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%\controldates{16-JUL-1996,16-JUL-1996,16-JUL-1996,23-JUL-1996}
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\newtheorem*{theorem6}{Corollary}
\newtheorem*{theorem7}{Theorem E}
\newtheorem*{theorem8}{Theorem F}
\newtheorem*{theorem9}{Theorem G}
\newtheorem*{theorem10}{Proposition H}

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\begin{document}

\title[Optimal regularity]{Optimal  regularity  for 
quasilinear equations in stratified nilpotent Lie groups 
and applications}
\author{Luca Capogna}
\address{Department of Mathematics, Purdue University, West Lafayette, 
IN 47907}
\email{capogna@math.purdue.edu}
\thanks{Alfred P. Sloan Doctoral Dissertation Fellow.}
%\issueinfo{2}{1}{}{1996}
\date{March 15, 1996}
\subjclass{Primary 35H05}
\commby{Thomas Wolff}
\begin{abstract}We announce  the optimal 
 $C^{1+\alpha }$ regularity of the gradient
of weak solutions to a class of quasilinear degenerate elliptic equations
in nilpotent stratified Lie groups of step two. 
As a consequence we also prove a Liouville type theorem for
 $1$-quasiconformal mappings between domains of the  Heisenberg group
$\mathbb{H}^{n}$.
\end{abstract}
\maketitle

\section*{Statement of the results}Consider the quasilinear elliptic equation
\begin{equation*}\sum _{i=1}^{n} \p _{x_{i}} A_{i}(x,\nabla u)=0, \tag{1}\end{equation*}
where
 $A_{i}(x,\xi ):\br ^{n+n}\to \br $, $i=1,\dots,n$, are  differentiable functions
satisfying $\lambda |\eta |^{2}\le \sum _{i,j=1}^{n} \p _{\xi _{j}}A_{i}(x,\xi ){\eta }_{i} 
{\eta }_{j} \le {\lambda }^{-1} |\eta |^{2}$, and $\sum _{i,j=1}^{n} \p _{x_{j}} A_{i}(x,\xi ) \le C(1+|\xi |)$, for every 
$\eta \in \br ^{n}$, and almost every $x,\xi \in \br ^{n}$.
The  sharp $C^{1+\a }$ regularity of weak solutions to (1) is 
one of the pillars on which the modern 
theory of quasilinear partial differential equations rests.
 The ideas on which  its  proof
is based form a recurring theme in nonlinear analysis: first use difference
quotients  to prove that the weak solutions
admit second (weak) derivatives, then differentiate the equations and 
observe that the derivatives of the solution are themselves solutions to
some linear partial differential equations, whose coefficients are not
very regular. At this point the regularity theory
for linear equations with nonsmooth coefficients provides
the final step
in the proof of
the H\"{o}lder continuity
of the gradient of weak solutions to (1).
The observation that one could reduce the
study of  quasilinear equations
to studying linear equations with ``bad'' coefficients
goes back to the pioneering work 
 of Morrey, and
has been developed by many mathematicians in the last thirty years.
Although (1) is a relatively simple elliptic equation, the
regularity theorem has  far-reaching  applications in 
calculus of variations (see, for instance, \cite{Gi})
 and in the theory of quasiconformal mappings in space \cite{G1}.

Two natural and inter-related questions arise: Can the ellipticity hypothesis
in (1) be somewhat weakened, and still expect regularity of the gradient
of the solutions? Is this ``new'' problem of some geometric relevance?
In this announcement we provide positive answers to both questions.

In the proof of his famous rigidity theorem \cite{Mo},  
Mostow introduced quasiconformal mappings in
the setting of stratified nilpotent Lie groups \cite{F}.
With this name one refers to the class of simply connected
Lie groups $G$ endowed with a stratification of the Lie algebra
$g=V^{1}\oplus \cdots \oplus V^{r}$, with $r\ge 1$ (the {\em step} of the group), 
such that 
$[V^{1},V^{j}]=V^{j+1}$, $j=1,\dots,r-1$, and $[V^{1},V^{r}]=0$.
The simplest example of a stratified Lie group is the
Euclidean space $\br ^{n}$, with $r=1$. A less trivial, and
genuinely non-Euclidean  
 example is provided by  the Heisenberg group $\bh ^{n}$, $n\ge 1$, whose
 Lie algebra is  $h^{n}=\br ^{2n}\oplus \br $.
The central role played by the Heisenberg group in many problems
of complex geometry, representation theory and partial differential
equations makes $\bh ^{n}$ the prototype {\em par excellence} of
stratified nilpotent Lie groups.
The theory of quasiconformal mappings between domains of the Heisenberg group
has been developed recently in a series of papers
 by Kor\'{a}nyi and Reimann \cite{KR1}--\cite{KR3} and by 
 Pansu \cite{P}.
As in the Euclidean case (see \cite{G1} and  \cite{R}),
this development led to
various questions concerning the regularity
of weak solutions to a class of  quasilinear equations similar to (1).

