ON THE REGULARITY OF p-harmonic FUNCTIONS IN THE HEISENBERG GROUP. by András Domokos Doctoral Degree, Babeş-Bolyai University, Romania, 1997

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1 ON THE REGULARITY OF p-harmonic FUNCTIONS IN THE HEISENBERG GROUP by András Domokos Doctoral Degree, Babeş-Bolyai University, Romania, 1997 Submitted to the Graduate Faculty of the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 004

2 UNIVERSITY OF PITTSBURGH DEPARTMENT OF MATHEMATICS This dissertation was presented by András Domokos It was defended on March, 004 and approved by Juan J. Manfredi, Department of Mathematics Frank H. Beatrous, Jr., Department of Mathematics Luis F. Chaparro, Electrical Engineering Department Thomas A. Metzger, Department of Mathematics Yibiao Pan, Department of Mathematics David Saunders, Department of Mathematics Dissertation Director: Juan J. Manfredi, Department of Mathematics ii

3 ON THE REGULARITY OF p-harmonic FUNCTIONS IN THE HEISENBERG GROUP András Domokos, PhD University of Pittsburgh, 004 In this thesis we first implement iteration methods for fractional difference quotients of weak solutions to the p-laplace equation in the Heisenberg group. We obtain that T u L p loc for 1 < p < 4, where u is a p-harmonic function. Then we give detailed proofs for HW, - regularity for p in the range p < 4 and HW,p -regularity in the case 17 1 p for ε-approximate p-harmonic functions in the Heisenberg group. These last estimates however are not uniform in ε. The method to prove uniform estimates is based on Cordes type estimates for subelliptic linear partial differential operators in non-divergence form with measurable coefficients in the Heisenberg group. In this way we establish interior HW, - regularity for p-harmonic functions in the Heisenberg group H n for p in an interval containing. We will also show that the C 1,α regularity is true for p in a neighborhood of. In the last chapter we extend our results to the more general case of Carnot groups. iii

4 Acknowledgements I wish to thank my advisor, Professor Juan J. Manfredi, for his continuous help and for the many hours spent talking on this topic. His encouragements during the inevitable ups and downs of my research work proved to be priceless. I wish to thank the members of the Dissertation Committee for their comments and suggestions. A special thanks to David Saunders for his careful reading of the manuscript. Also, I am indebted to the Department of Mathematics, for its support and hospitality. iv

5 TABLE OF CONTENTS 1.0 INTRODUCTION DIFFERENTIABILITY ALONG THE T-DIRECTION Preliminaries Fractional difference quotients Iterations in the T-direction for p Iterations in the T-direction for 1 < p < SECOND ORDER HORIZONTAL DIFFERENTIABILITY OF THE APPROXIMATING p-harmonic FUNCTIONS Case p Case 1 < p < CORDES CONDITIONS AND UNIFORM ESTIMATES IN THE HEISEN- BERG GROUP Bounding the second order horizontal derivatives by the subelliptic Laplacian Cordes conditions for second order subelliptic PDE operators in non-divergence forms with measurable coefficients HW, -interior regularity for p-harmonic functions in H n C 1,α -regularity for p-harmonic functions in the Heisenberg group for p near REGULARITY OF p-harmonic FUNCTIONS IN CARNOT GROUPS Basic facts about Lie groups Nilpotent Lie groups Differentiability along vector fields from the center of the Lie algebra The case: p < v

6 5.3. The case 1 < p < Differentiability along horizontal vector fields The case: p < ν ν The case 1 < p < in a Carnot group of step Cordes conditions in Carnot groups C 1,α regularity of p-harmonic functions for p close to The case p The case p in a Carnot group of step BIBLIOGRAPHY

7 1.0 INTRODUCTION The Heisenberg group plays an important role in several branches of mathematics such as representation theory, harmonic analysis, complex variables, partial differential equations and quantum mechanics. It can be constructed in many different ways, for example, as a group of unitary operators acting on L R n, or it can be identified with the group translations of the Siegel upper half space in C n, or it can be realized as a group of unitary operators generated by the exponentials of the position and momentum operators in quantum mechanics. In the Heisenberg groups we find an abstract form of the commutation relations for the quantum-mechanical position and momentum operators. The commutation relations will be present in the form of the noncommutatitvity of first order differential operators, more exactly of the horizontal left invariant vector fields. The number of the horizontal vector fields we use is n in a n + 1 dimensional space. The horizontal vector fields and their commutators span the tangent space at any point, so they form a completely nonholonomic or bracket-generating family. According to the Rashevsky-Chow theorem, we can connect any two points in the Heisenberg group using curves that have tangent vectors at each point in the subspace generated by the horizontal vector fields. This is a very important fact in control theory and has important consequences in the regularity of weak solutions of partial differential equations. The study of regularity of weak solutions is needed because it is difficult to find classical solutions that match real world situations. Therefore, we have to extend the search and first get solutions in a very general class of functions. After that one has to show that it has the required properties. Let us consider the Heisenberg group H n as R n R n R endowed with the group

8 multiplication x 1,..., x n, t y 1,..., y n, u = x 1 + y 1,..., x n + y n, t + u 1 n x n+i y i x i y n+i. With respect to this operation the neutral element is 1 = 0,..., 0 and the inverse is given by x 1,..., x n, t 1 = x 1,..., x n, t. The conjugation by x = x 1,..., x n, t is defined as Adx 1,..., x n, t : H n H n, n Adx 1,..., x n, ty 1,..., y n, s = y 1,..., y n, s x n+i y i x i y n+i. The tangent space at 1 and at the same time the Lie algebra of the Heisenberg group is R n+1, hence the differential of Adx 1,..., x n, t at 1 is Adx 1,..., x n, t = D 1 Adx 1,..., x n, t : R n+1 R n+1 i=1 i=1 given in matrix form by Adx 1,..., x n, t = x n+1... x n x 1... x n 1 Therefore we can consider the mapping Ad : R n+1 GLR n+1 and its differential at 0, ad = D 0,...,0,0 Ad : R n+1 LR n+1, R n+1 given by adx 1,..., X n, T = X n+1... X n X 1... X n 0 3

9 The Lie bracket or commutator of X, Y R n+1 is given by n [X, Y ] = adxy = 0,..., 0, X n+i Y i X i Y n+i. The left multiplication by x = x 1,..., x n, t is defined by L x : H n H n, i=1 L x y = x y = x 1 + y 1,..., x n + y n, t + u 1 n x n+i y i x i y n+i, i=1 and its differential at 1 is D 1 L x = x n x n 1 x x n 1 For each v = v 1,..., v n, s R n+1 corresponds a left invariant vector field X v given by X v x = D 1 L x v = = v v n + s 1 x 1 x n n x n+i v i x i v n+i t. Therefore, if i {1,..., n} and e i R n+1 is the vector with the i th component 1 and the others 0, we have the corresponding left invariant vector field For e n+i we have while for e n+1 = 0,..., 0, 1 we have X i x = X n+i x = i=1 x n+i x i t. + x i x n+i t, T x = t. 4

