group（denoted by Hn）［4］，which is the simplest non-abelian nilpotent Lie group.The Kohn Laplace equation in the Heisenberg group is a classical sub-elliptic equaion.The second author and his collaborator derived the Schauder estimates to the Kohn Laplace equation

Analogously to the definition of Dini continuous in one direction［20］，we say that f is Dini continuous in

This definition of Dini continouous for a function in a plane can be regarded as an extension from the classical concept of Dini continouous.Indeed，since the space Hncan be spanned by a collection of planes，there exists a pointsuch thatHence，

In this paper，we assume that the solution is smooth，for exampleBy approximation，the following estimates hold for weak solutions.

Theorem 1.1Let u be a solution of（1.1）.If f is Dini continuous in a plane Pm，then

Here，we assume that f is Dini contiouous in the plane Pmonly.By（1.4），we see that the conclusions about Schauder estimates in［10］is a corollary of Theorem 1.1.The proof of Theorem 1.1 follows the same ideas as in［20］.In brief，we decompose the differenceas the sum of a Newton potential and a sequence of ΔHn-harmonic functions.Since the ΔHnharmonic function is sufficiently smooth，the main difficulties of this paper are to establish a reasonable decomposition ofand to study the smoothness of the Newton potential（see Lemma 3.1 below）.

The above estimates imply the partial Schauder estimates for the Kohn Laplace equation. To precisely describe the partial Schauder estimates for the Kohn Laplace operator，we need a precise definition as follows.

Definition 1.2（Partial H¨older space）Let Pmbe a plane inandbe a continuous function.We say that v is H¨older continuous in the plane Pmwith H¨older exponent α if

which means that v is H¨older continuous in Ω with exponent α，i.e.，

By Definition 1.2 and a simple calculation，we see that the termsin（1.5）and（1.6）can be controlled by the H¨older norm（1.7）with v=f（see（3.28）and（3.29）below）.Then we have the following partial Schauder estimate for the solution of（1.1）.

In this theorem，f is assumed to be H¨older continuous in a plane Pm（m=1，2，···，2n）only.But ZiZmu and ZmZiu are H¨older continuous in all variables for all i=1，2，···，2n. If f is H¨older continuous in Hn，then it is H¨older continuous in the plane Pmfor all m= 1，2，···，2n.By using Theorem 1.4 we get that ZiZju are H¨older continuous in all variables for i，j=1，2，···，2n，which is the classical regularity for sub-elliptic equations and hence extends the results in［10］.Moreover，if f is H¨older continuous in planes Pk，Pk+1，···，P2n，then ZiZju is H¨older continuous for all i，j≥k.Similar results for elliptic and parobolic equations in Euclidean space were derived in［9］by using the maximum principle and the Krylov-Safonov theory.We should address that the partial regularity for elliptic operators and parobolic operators were proved by Wang and Tian in［20］，where the commutativity of the gradient operators in Euclidean spaces helps getting the smooth estimations by mainly usingthe Maximum principle and a priori estimates for the derivatives of a harmonic function.The gradient operators in the Heisenberg group are non commutative，see（2.1）below.It seems impossible to get a harmonic function by differentiating an auxiliary equation as it is in［20］. So，it is difficult to get the smooth estimates by only using the Maximum principle and the property of harmonic functions.In this paper，we need more estimates to the Newton potential as it is in Lemma 3.1 below to overcome this difficulty.

The rest of the paper is organized as follows:Section 2 presents the notions of the Heisenberg group and some preliminary lemmas.Section 3 is devoted to the proof of Theorems 1.1 and 1.4.

2　Preliminaries

In this section，we introduce some notations.We denote the points of the Heisenberg groupThe group law on Hnis given by

A natural group of dilations on Hnis given byThe Jacobian determinant ofis called the homogeneous dimension of Hn. The operatorsare invariant with respect to the left translationsand homogeneous with respect to the dilations δλof degree one and of degree two，respectively.A remarkable analogy between the Kohn Laplacian and the classical Laplace operator，given in［12］，is that a fundamental solution ofwith pole at zero is given by

where cQis a suitable positive constant and

We first present a priori estimate of the derivatives of the ΔHn-harmonic function u，which plays an important role in the paper.