The notion of quasiconformality is a metric one, and in this
setting it is related to   the Carnot-Carath\'{e}odory
metric associated  to a basis $X^{1}_{1},\dots,X^{1}_{m}$, $m=m^{1}=$dim$(V^{1})$ of $V^{1}$
(with our notation we do not distinguish between elements of $g$ and left invariant vector fields).
Since this metric is nonisotropic, it is natural to expect some nonisotropic structure in the 
relevant equations. 
In order to be more precise we need to recall that the 
exponential mapping $\text{exp}:g\to G$ is a diffeomorphism, and so we can use
exponential coordinates 
$p=(p_{1}^{1},\dots,p^{1}_{m},p^{2}_{1},\dots,p^{2}_{\text{dim}(V^{2})},
\dots)$ on $G$.
 Let $Xu=(X_{1}^{1}u,\dots,X^{1}_{m} u)$ denote the {\em horizontal} gradient
of the function $u$.  Consider  the equation
\begin{equation*}\sum _{i=1}^{m} X_{i}^{1} A_{i}(p,Xu)=f(p), \tag{2}\end{equation*}
where $A_{i}(p,\xi ):G\times \br ^{m} \to \br $, $i=1,\dots,m$, 
are differentiable functions satisfying
\begin{equation*}\lambda |\eta |^{2} \le \sum _{i,j=1}^{m} \p _{\xi _{j}} A_{i}(p,\xi ) \eta _{i} \eta _{j} 
\le \lambda ^{-1}|\eta |^{2},\end{equation*}
and 
\begin{equation*}\sum _{i=1}^{m} \sum _{j=1}^{\text{dim}(g)} \p _{p_{j}} A_{i}(p,\xi ) \le C(1+|\xi |), \end{equation*}
for any $\eta \in \br ^{m}$, for almost every $p\in G$ and $\xi \in \br ^{m}$, and
for some positive $\lambda , C$. 
The models we have in mind are
\begin{equation*}A_{i}(p,\xi )= \sum _{j=1}^{m^{1}} a_{ij}(p) \xi _{j} + a_{i}(p),\end{equation*}
or
\begin{equation*}A_{i}(p,\xi )=|\xi |^{p-2}\xi _{i},\end{equation*}
for $02$.
Let $\OXYZii \subset G$ be an open set, and 
let $S^{k,p}_{\loc }(\OXYZii )$ denote the space of $L^{p}_{\loc }(\OXYZii )$ functions
that are weakly differentiable $k$ times in the horizontal directions, and such that
their horizontal derivatives up to order $k$ lie  in $L^{p}_{\loc }(\OXYZii )$.
A function $u\in S^{1,2}_{\loc }(\OXYZii )$ is a weak solution to (2) if and only if
\begin{equation*}\int _{\OXYZii }\sum _{i=1}^{m} A_{i}(p,Xu)  X_{i}^{1} 
\phi\, dp=\int _{\OXYZii } f(p) \phi (p) dp,\end{equation*}
for every $\phi \in C^{\infty }_{0}(\OXYZii )$.
The regularity of the weak solutions to (2) is measured
in terms of the H\"{o}lder continuity with respect to
 the Carnot-Carath\'{e}odory distance $d(x,y)$  associated
to $X_{1}^{1},\dots,X_{m}^{1}$. More precisely, 
for $0<\alpha <1$ we define the Folland-Stein class
\begin{equation*}\Gamma ^{\a }(\OXYZii )=\left\{u\ 
\Big| \sup _{x,y\in \OXYZii } \frac{|u(x)-u(y)|}{d(x,y)^{\a }}
<\infty\right \},\end{equation*}
and its local version $\Gamma ^{\a }_{\loc }(\OXYZii )=\{u\mid \eta 
u \in \Gamma ^{\a }(
\OXYZii ) $ for some  $\eta \in C^{\infty }_{0}(\OXYZii )\}.$
If $k\in \bn $, the symbol $\G ^{k,\a }_{\loc }(\OXYZii )$, will denote the set
of functions having horizontal derivatives up to order $k$ in $\G ^{\a }_{\loc }(\OXYZii )$.