10 The commutators of the horizontal vector fields X i satisfy [X i, X n+i ] = T, otherwise [X i, X j ] = 0. Therefore the horizontal vector fields X i and their commutators span the tangent space of H n at each point and hence satisfy the Hörmander s condition of hypoellipticity. Let be a domain in H n and let p > 1. Recall that the Haar measure in H n is the Lebesque measure of R n+1, therefore the space L p is defined in the usual way. Consider the following Sobolev space with respect to the horizontal vector fields X i HW 1,p = { } u L p : X i u L p, for all i {1,..., n}. HW 1,p is a Banach space with respect to the norm u HW 1,p = u L p + n i=1 X i u L p. We denote by HW 1,p 0 the closure of C 0 in HW 1,p. We will also use the local Sobolev space } HW 1,p loc {u = : R : ηu HW 1,p, for all η C0. Consider the p-laplace equation: where Xu = X 1 u,..., X n u is the horizontal gradient of u. n i=1 X i Xu p X i u = 0, in A function u from the horizontal Sobolev space HW 1,p loc is called a p-harmonic function if it is a weak solution of equation 1.0.1, that is Xux p Xux, Xϕx dx = 0, for all ϕ HW 1,p Together with equation we will consider for ε > 0 small the approximating equations n i=1 ε X p i + Xu X i u = 0, in and their weak solutions u ε HW 1,p loc which we will call ε-approximate p-harmonic functions. 5

11 In the case p = the left hand side of equation is the Kohn-Hörmander Laplacian and the C -regularity of the weak solutions u and u ε follows from Hörmander s celebrated theorem [1]. In the case p the equation degenerates. In the classical Euclidean case we know that u ε C and u C 1,α loc, for 1 < p < and u Wloc for p close to. In the case of the Heisenberg group or in general in the subelliptic case there are no definite answers yet. We can mention the results from the papers of Capogna [, 3], Capogna and Garofalo [4] and Marchi [17, 18, 19]. In the papers [, 3, 4] the a priori assumption on the boundedness of the horizontal gradient allows the use of some aspects of linear theory like L spaces or fractional derivatives defined via Fourier transform to gain control on difference quotients and prove interior C regularity for the weak solutions of Due to the noncommutativity of the horizontal vector fields in the Heisenberg group, the first thing to be proved is the differentiability in the non-horizontal direction T. Under the boundedness condition of the horizontal gradient it is possible to prove for any p not just that T u ε L loc but T u ε HW 1, loc. This opens the way to the proof of u ε HW, loc and then differentiating equation we can prove C -regularity. In the general case proving T u L p loc is more difficult. Marchi [17, 18, 19] proved this for < p < She used the fractional difference quotients to show that a weak solution is in some truncated versions of fractional Besov and Bessel-potential spaces. Marchi used the embedding among these spaces see [1, 3, 4, 5] to obtain more information on the differentiability of weak solutions. It is clear that the way we manage the fractional difference quotients constitutes a key point in the further development of this theory. We propose a direct method to bound the first order difference quotients. Using the semi-group properties hidden in the second order difference quotients we will be able to control the first order fractional difference quotients and hence to get a complete nonlinear treatment of the regularity problems. Among our main contributions are Lemma..1 and the implementation of several iteration schemes on fractional difference quotients. The point here is that using an appropriate test function, and exploiting the geometry of vector fields in the Heisenberg group described by the Baker- Campbell-Hausdorff formula, we get information on the second order difference quotients. 6

12 Using Lemma..1 we transfer this information to the first order difference quotients and do our iterations. In this way first we will extend Marchi s results by proving that T u L p loc for 1 < p < 4. Our method can be used also to give a new proof of T u HW 1, loc for 1 < p < under the boundedness assumption of the papers [, 3, 4]. Once we have the differentiability in the T direction we can prove second order differentiability in the horizontal directions. We do modified, and at the same time relatively simple versions of Marchi s proofs, that are independent of the embedding properties of Besov and Bessel-potential spaces. We remark that our HW, estimates for p < 4 and the HW,p estimates for 17 1 p are essential to be able to differentiate equation 1.1 and use the Cordes conditions in order prove uniform HW, bounds, which leads to interior HW, and C 1,α regularity of p-harmonic functions in intervals that contain p = and depend on n. Here is the plan of this thesis. In the next chapter we prove that T u L p loc for 1 < p < 4. Our main contributions are Lemma..1 and the implementation of several iteration schemes in the T-direction. Lemma..1 presents a direct proof based on a classical argument of A. Zygmund used for Hölder-Zygmund spaces of one variable functions [30]. In the third chapter we prove HW, estimates for p < 4 and the HW,p estimates for 17 1 < p of the ɛ-approximate p-harmonic functions. In the fourth chapter we use the Cordes condition [5, 8] and Strichartz s spectral analysis [7] to establish HW, estimates for linear subelliptic partial differential operators with measurable coefficients. As an application we obtain uniform HW, bounds for the ε- approximate p-harmonic functions for p in a range that depends on the dimension of the Heisenberg group H n. Using a stronger version of Cordes condition we prove C 1,α regularity of the p-harmonic functions for p close to. In the last chapter we extend the results from the previous chapters to the case of Carnot groups of an arbitrary step. 7

13 .0 DIFFERENTIABILITY ALONG THE T-DIRECTION.1 PRELIMINARIES In this section we introduce the first and second order difference quotients and state the first results involving them. In the next section we prove the lemma about the connection between second order and first order fractional difference quotients. The third section is devoted to the iteration scheme in the T-direction for p < 4, while in the fourth section we discuss the case 1 < p <. where Let us rewrite equation in the following way n i=1 X i a i Xu = 0, in.1.1 a i ξ = ξ p ξ i, for all ξ R n. A p-harmonic function u HW 1,p loc is a weak solution of equation.1.1, i.e. n i=1 a i Xux X i ϕxdx = 0, for all ϕ HW 1,p For ε > 0 small the ε-approximating equation to.1.1 is n i=1 X i a ε i Xu = 0, in.1.3 where a ε i ξ = ε + ξ p ξ i, for all ξ R n. We will use the following properties of the functions a i and a ε i : 8

14 i There exists a constant c > 0 such that and c ξ p q n i,j=1 c ε + ξ p q ii there exists a constant c > 0 such that and a i ξ ξ j q i q j, for all ξ, q R n.1.4 n i,j=1 If Z is a left invariant vector field then for some a ε i ξ ξ j q i q j, for all ξ, q R n..1.5 a i ξ ξ j c ξ p, for all ξ R n.1.6 a ε i ξ ξ j c ε + ξ p, for all ξ R n..1.7 z = z H, z T = z 1,..., z n, z T we can write Z = n i=1 z i X i + z T T. The exponential mapping in canonical coordinates is defined by e Z = z. In particular, e X 1 = 1, 0,..., 0, 0,..., e X n = 0, 0,..., 1, 0, and e T = 0, 0,..., 0, 1. Recall that in the Heisenberg group the Baker-Campbell-Hausdorff formula for two left invariant vector fields Z = n i=1 z ix i + z T T and V = n i=1 v ix i + v T T is e Z e V = e Z+V + 1 [Z,V ] = z v. 9

15 Let H n be a bounded domain. For x, a left invariant vector field Z, s R sufficiently small, 0 < α, θ 1, and u : R let us define: Then Z,s ux = ux e sz ux, Z,sux = ux e sz + ux e sz ux, D Z,s,θ ux = ux esz ux s θ, D Z, s,θ ux = ux e sz ux s θ. D Z, s,α D Z,s,θ ux = D Z,s,θ D Z, s,α ux = ux esz + ux e sz ux s α+θ = Z,s ux s α+θ. We will use the following result [, 1]: Proposition.1.1. Let H n be an open set, K a compact set included in, Z a left invariant vector field and u L p loc. If there exist σ and C two positive constants such that then Zu L p K and Zu L p K C. sup D Z,s,1 ux p dx C p 0< s <σ K Conversely, if Zu L p K then for some σ > 0 p sup D Z,s,1 ux p dx Zu L K p. 0< s <σ K The following result is a direct consequence of the Baker-Campbell-Hausdorff formula see [, 1]. We will use the notation s = 0,..., 0, s and D s,α ux = D T,s,α ux. Proposition.1.. Let H n be an open set, 1 p <, u HW 1,p loc, x 0 and r > 0 such that Bx 0, 3r. Then there exists a positive constant c independent of u such that Bx 0,r D s, 1 ux p dx c Bx 0,r u p + Xu p dx