Lemma 2.1（see［21］，Proposition 2.1）Let Ω be an open subset of Hn，and let u solve，we have

Lemma 2.1 gives a classical estimate to the horizontal gradient of ΔHn-harmonic functions. In this paper，we also need the estimates to the vertical derivatives of ΔHn-harmonic functions as follows.

Lemma 2.2Under the same assumption of Lemma 2.1，we have

ProofBy using（2.1），we have

Then by applying Lemma 2.1 in（2.6），we get estimate（2.4）.Using a similar argument for（2.4）by replacing u with Zku，we derive（2.5）.

We also need the following Maximum principle for the solution u

Lemma 2.3Let Ω be a bounded open subset of，then u belongs toand one has the estimate

We can apply the mean value theorem in the homogeneous Carnot group［2］to our case to obtian the following result.

Lemma 2.4There exists a constant c1＞0，depending only on Hnand on the homogeneous norm d，such that for all

In order to prove our main result，we also need to use the Taylor formula in［1］，which is established in Hnand is given by the lemma below

3　Proof of the Main Results

In this section，we prove our main results（Therem 1.1）by using the perturbation argument established in［22］.To get the smooth estimate of the difference u（ξ）-u（η），we decompose it as the sum of a Newton potential and a sequence of ΔHn-harmonic functions.First，we derive the estimates for the Newton potential by using the ideas in［20，22］.

Lemma 3.1Let f（ξ）be an integrable function on B1and

ProofWithout loss of generality，letandBy the definition of d in（2.2），a simple calculatun implies that and hence，

Based on this observation，we prove（3.1）by the following two cases.

We see that

It follows from（3.4）and（3.5）that

Similarly we have

By applying（3.6），（3.7）in（3.3），we get（3.1）.LetThen proceeding as for（3.1），we have

As for derivating（3.3），estimate（3.8）helps us getting that

By using（3.7），（3.9）and the commutation relations of the left-invariant vector fields（2.1），we have

Then（3.2）is obtained by combining（3.9）and（3.10）.

Now，we give the proof of Theorem 1.1.The method is similar to［20］and［22］，but we have to use Lemma 3.1 instead of harmonic function to estimate the Newton potential due to the non-commutativity of the horizontal gradient operators on Hn.For convenience of the reader，we present it entirely here.

Proof of Theorem 1.1Without loss of generality，let m=n.For any given point η near the origin，we have

where ukis the solution of

By using Lemma 2.5 and（3.11），we have

Now we can estimate I3in a similar way.Let vlbe the solution of

Similar to（3.15），we have

By using Theorem 2.1 and Corollary 2.8 in［14］with，we get

be obtianed similarly as（3.11）.Therefore，

This together with Lemma 2.1 implies that

Combining（3.18）-（3.21），we have

Then，by（3.16），（3.17）and（3.22），we have

Next we estimate I1.Letby（3.12），Lemmas 2.1 and 2.4，we see that

Similarly to（3.22），we have

Finally，（1.5）can be obtained by combining（3.15），（3.23）and（3.24）.And（1.6）can be obtained by using（1.5）and the commutation relations of the left-invariant vector fields.Indeed，from（2.1）we have

whose last inequality is obtianed by applying the same argument for（1.5），since we have estimates（2.4），（2.5）and（3.7）.

Proof of Theorem 1.4Let r∈（0，1），by the defintion（1.3）we have a sequencein the set

This together with the definition of the partial H¨older norm（1.7）shows that

Then（1.8）follows from（1.5），（1.6），（3.28）and（3.29）.

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?April 28，2015;revised September 29，2015.This work was supported by the NSFC（11201486，11326153）.The

was supported by“the Fundamental Research Funds for the Central Universities（31541411213）”.

?Yongsheng JIANG.