The local H\"{o}lder regularity of weak solutions to (2) has been established
independently in \cite{X1} and \cite{CDG1}.
The boundary regularity was studied in \cite{D}.
One would like to have an analogue of the Euclidean  $C^{1+\a }$ regularity
of weak solutions in this setting also, namely the $\G ^{1,\a }$ regularity.
  However, the classical argument
 breaks down at its first step and a new method is required.
In fact, one should observe that if one differentiates (even formally)
 the equation
along a vector field,  terms containing derivatives of the solution
along the commutators arise. There is no a priori control over these
derivatives, in fact the notion of weak solution implies only differentiability
in the horizontal directions $X_{1}^{1},\dots,X_{m}^{1}$.

In the special case of stratified nilpotent Lie groups of step two, 
we have been able to overcome these difficulties.
We prove 

\begin{theorem1} Let $\OXYZii \subset \bh ^{n}$ be an open set, 
and $u\in S^{1,2}_{\loc }(\OXYZii )$ a weak solution of
\begin{equation*}\sum _{i=1}^{m} X_{i}^{1} A_{i}(p,Xu)=0 \end{equation*}
 in $\OXYZii $. There exists $\lambda >0$ such that
\begin{equation*}u\in \G ^{1,\lambda }_{\loc }(\OXYZii ).\end{equation*}
\end{theorem1}


This result is only a particular case of the following

\begin{theorem2} Let $G$ be a stratified nilpotent Lie group of step
two,  $\OXYZii \subset G$ be an open set, and $u\in S^{1,2}_{\loc }(\OXYZii )$ be a weak solution to (2) in $\OXYZii $.
Denote by 
 $X^{2}_{\nu }$, $\nu =1,\dots,m^{2}=\dim(V^{2})$, a basis of $V^{2}$.
If we let $Q=m+2m^{2}$ denote the homogeneous dimension of $G$, and
 $\sum _{\nu =1}^{m^{2}}|X^{2}_{\nu }f| \in L^{s}_{\loc }(\OXYZii )$ with $s>Q/2$,
then
$X^{2}_{\nu }u\in \G ^{\lambda }_{\loc }(\OXYZii )$ 
for every $\nu =1,\dots,m^{2}$. If we also assume
$ \sum _{i=1}^{m^{1}}|X_{i}^{1}f|\in L^{s}_{\loc }(\OXYZii )$,  
 then \begin{equation*}u\in \G ^{1,\lambda }_{\loc }(\OXYZii )\end{equation*} 
for every $i=1,\dots,m^{1}$.

\end{theorem2}
Theorem B rests on 
  the following crucial result:

\begin{theorem3} Using the notation of Theorem B,
 if $f, \sum _{\nu =1}^{m^{2}}|X_{\nu }^{2} f| \in L^{2}_{\loc }(\OXYZii )$,   then
\begin{equation*}X^{2}_{\nu }u \in S^{1,2}_{\loc }(\OXYZii )\end{equation*}
and
\begin{equation*}u\in S^{2,2}_{\loc }(\OXYZii ), \ \ i=1,\dots,
m^{1}.\end{equation*}
\end{theorem3}


The strategy for the proof of Theorem  C is  described in the next section.

In view of Theorem B one can see that (surprisingly!) the sharp regularity
of the weak solutions is not $\G ^{1,\a }$ but the H\"{o}lder continuity of
the full gradient of $u$.

\begin{theorem4}  If $u$ is a weak solution to (2) in some open set
$\OXYZii \subset G$, and the hypotheses in Theorem B 
above are satisfied, then
there exists $\a >0$ such that \begin{equation*}u\in C^{1+\a }_{\loc }(\OXYZii ).\end{equation*} 
\end{theorem4}


Several remarks are in order:
(1) The results in Theorems B and  C are
new even for linear equations of the form
\begin{equation*}\sum _{i,j=1}^{m^{1}} X_{i}^{1}\bigg (a_{ij} X_{j}^{1}u + a_{i}\bigg )=f,\end{equation*}
with $a_{ij}$ a positive definite $m^{1}\times m^{1}$ matrix with
Lipschitz coefficients, and $a_{i}$ Lipschitz functions. However, in this
case, the local $\G ^{\a }$ regularity of the horizontal gradient is much easier 
to prove;
in fact it can be done by using the freezing techniques, without
differentiating the equation. This approach does not work when
nonlinear dependence from the horizontal gradient is allowed on the
left-hand side of the equation.
 