16 Remark.1.1. Let us observe that if g is a cut-off function between Bx 0, r and Bx 0, r then Bx 0,r D s, 1 ux p dx Bx 0,r D s, 1 g ux p dx c Bx 0,r u p + Xu p dx FRACTIONAL DIFFERENCE QUOTIENTS In this section we will prove a lemma that will help us handle the second order fractional difference quotients. The classical method is to use the interpolation properties or equivalent norms of Besov or Lipschitz spaces, and Bessel potential or Triebel-Lizorkinspaces. However, our approach requires a truncated version of these spaces. Rather than referring the reader to a modified version of Theorem.5.1 on page 189 [4], we present a direct proof based on a classical argument of A. Zygmund Theorem 3.4 [30]. Let us continue to denote by s = 0,..., 0, s R n+1. Although our lemma will be stated in R n+1 we will be able to use it in the Heisenberg group, because the group multiplication by s is just the addition in the last variable. Let us observe that a similar proof can be carried out if we replace the Euclidean space by a nilpotent stratified Lie group and the translations by the flow of a left invariant vector field. Let us recall our notations for the following lemma: s ux = ux + s ux sux = ux + s + ux s ux. Lemma..1. Let u L p R n+1, 0 < α, 0 < σ and M 0. Suppose that su L p sup M...1 0< s σ s α Then for all 0 < β min{1, α} if α 1 and for all 0 < β < 1 if α = 1 there exists c > 0 independent of u and 0 < σ σ such that sup 0< s σ su Lp s β c u L p + M α... 11

17 Proof. Using u L p R n+1 we have that s u L p R n+1 and s u L p u L p for all 0 < s σ. Let us denote gsx = ux + s ux. Condition..1 implies that u + s + u s u L p M s α. Without loss of generality we can work just with s > 0. Replacing s by s, denoting M = m α and then changing the variables x x + s in the integral gives u + s + u u + s M s α, Lp and hence Replacing s by s in formula..3 we get s gs g M s α...3 Lp s s g g M sα, Lp α and hence s s g g M s α 1 α...4 Lp Repeating this procedure we get Adding the above inequalities we get If 0 < α < 1 then s s n 1 g n g M s α 1 αn n 1 n L p s g s n g M s α n L p n 1 1 αk...6 k=0 s g s n g M s α 1 αn 1 n L p 1 α 1 M s α 1 αn 1 α 1 and hence s g 1 n L p u n L p + cm s α αn. 1

18 Consider now 0 < a < σ n N and s [ a, a] such that h = s n. Then fixed and s [ a, a]. For all h > 0 sufficiently small there exist gh L p 4h a u L p + cm h α. Dividing this last inequality by h α we get... If α = 1, then inequality..6 implies that s g s n g M s n...7 n L p Consider now h = s n in a similar way as for the previous case and observe that n = Olog h to get and hence we can use any β < 1 to get... If α > 1 then inequality..6 implies that gh L p h u L p + holog h,..8 s g s n g M s α n L p 1 1 α Therefore, we have s g 1 n u n L p + 1 M s α 1 n 1, 1 α and hence for h = s and s [ a, a] we obtain n gh L p 4h a u L p + h a M α aα...10 Now we can use β = 1 to get... Remark..1. Proposition.1.1 together with Lemma..1 implies that if u has compact support K and..1 is satisfied with α > 1, then T u L p K. 13

19 .3 ITERATIONS IN THE T-DIRECTION FOR P. We prove a general lemma, that constitutes the key step in our iteration. In an informal way, we can say that if u ε has locally 1 + α derivatives in the T direction, then it also has α derivatives in the same direction. p p Lemma.3.1. Let u ε HW 1,p loc be a weak solution of.1.3, x 0, r > 0 such that Bx 0, 3r. Let us suppose that there exists constants c > 0, σ > 0 and α [0, 1 such that sup 0 s σ 1 D s, Bx 0,r +αu ε p dx c Bx 0,r ε + Xuε x p + u ε x p dx..3.1 If we have 1 + α p then for possibly different c > 0, σ > 0 holds < 1 sup 0 s σ In the case we have that Otherwise, 1 D s, Bx 0, r and we have that + 1 p + αu ε p dx p T u ε x p dx c Bx 0, r T u ε x p dx c Bx 0, r 4 ε c + Xuε x p + u ε x p dx..3. Bx 0,r Bx 0,r Bx 0,r 1 + α p > 1 ε + Xuε x p + u ε x p dx α p = 1 ε + Xuε x p + u ε x p dx

20 Proof. Consider γ = 1 + α, and let g be a cut-off function between Bx 0, r and Bx 0, r. We use now the test function ϕ = D s,γ g D s,γ u ε.3.5 to get n i=1 a ε i Xu ε x X i D s,γ g D s,γ u ε x dx = 0 and from here, by the fact that X i commutes with D s,γ n i=1 D s,γ a ε i Xu ε x g x D s,γ X i u ε x dx + n i=1 and D s,γ, we obtain D s,γ a ε i Xu ε x D s,γ u ε x gx X i gx dx = We can use now similar arguments as in Marchi s proof [17, 19], involving the properties of the functions a ε i and Lemma 8.3 [11] to get Bx 0,r g x ε + Xu ε x + Xu ε x s p D s,γ Xu ε x dx c ε + Xuε x + Xu ε x s p D s,γ Xu ε x Bx 0,r D s,γ u ε x gx Xgx dx. Using the fact that p we get Bx 0,r g x ε + Xu ε x + Xu ε x s p D s,γ Xu ε x dx c ε + Xuε x + Xu ε x s p D s,γ u ε x Xgx dx..3.7 Bx 0,r Denoting by RHS the right hand side of.3.7 we have that ε RHS c + Xuε x + Xu ε x s p + D s,γ u ε x p dx. Bx 0,r 15

21 Using.3.1 we get that RHS c and therefore Bx 0,r g x From the inequality Bx 0,r ε + Xu ε x + Xu ε x s p ε + Xuε x p + u ε x p dx D s,γ Xu ε x dx c ε + Xuε x p + u ε x p dx..3.8 Bx 0,r s γ D s,γ Xu ε x ε + Xu ε x + Xu ε x s we get Since Bx 0,r g x s p γ D s,γ Xu ε x p dx c ε + Xuε x p + u ε x p dx. Bx 0,r D s,γ Xg u ε x = D s,γ Xg x u ε x s + Xg x D s,γ u ε x + D s,γ g x Xu ε x s + g x D s,γ Xu ε x it follows that Bx 0,r D s, γ p Xg u ε x p dx c Bx 0,r ε + Xuε x p + u ε x p dx..3.9 Let us denote the right hand side of.3.9 by M p. Using Proposition.1. we get g u ε x p dx M p Bx 0,r D s, 1 D s, γ p Therefore, for all s sufficiently small we have so there exists σ > 0 such that sg u ε L p H n s α p M, sup 0< s σ sg u ε L p H n s α p M

22 If it happens that then by Lemma..1 we get.3.. If we have 1 + α p 1 + α p < 1 > 1 then by Lemma..1 we have T u L p loc and estimate.3.3 is valid. In the remaining case and then using that α [0, 1 we get 1 + α p = 1 0 p 4 < 1 which gives p < 4. Lemma..1 implies that we can use.3.1 with α arbitrarily close to 1, in particular α > p, to get back.3.11 with 4 and then use the previous case. 1 + α p Proposition.1. implies that we can start with α 0 = 0 in the assumption.3.1 to get > 1 α 1 = 1 p in.3.. Now we can use α 1 in.3.1 to get α = 1 p + p α 1 such that estimate.3. is true. In general, if we already found α 1,..., α k, then we get α k+1 = 1 p + p α k = 1 p k p k 1 + k 1 p α k 1 1 = 1 k 1 i = 1 p p p Therefore, for a given p > the supremum for the numbers α k, k N is given by 1 p. i=0 1 p 1 p Hence, for p [, 4, after a number sufficiently large of k iterations, we get that α k 1 and this means that T u ε L p loc. k. 17

23 Remark.3.1. If we ask for α 1 then we get the inequality p p 4 0 that leads to Marchi s result p [, We can summarize our results from this section by the following theorem that extends the results of Marchi [17]. Theorem.3.1. If p < 4, then for any weak solution u ε T u ε L p loc with bounds locally independent of ε. of.1.3 we have that In the case p 4 our proof gives the following result. Theorem.3.. For p 4 and weak solutions u ε of.1.3 we have 1 D s, +α u ε p dx sup 0 s σ Bx 0, r k ε c + Xuε x p + u ε x p dx..3.1 Bx 0,r for c > 0 independent of ε, α less then, but arbitrarily close to 1, and a corresponding p number k of iterations..4 ITERATIONS IN THE T-DIRECTION FOR 1 < p <. Theorem.4.1. Let 1 < p < and u ε HW 1,p loc be a weak solution of.1.3. Then T u ε L p loc with bounds locally independent of ε. Proof. Let us consider arbitrary x 0, r > 0 such that Bx 0, 3r and let g be a cut off function between Bx 0, r and Bx 0, r. We can follow then the proof of Lemma.3.1 for α = 0 and γ = 1 until we get g x ε + Xu ε x + Xu ε x s p D s,γ Xu ε x dx Bx 0,r c ε + Xuε x + Xu ε x s p D s,γ Xu ε x Bx 0,r D s,γ u ε x gx Xgx dx

24 Let us denote by RHS the right hand side of.4.1. We will keep using γ instead of 1 to get a general iteration formula. Then RHS c ε + Xuε x + Xu s γ ε x s p Xu ε x s Xu ε x D s,γ u ε x dx Bx 0,r c ε + Xuε x + Xu s γ ε x s p Bx 0,r ε + Xu ε x + Xu ε x s 1 D s,γ u ε x dx = c ε + Xuε x + Xu s γ ε x s p 1 D s,γ u ε x dx Bx 0,r c ε + Xuε x + Xu s γ ε x s p 1 p p dx Bx 0,r D s,γ u ε x p dx Bx 0,r 1 p c ε + Xuε x + Xu s γ ε x s p 1 p p dx Bx 0,r 1 u ε p + Xu ε p p dx Bx 0,r c ε + Xuε x p + u s γ ε x p dx. Bx 0,r Therefore, Bx 0,r g x ε + Xu ε x + Xu ε x s p Xu ε x s Xu ε x dx c s γ Bx 0,r ε + Xuε x p + u ε x p dx

25 We need the following inequalities used initially in the Euclidean case see [16]. ε + Xuε x + Xu ε x s p ε + Xu ε x + Xu ε x s p 1 ε + Xuε x + Xu ε x s ε + Xu ε x + Xu ε x s p 1 ε + Xu ε x + Xu ε x s Xu ε x 3 ε + Xu ε x p + 3 ε + Xu ε x + Xu ε x s p 1 Xu ε x s Xu ε x We can suppose s 1 and then Bx 0,r 3 c g x ε + Xu ε x + Xu ε x s p dx g x ε + Xu ε x p dx + c ε + Xuε x p + u ε x p dx Bx 0,r ε + Xuε x p + u ε x p dx. Bx 0,r Bx 0,r Also, by Hölder s inequality we get g x Xu ε x s Xu ε x p dx Bx 0,r = Bx 0,r g x ε + Xu ε x + Xu ε x s p 1 Xu ε x s Xu ε x p g 4 p x ε + Xuε x + Xu ε x s 1 p p dx g x ε + Xu ε x + Xu ε x s p Bx 0,r Bx 0,r p 1 Xu ε x s Xu ε x dx g 4 p x ε + Xuε x + Xu ε x s p 1 p dx 0

26 c s γ ε + Xuε x p p + u ε x p dx Bx 0,r g x ε + Xu ε x + Xu ε x s p Bx 0,r 1 p dx c s p γ Bx 0,r ε + Xuε x p p + u ε x p dx Bx 0,r ε + Xuε x p 1 p + u ε x p dx c s p γ Bx 0,r ε + Xuε x p + u ε x p dx. Therefore, Bx 0,r g x D s, γ Xu εx p dx c Bx 0,r ε + Xuε x p + u ε x p dx. In the same way as we obtained inequality.3.9, we get γ D s, Xg u ε x p dx c ε + Xuε x p + u ε x p dx..4.3 Bx 0,r Proposition.1. implies that Bx 0,r D s, 1 D s, γ Bx 0,r g u ε x p dx c and this means for a sufficiently small σ sg u ε L sup p H n c 0< s σ s 1 + γ Bx 0,r Bx 0,r We started with γ = 1 therefore in.4.3 we have a power of 1 4 ε + Xuε x p + u ε x p dx,.4.4 ε + Xuε x p + u ε x p dx..4.5 for s while in.4.5 we have a power of 3. Using Lemma..1 we can do iterations to obtain after k steps and 4 corresponding cut off functions between Bx 0, r k and that Bx 0, r k 1 that D s, r k 1 Bx 0, k 1 Xg u ε x k+1 p dx c Bx 0,r ε + Xuε x p + u ε x p dx,.4.6 1

27 and sup 0< s σ s g u ε L p H n c s k+1 1 k+1 Bx 0,r ε + Xuε x p + u ε x p dx..4.7 Let us consider now k N such that 1 k 1 < p 1. Then for we have a = k 1 and b = k+1 1 k+1 k+1 ap 1 + b > 1. Let us consider now γ = ap 1 + b > 1 and return to.4.1 with a cut off function g between Bx 0, r and Bx k+1 0, r. Then k RHS c ε + Xuε x + Xu ε x s p Xu ε x s Xu ε x p Bx 0, r k Xu εx s Xu ε x p 1 s ap 1 D s,b u ε x dx Xu ε x s Xu ε x p 1 Bx 0, r k s ap 1 D s,b u ε x dx Xu ε x s Xu ε x p dx Bx 0, r k s ap p 1 p Bx 0, r k D s,b u ε x p dx 1 p c Bx 0,r ε + Xuε x p + u ε x p dx. Therefore, Bx 0, r k g x ε + Xuε x + Xu ε x s p Xu ε x s Xu ε x dx

28 c s γ Bx 0,r ε + Xuε x p + u ε x p dx..4.8 Doing a similar proof as we did starting from formula.4. we get that sup 0< s σ sg u ε L p H n s 1 +γ c Bx 0,r ε + Xuε x p + u ε x p dx..4.9 Using the fact that 1 + γ > 1, Lemma..1 implies now that ε T u ε x p dx c + Xuε x p + u ε x p dx.4.10 r Bx 0, k+1 Bx 0,r and therefore T u ε L p loc. 3

29 3.0 SECOND ORDER HORIZONTAL DIFFERENTIABILITY OF THE APPROXIMATING p-harmonic FUNCTIONS 3.1 CASE p In this section we prove the HW, regularity of the approximate p-harmonic functions u ε. As immediate consequences of the results from the previous section we can prove that: Proposition With estimates depending on ε we have the following two regularity properties. 1 For all p we have T u ε L loc. For p < 4 we have that also XT u ε L loc. Proof. For p < 4 we know that T u ε L p loc L loc. Theorem.3. implies that for all p 4, x 0 and r > 0 sufficiently small we can choose an α > 0, a cut off function g between Bx 0, r and Bx k+1 0, r and repeat the proof of Lemma.3.1 until we obtain for k γ = 1 + α and M p = we have From here we obtain ε p Bx 0, r k Proposition.1. implies that Bx 0, r k Bx 0,r ε + Xuε x p + u ε x p dx Bx 0, r k g x D s,γ Xu ε x dx cm p. D s,γ X g u ε x dx cε p M p + M D s, 1 D s,γ g u ε x dx cε p M p + M dx 4

30 and hence for some σ > 0 holds sg u ε L sup H n 0< s σ s 1+α Lemma..1 gives now that T u ε L loc. cε p 1 M p + M To prove the second part let us observe that in the case p < 4 we can start the proof with γ = 1 and get Bx 0, r k D s,1 X g u ε x dx cε p M p + M, 3.1. and hence by Proposition.1.1 we have T Xu ε = XT u ε L loc. Theorem Let p < 4 and u ε HW 1,p loc be a weak solution of.1.3. Also consider x 0, r > 0 such that Bx 0, 3r and let k be the number of iterations depending only on p. Then we have. r Bx 0, k+ ε + Xuε x p X u ε x p dx c ε + Xux + ux p dx. Bx 0,r Proof. For the proof we use a simplified version of Marchi s method [17] and use the extended range of p < 4 obtained in the previous chapter. For i 0 {1,..., n}, h > 0, let us denote h i0 = 0,..., h,...0, 0 H n with the h in the i 0 th place. We will use the notation D hi0 = D Xi0,h,1 and D hi0 = D Xi0, h,1 and the test function where g is a cut-off function between Bx 0, For i i 0 + n we have ϕ = D hi0 D hi0 g 4 u ε r r and Bx k+ 0,. k+1 X i D hi0 D hi0 g 4 u ε = D hi0 D hi0 X i g 4 u ε, 5

31 while for i = i 0 + n we have X i0 +n D hi0 D h i0 g 4 u ε x = D hi0 D h i0 X i0 +n g 4 u ε x T g 4 u ε x hi0 T g 4 u ε x h 1 i h 0. To see that formula is true it is enough to observe that X i0 +n g 4 u ε x hi0 = X i0 +n g 4 u ε x hi0 ht g 4 u ε x hi0 and X i0 +n g 4 u ε x h 1 i 0 = X i0 +n g 4 u ε x h 1 + ht g 4 u ε x h 1 i 0. i 0 Using the test function ϕ in the weak form of the equation.1.3 we get n i=1 a ε i Xuε x D hi0 D h i0 X i g 4 u ε x dx = = a ε i 0 +n Xuε x 1 h T g 4 u ε x hi0 T g 4 u ε x h 1 i 0 dx Therefore, n i=1 D hi0 a ε i Xuε x D hi0 X i g 4 u ε x dx = = a ε i 0 +n Xuε x D T hi0 g 4 u ε x + D hi0 T g 4 u ε x dx. 6

32 We use that D hi0 X i g 4 u ε x = Dhi0 4g 3 x X i gx u ε x + g 4 x X i u ε x = = 4D hi0 gx g x h i0 X i gx h i0 ux h i0 + 4gx D hi0 gx gx h i0 X i gx h i0 u ε x h i0 + 4g x D hi0 gx X i gx h i0 u ε x h i0 + 4g 3 x D hi0 X i gx ux h i0 + 4g 3 x X i gx D hi0 u ε x + D hi0 gx g 3 x h i0 X i u ε x h i0 + gx D hi0 gx g x h i0 X i u ε x h i0 + g x D hi0 gx gx h i0 X i u ε x h i0 + g 3 x D hi0 gx X i u ε x h i0 + g 4 x D hi0 X i u ε x Therefore, equation has the form n i=1 D hi0 a ε i Xuε x D hi0 X iu ε x g 4 x dx = L1 = D hi0 a ε i 0 +n Xuε x T g 4 u ε x dx + D hi0 a ε i 0 +n Xuε x T g 4 u ε x dx R1 n i=1 D hi0 a ε i Xuε x 4D hi0 gx g x h i0 X i gx h i0 ux h i0 dx R 7

33 n i=1 D hi0 a ε i Xuε x 4gx D hi0 gx gx h i 0 X i gx h i0 u ε x h i0 dx R3 n i=1 D hi0 a ε i Xuε x 4g x D hi0 gx X igx h i0 u ε x h i0 dx R4 n i=1 D hi0 a ε i Xuε x 4g 3 x D hi0 X igx ux h i0 dx R5 n i=1 D hi0 a ε i Xuε x 4g 3 x X i gx D hi0 u εx dx R6 n i=1 D hi0 a ε i Xuε x D hi0 gx g3 x h i0 X i u ε x h i0 dx R7 n i=1 D hi0 a ε i Xuε x gx D hi0 gx g x h i0 X i u ε x h i0 dx R8 n i=1 D hi0 a ε i Xuε x g x D hi0 gx gx h i 0 X i u ε x h i0 dx R9 8

34 n i=1 D hi0 a ε i Xuε x g 3 x D hi0 gx X iu ε x h i0 dx R10 We estimate now each of the above lines. We will use δ > 0 as a sufficiently small number. L1 c ε + Xuε x + Xu ε x h i0 p Dhi0 Xu ε x g 4 x dx. R1 c + c + c + c ε + Xuε x + Xu ε x h 1 i 0 p ε + Xuε x + Xu ε x h 1 i 0 p ε + Xuε x + Xu ε x h i0 p ε + Xuε x + Xu ε x h i0 p D hi0 Xu ε x g 4 x T u ε x dx D hi0 Xu ε x 4 g 3 x T gx u ε x dx Dhi0 Xu ε x g 4 x T u ε x dx Dhi0 Xu ε x 4 g 3 x T gx u ε x dx δ + cδ + cδ + δ + cδ + cδ ε + Xuε x + Xu ε x h 1 i 0 p D hi0 Xu ε x g 4 x dx ε + Xuε x + Xu ε x h 1 i 0 p g 4 x T u ε x dx ε + Xuε x + Xu ε x h 1 i 0 p g x T gx u ε x dx ε + Xuε x + Xu ε x h i0 p Dhi0 Xu ε x g 4 x dx ε + Xuε x + Xu ε x h i0 p g 4 x T u ε x dx ε + Xuε x + Xu ε x h i0 p g x T gx u ε x dx 9

35 R c ε + Xuε x + Xu ε x h i0 p + c ε + Xuε x + Xu ε x h i0 p h Dhi0 Xu ε x Dhi0 gx g x Xgx h i0 u ε x h i0 dx Dhi0 Xu ε x Dhi0 gx g x h i0 g x h Xgx h i 0 u ε x h i0 dx δ + cδ + c ε + Xuε x + Xu ε x h i0 p ε + Xuε x + Xu ε x h i0 p ε + Xuε x + Xu ε x h i0 p 1 Dhi0 Xu ε x g 4 x dx Dhi0 gx Xgx h i0 u ε x h i0 dx D hi0 gx Xgx h i0 u ε x h i0 dx The estimates for R3 - R5 are similar to that of R. R6 δ + cδ ε + Xuε x + Xu ε x h i0 p Dhi0 Xu ε x g 4 x dx ε + Xuε x + Xu ε x h i0 p g x Xgx D hi0 u ε dx R7 c ε + Xuε x + Xu ε x h i0 p + c ε + Xuε x + Xu ε x h i0 p Dhi0 Xu ε x Dhi0 gx g 3 x Xu ε x h i0 dx Dhi0 Xu ε x Dhi0 gx h g 3 x h i0 g 3 x h Xu εx h i0 dx 30

36 δ + cδ + c ε + Xuε x + Xu ε x h i0 p Dhi0 Xu ε x g 4 x dx ε + Xuε x + Xu ε x h i0 p Xu ε x h i0 Dhi0 gx g x dx ε + Xuε x + Xu ε x h i0 p 1 Dhi0 gx Xu ε x h i0 dx The estimates for R8-R10 are similar to that of R7. beginning of the proof and use a test function We can go back now to the ϕ = D hi0 D hi0 g 4 u ε to get similar results with x h i0 changed to x h 1 i 0. Adding the two inequalities, embedding the terms with δ coefficient into the left hand side and using Theorem.3.1 we get that for all h > 0 sufficiently small we have + c ε + Xuε x + Xu ε x h i0 p ε + Xuε x + Xu ε x h 1 i 0 Bx 0,r ε + Xux p p + ux p dx Dhi0 Xu ε x g 4 x dx D hi0 Xu ε x g 4 x dx We can repeat the proof for n < i 0 n and then we get that X u ε L loc and this leads to Remark Theorem 3.3 shows that u ε HW, loc, even if for this case the bounds for the second order horizontal derivatives have bounds dependent on ε. 31

37 3. CASE 1 < p <. Let us use in equation.1.3 a test function where g is a cut-off function between Bx 0, ϕx = s g x s u ε x r r and Bx k+ 0, to get k+1 g x ε + Xu ε x + Xu ε x s p Xu ε x s Xu ε x dx c ε + Xuε x + Xu ε x s p Xu ε x s Xu ε x gx Xgx u ε x s u ε x dx 3..1 Following a method from [11, 18] and using Young s inequality we estimate the right hand side as follows. RHS = c ε + Xuε x + Xu ε x s p + p p ε + Xuε x + Xu ε x s p p Xu ε x s Xu ε x gx Xgx u ε x s u ε x dx c ε + Xuε x + Xu ε x s p p 1 p Xu ε x s Xu ε x p 1 p gx Xgx u ε x s u ε x dx δ + cδ ε + Xuε x + Xu ε x s p Xu ε x s Xu ε x g x dx gx p Xgx p u ε x s u ε x p dx 3

38 Therefore, g x ε + Xu ε x + Xu ε x s p Xu ε x s Xu ε x dx c u ε x s u ε x p dx. r Bx 0, k+1 The method used in the previous section for handling the left hand side gives g x Xu ε x s Xu ε x p dx c u ε x s u ε x p dx r Bx 0, k+1 p Using Theorem.4.1 and Proposition.1.1 we get that p D s, Xu εx p M p 3.. where we denote p M p = c Λ + Xux + ux p dx. Bx 0,r This shows that Xu ε has locally p derivatives in the T direction. Now we use Proposition.1. to get that for a sufficiently small σ > 0 we have sup 0<s<σ sg u ε L p s 1+p M We will use the fact that for a for small δ > 0 we have u ε is locally C δ see [1] and that for 17 1 p we have p p 0. Therefore, for all 0 < s < σ and for δ = δ p we have = sg u ε x dx s +δ sg u ε x p s p + p sg u ε x p s +δ p p dx c M p g u ε p C δ. Theorem..1 shows now that T u ε L loc. Therefore we have proved the following lemma: 33

39 Lemma Let 17 1 p < and u ε HW 1,p loc be a weak solution of 1.1. Let x 0, r > 0 such that Bx 0, 3r, and let k be the number of iterations from the proof of Theorem.4.1. that depends only on p. Then we have T u ε L loc and T u ε x dx r Bx 0, k+ c u p C δ r Bx 0, k+1 Bx 0,r ε + Xuε x p + u ε x p dx + u ε L Bx 0, r k As an immediate corollary of the above lemma we have: Corollary For 17 1 p < we have XT u ε L p loc with bounds depending on ε. Proof. Lemma 3..1 allows us to estimate the right hand side of 3..1 in the following way. RHS δ Therefore, g x ε + Xu ε x + Xu ε x s p Xu ε x s Xu ε x dx + cδ Xgx ε + Xu ε x + Xu ε x s p u ε x s u ε x dx. g x ε + Xu ε x + Xu ε x s p Xu ε x s Xu ε x dx cε Xgx u ε x s u ε x dx and hence p g x Xu ε x s Xu ε x p dx cε u ε x s u ε x dx Bx 0,r 3..5 which gives XT u ε L p loc. We will prove now a theorem on estimates of the second order horizontal derivatives. 34

40 Theorem Let 17 1 p and u ε HW 1,p loc be a weak solution of 1.1. Consider x 0, r > 0 such that Bx 0, 3r, and let k be the number of iterations from the proof of Theorem.4.1 that depends only on p. Then for each i 0 {1,..., n} and s > 0 sufficiently small we have r Bx 0, k+3 c ε + Xuε x + Xu ε x h i0 p ε p u p +ε p C δ r Bx 0, k+1 and hence u ε HW,p loc. Bx 0,r u L r Bx 0, Bx k+1 + c 0,r Proof. Let g be a cut-off function between Bx 0, Dhi0 Xu ε x dx ε + Xuε x p + u ε x p dx ε + Xuε x p + u ε x p dx, 3..6 r r and that Bx k+3 0,. The proof k+ begins in the same way as the proof of Theorem 3.1.1, until we get the extended form of our inequality with the lines L1 and R1-R10. We can remark that although we could use a test function ϕ = D hi0 g D hi0 u ε, we cannot avoid estimates similar to that of line R6. For the line L1 the estimate is the same as in the proof of Theorem For the lines R1-R5 we keep again the same estimates and use Lemma 3..1 with the facts that for p < we have For R7 we have ε + Xuε x + Xu ε x s p ε p. R7 c ε + Xuε x + Xu ε x h i0 p 1 = c + c ε + Xuε x + Xu ε x h i0 p 1 ε + Xuε x + Xu ε x h i0 p 1 Dhi0 Xu ε x Dhi0 gx g 3 x h i0 dx Dhi0 Xu ε x Dhi0 gx g 3 x dx Dhi0 Xu ε x Dhi0 gx h g 3 x h i0 g 3 x h dx 35

41 = c ε + Xuε x + Xu ε x h i0 p 4 + c ε + Xuε x + Xu ε x h i0 p 1 δ + cδ + c ε + Xuε x + Xu ε x h i0 p ε + Xuε x + Xu ε x h i0 p ε + Xuε x + Xu ε x h i0 p Dhi0 Xu ε x g x ε + Xuε x + Xu ε x h i0 p 4 Dhi0 Xu ε x Dhi0 gx h D hi0 Xu ε x g 4 x dx Dhi0 gx g x dx Dhi0 gx 3 dx Dhi0 gx gx dx g 3 x h i0 g 3 x h dx The estimates for R8-R10 are similar. It is left to estimate the line R6. Following the methods in [11, 18] we consider for small h > 0 and a.e. x Bx 0, 4r α i x = 1 0 a ε i Xuε x th i0 dt and Y x = In the distributional sense we have 1 0 p 1 ε + Xuε x th i0 dt. D hi0 a ε i Xuε x = X i0 α i x. Also, α i x Y x, a.e x Bx 0, 4r. 36

42 Therefore, we can estimate R6 in the following way. R6 = = 4 = 4 n i=1 n i=1 n i=1 n i=1 n i=1 D hi0 a ε i Xuε x 4g 3 x X i gx D hi0 u εx dx α i x X i0 g 3 x X i gx D hi0 u ε x dx α i x 3g x X i0 gx X i gx D hi0 u ε x dx α i x g 3 x X i0 X i gx D hi0 u ε x dx α i x g 3 x X i gx D hi0 X i0 u ε x dx c g x Y i x Dhi0 u ε x dx R61 + c g 3 x Yi x Dhi0 Xu ε x dx R6 Because of Y i L p p 1 loc and Xu ε L p loc we get that R6 1 is finite. To estimate R6 we follow the method from [11]. Therefore, R6 = δ δ g x ε + Xu ε x + Xu ε x h i0 p 4 g 4 x ε + Xu ε x + Xu ε x h i0 p Dhi0 Xu ε x gx Y i x ε + Xu ε x + Xu ε x h i0 p 4 dx Dhi0 Xu ε x dx + cδ g x Yi x ε + Xu ε x + Xu ε x h i0 p dx g 4 x ε + Xu ε x + Xu ε x h i0 p + cδ g x Y p p 1 i Dhi0 Xu ε x dx x + ε + Xu ε x + Xu ε x h i0 p dx 37

43 We can now continue the proof in the same way as we did in the case p, going back to the beginning of the proof and using a test function ϕ = D hi0 D hi0 g 4 u ε, then adding the two inequalities and embedding the terms with δ coefficients into the left hand side. Therefore, we get g 4 x ε + Xu ε x + Xu ε x h i0 p Dhi0 Xu ε x dx Mε where by Mε we denote the right hand side of the inequality Quoting again the method in section.4 we get that g 4 x D hi0 Xu ε x p dx Mε, 3..7 and this proves that X u ε L p loc. 38

44 4.0 CORDES CONDITIONS AND UNIFORM ESTIMATES IN THE HEISENBERG GROUP 4.1 BOUNDING THE SECOND ORDER HORIZONTAL DERIVATIVES BY THE SUBELLIPTIC LAPLACIAN We denote by X u the matrix of the second order horizontal derivatives and by H u = n i=1 X ix i u the subelliptic Laplacian associated to the horizontal vector fields X i. Lemma For all u HW, 0 we have X u L c n H u L, where c n = 1 + n, and it is a sharp constant when = H n. Proof. We follow the spectral analysis of H developed by Strichartz [7]. Let us recall the fact that H and it commute, and share the same system of eigenvectors Φ λ,k,l z, t = λ n exp ilλt π n+1 n + k n+1 n + k exp λ z λ z L n 1 k, 4n + k n + k where l = ±1, k {0, 1,,...} and L n 1 k is the Laguerre polynomial L n 1 k t = et t n 1 1 k! dk dt k e t t k+n 1. 39

45 For the eigenvalues, we have the following relations it u Φ λ,k,l = lλ n + k u Φ λ,k,l H u Φ λ,k,l = λu Φ λ,k,l, 4.1. where denotes the group convolution. Therefore, the spectral decomposition of H u for u C 0, the Plancherel formula, and relations give H u L = π = π k=0 k=0 n + k l=±1 n + k l=±1 n T u L 0 0 H u Φ λ,k,l z, 0 dzdλ C n n + k it u Φ λ,k,l z, 0 l C n dzdλ Therefore, for all u C 0 we have T u L 1 n Hu L In the following we will use the fact that the formal adjoint of X k is X k. Let u C0. For k {1,..., n} and j k + n, X k and X j commute, therefore X k X j ux dx = X k X k ux X j X j uxdx. For j = k + n we have X k X j ux dx = X k X j ux X j X k ux + T ux dx = X k X j ux X j X k ux dx + X k X j ux T ux dx = X j ux X k X j X k ux dx + X k X j ux T ux dx = X j ux X j X k + T X k ux dx + X k X j ux T ux dx = X j ux X j X k X k ux dx + X k X j ux T ux dx = X k X k ux X j X j ux dx + X k X j ux T ux dx. 40

46 Similarly, Therefore, = X j X k ux dx X k X k ux X j X j ux dx X j X k ux T ux dx. = = X u L = n k,j=1 n k=1 n k,j=1 X k X j u L = X k X k ux X j X j ux dx + X k X k ux dx + n n k=1 T ux dx 1 + n 1n H u L = 1 + H u L n. [X k, X k+n ]ux T ux dx The constant 1 + n is sharp when = Hn, because for v = Φ λ,0,1 we have T v = i n Hv. 4. CORDES CONDITIONS FOR SECOND ORDER SUBELLIPTIC PDE OPERATORS IN NON-DIVERGENCE FORMS WITH MEASURABLE COEFFICIENTS Let us consider now Au = n i,j=1 a ij xx i X j u where the functions a ij L. Let us denote by A = a ij the n n matrix of coefficients. Definition [5, 8] We say that A satisfies the Cordes condition K ε,σ if there exists ε 0, 1] and σ > 0 such that 0 < 1 σ n i,j=1 a ijx n 1 a ii x, a.e. x n 1 + ε i=1 41

47 Theorem Let 0 < ε 1, σ > 0 such that γ = 1 ε c n < 1 and A satisfies the Cordes condition K ε,σ. Then for all u HW, 0 we have X u L n 1 γ α L Au L, 4.. where αx = Ax, I Ax. Proof. We denote by I the identity n n matrix, by A, B = n product and by A = n The Cordes condition K ε,σ implies that i,j=1 a ijb ij the inner i,j=1 a ij the Euclidean norm in Rn n for matrices A and B. for all x, where the Lebesgue measure of \ is 0. Ax, I Ax n 1 ε 4..3 Let be now x arbitrary, but fixed. Consider the quadratic polynomial P α = Ax α Ax, I α + n 1 ε. Inequality 4..3 shows that Therefore there exists min P α = P α R such that P αx 0. Observing that αx = Ax, I Ax Ax, I Ax 4..5 I αxax = Ax α x Ax, I αx + n we get that 4..4 implies that I αxax 1 ε, which is equivalent to I αxax, M 1 ε M, for all M M n R

48 Condition 4..6 can be written also as n m ii αx i=1 n a ij xm ij n 1/ 1 ε mij 4..7 i,j=1 i,j=1 for all M M n R. Formula 4..7 and Lemma imply that for all u HW, 0 we have H ux αxaux dx 1 ε n i,j=1 X i X j ux dx 1 εc n Therefore, for γ = 1 ε c n < 1 we get H ux dx. H u αau L γ H u L which shows that X u L c n H u L c n 1 γ αau L c n 1 γ α L Au L. 43

49 4.3 HW, -INTERIOR REGULARITY FOR p-harmonic FUNCTIONS IN H n Let H n be a domain, h HW 1,p and p > 1. Consider the problem of minimizing the functional Φu = Xux p dx over all u HW 1,p such that u h HW 1,p 0. The Euler equation for this problem is the p-laplace equation n i=1 X i Xu p X i u = 0, in A function u HW 1,p is called a weak solution for if n i=1 Xux p X i ux X i ϕxdx = 0, ϕ HW 1,p Φ is a convex functional on HW 1,p, therefore weak solutions are minimizers for Φ and viceversa. For m N let us define now the approximating problems of minimizing functionals Φ m u = and the corresponding Euler equations p 1 m + Xux dx n i=1 p 1 X i m + Xu X i u = 0, in The weak form of this equation is n i=1 p 1 m + Xux X i ux X i ϕxdx = 0, for all ϕ HW 1,p The differentiated version of equation has the form n i,j=1 a m ij X i X j u = 0, in

50 where a m ij x = δ ij + p X iux X j ux 1 +. m Xux Let us consider a weak solution u m HW 1,p loc of equation Then am ij Define the mapping L m : HW, 0 L by L. L m vx = n i,j=1 a m ij xx i X j vx and Denote We will check the validity of Theorem 4..1 for L m. We have n i,j=1 n i=1 a m Xu m ii x = n + p 1 + Xu m m, a m ij x Xu m = n + p 1 + Xu m m + p Xu m Xu m m. Therefore, for an ε 1 1, 1 we need c n This leads to Xu m p 1 + Xu m m = Λ. n + Λ + Λ 1 n 1 + ε n + Λ. n 1Λ 1 ε n + Λ + Λ < 1 c n n + Λ + Λ. Hence, n 1c n 1 Λ Λ n < 0. Solving this inequality we get 1 n n 1 c Λ n 1 + 1, 1 + n n 1 c n n 1c n 1 n 1c n 1 Using c n = n+ n and the fact that Xu m 1 m + Xum n n 4n + 4n 3 p n + n < 1 we have that for all m N we have, n + n 4n + 4n 3, n + n 45

51 and that the operators L m satisfies the assumptions of Theorem 4..1 uniformly in m. Let us remark that in the case n = 1 our methods gives 5 5 p, Theorem Let p < + n + n 4n + 4n 3 n + n If u HW 1,p is a p-harmonic function then u HW, loc. Proof. The case p = it is well known, so let us suppose p. Consider x 0 and r > 0 such that B 4r = Bx 0, 4r. We need a cut-off function η C 0 B r such that η = 1 on B r. Also consider minimizers u m for Φ m on HW 1,p B r subject to the condition u m u HW 1,p 0 B r. Then u m u in HW 1,p B r as m. By Theorem we get that for p < 4 we have u m HW, loc, but with bounds depending on m, and also that u m satisfies equation L m u m = 0 a.e. in B r. So, in B r we have a.e.. X i X j η u m = X i X j η u m + X j η X i u m + X i η X j u m + η X i X j u m and hence L m η u m = u m L m,um η + n i,j=1 a m ij x X j η X i u m + X i η X j u m. By Theorem 4..1 it follows that X u m L B r X η u m L B r c L m η u m L B r c u m HW 1,p B r c u HW 1,p B r where c is independent of m. Therefore, u HW, B r. Remark Observe that the range for p given by Theorem is shrinking from [, 5+ 5 to [, 3] as n increases from 1 to. For the case p < we need the following lemmas. The first lemma is an interpolation result and its proof is based on integration by parts. 46

52 Lemma For all u C 0 and for all δ > 0 there exists cδ > 0 such that Xu L δ X u L + cδ u L. Proof. n Xu L = = δ i=1 X i ux X i ux dx = n i=1 ux H ux dx δ H ux dx + cδ n X ux dx + cδ u x dx ux X i X i uxdx = u x dx From Lemma and the higher order extension results available for the Sobolev spaces on the Heisenberg group [15, 0] we get the following result. Lemma For all u HW, B r and all δ > 0 there exists cδ > 0 such that Xu L B r δ X u L B r + cδ u L B r. By Lemmas and 4.3. we can use a method similar to the proof of Theorem 9.11 [10] to get the following result. Lemma Let us suppose that the operator A satisfies the assumptions of Theorem 4..1 and that B 3r. Then X u L B r c Au L B r + u L B r, for all u HW, loc B 3r. 47

53 Proof. Let η C 0 B r, 0 < σ < 1 and σ = 1+σ such that η is a cut-off function between B σr and B σ r satisfying Xη 1 σr and X η Then we can use Theorem 4..1 for ηu to get 4 1 σ r. X u L B σr X ηu L B r c Aηu L B r n = c ηau + u Aη + a ij X j ηx i u + X i ηx j u c Au L B r + i,j=1 1 1 σr Xu L B σ r + L B r 1 1 σ r u L B σ r For k {0, 1, } let us use the seminorms u k = sup 1 σ k r k X k u L B σr. 0<σ<1 Then u c r Au L Br + u 1 + u 0. Lemma 4.3. implies that for δ > 0 small we have u 1 δ u + cδ u 0. Therefore, u c r Au L Br + u 0 and hence For σ = 1 X u L B σr we get the desired inequality. c r Au 1 σ r L B r + u L B r. Theorem Let us consider the Heisenberg group H 1 and 17 1 p. If u HW 1,p is a p-harmonic function then u HW, loc. 48

54 Proof. We start the proof in the same way as we did in the proof of Theorem Consider x 0 and r > 0 such that B 4r = Bx 0, 4r. We need a test function η C 0 B 3r. Also consider minimizers u m for Φ m on HW 1,p B 3r subject to the condition u m u HW 1,p 0 B 3r. Then u m u in HW 1,p B 3r as m. We use the facts that 4 3 < 5 5 < 17 1 <, the homogeneous dimension of H 1 is Q = 4, and 4p 4 p for all 4 3 p <. The Sobolev embeddings result in the subelliptic setting [1] says that HW 1,p 0 B3r L q 4p B 3r, for 1 q 4 p. Therefore, u m u in L B 3r. Also, using Theorem 3..1 we have for 17 1 p < that u m HW,p loc B3r we get that Xum L loc B3r. Let us remark that these bounds of X u m in L p may depend on m and that L m u m = 0 a.e. in B 3r. Moreover, L m η u m L B 3r = c u m L m η + n i,j=1 a m,u ij x X j η X i u m + X i η X j u m L B3r c u m L suppη + Xu m L suppη < +. and hence u m HW, loc B3r. By Lemma for all m sufficiently large we have X u m L B r c u m L B r c u L B r which shows that X u m is uniformly bounded in HW, B r, hence u HW, B r. 49

55 4.4 C 1,α -REGULARITY FOR p-harmonic FUNCTIONS IN THE HEISENBERG GROUP FOR p NEAR In this section we use previous results regarding the Calderón-Zygmund theory in Heisenberg group see [9, 13, 14], the HW, regularity of p-harmonic functions from Chapter 3 and the properties of second order PDE operators that are near to the subelliptic Laplacian, to prove C 1,α regularity for p-harmonic functions in the Heisenberg group for p in a neighborhood of. In the Euclidean case this result is known for 1 < p <, while in the Heisenberg group there is no definite answer yet. Our result constitutes the first indication that the C 1,α regularity for p-harmonic functions in the Heisenberg group is possible. We keep the general setting from the previous section given by formulas and update the working methods from those corresponding to L to those corresponding to L s with s > 1. in [9]. The Calderón-Zygmund theory gives the following lemma see the theorem on page 917 Lemma For all 1 < s < there exists C n,s 1 such that for all u HW,s 0 we have Recall that in the case s = we have X u L s C n,s X u L s. C n, = 1 + n and this is a sharp constant as shown in the previous section. Let us consider now Au = n i,j=1 a ij xx i X j u where the functions a ij L and denote by A = a ij the n n matrix of coefficients. Theorem Let 0 < ε 1, such that ε C n,s < 1 and suppose that X ux Aux ε X ux