(2)
In the setting of nilpotent stratified Lie groups of step two,
the higher differentiability of  Theorem C gives a ``nonlinear'' analogue
of the $L^{2}$ estimates proved in \cite{K}, \cite{FS},  and \cite{F} 
for the sub-Laplacian
$\mathcal{L}=\sum _{i=1}^{m^{1}}{X_{i}^{1}}^{2}  $, but it does not generalize them.
In fact, here we need the extra differentiability  assumption
$X^{2}_{\nu }f\in L^{2}_{\loc }(\OXYZii )$ for every $\nu =1,\dots,m^{2}$.
However,  differentiability  of $f$ in the commutators direction
is necessary to obtain
 $X^{2}_{\nu }u\in S^{1,2}_{\loc }(\OXYZii )$, which is, in turn, a crucial ingredient of
the proof of Theorem B.
It seems worthwhile to observe also that in the Euclidean case
$m^{2}=0$, and so   Theorems B and C  do not require more hypotheses 
on $f$ than the classical $L^{2}$ estimates.
 
(3) Theorem B  is sharp, in the sense that for equations
of the form
\begin{equation*}\sum _{i=1}^{m^{1}} X_{i}^{1} A_{i}(p,Xu)=0,\end{equation*}
the local  H\"{o}lder regularity of the gradient
is the best one can expect unless more differentiability is required
 on the
 on the
$A_{i}$'s.

(4) If one assumes that the weak  solution to (2) is 
$\G ^{\a }_{\loc }(\OXYZii )$, then
the  conditions on the Lebesgue norm of  $X^{k}_{i} f$ in Theorem B can be 
weakened to conditions
on the norm of $X^{k}_{i} f$ in some {\em ad hoc}   Morrey spaces.

In view of the applications to the theory of quasiconformal mappings, it is
important to develop a higher regularity theory for (2).
The cornerstone of such theory is given by Theorem B,  but two
other ingredients from the linear theory are needed. The first is the subelliptic analogue
of the interior Schauder theory, developed in \cite{X2}; the second is the 
important $L^{2}$-estimates established in \cite{K}, \cite{FS}, and \cite{F}.

\begin{theorem5}
 Let $u$ be a weak solution
to (2), and $\lambda $ as in Theorem B.
Assume $X^{2}_{\nu }f\in L^{\infty }_{\loc }(\OXYZii )$, for $\nu =1,
\dots,m^{2}$.
If $k>0$, and  $\p _{\xi }A_{i},f, \p _{p^{l}_{\nu }}A_{i}(p,Xu)\in \G ^{k,\lambda }_{\loc }(\OXYZii )$, for $l=1,2$ and 
$\nu =1,...,m^{l}$,
then $u\in \Gamma ^{k+2,\lambda }_{\loc }(\OXYZii )$. In particular, if $A_{i}(p,\xi ),f(p)$ are smooth functions of $p$ and $\xi $,
then $u$ is smooth.
\end{theorem5}


Theorem D has the following important 
 
\begin{theorem6} 
For $p\ge 2$, consider 
a weak solution, $u$,  to
\begin{equation*}\sum _{j=1}^{m^{1}} {X_{j}^{1}}\bigg (|Xu|^{p-2}X_{j}^{1}u\bigg )=0\end{equation*}
in an open set  $\OXYZii \subset G$. If there exists $M>0$ such that $M\le |Xu|\le M^{-1}$ in $\OXYZii $,
then $u\in C^{\infty }(\OXYZii )$.
\end{theorem6}

 
The previous corollary
may be applied (in the case $p=Q$) to the theory
of quasiconformal mappings in the Heisenberg group.
 
In \cite{G1}, Gehring showed that the only $1$-quasiconformal
mappings in $\br ^{3}$ are the M\"{o}bius transformations.
In his famous argument, a regularity theorem, analogous
to Theorem D,  plays a crucial   role.
In \cite{KR1}, Kor\'{a}nyi and Reimann show that
$C^{4}$ $1$-quasiconformal mappings between  domains of $\bh ^{1}$ are obtained
by taking the restriction
of $SU(1,2)$  group actions. They conjecture that the same should be true
without the regularity assumption.
 Using the corollary,
we can
prove the conjecture and  identify
 all the $1$-quasiconformal mappings of the Heisenberg group.

\begin{theorem7}  If
$f:D\to f(D)$ is a $1$-quasiconformal mapping defined
on an open connected set $D\subset \bh ^{n}$, then
$f$ is the action of a group element in $SU(1,n+1)$, restricted to $D$.
\end{theorem7}


Once one has Theorem D, the proof of
Theorem E is similar to that in \cite{G1}, but here one  has to
take into account another ``pathology'' of the Heisenberg group: the
horizontal gradient of the gauge distance vanishes on the center of the group.
The deep results of Pansu \cite{P}, and Kor\'{a}nyi and Reimann 
\cite{KR1}--\cite{KR3}, as well 
as some theorem from \cite{CDG2}, play
a fundamental  role in the adaptation of Gehring's argument.
 
This Liouville type theorem has been recently and independently proved by
Tang \cite{T}
in the case of three-dimensional strongly pseudoconvex CR manifolds.
However, his proof is completely different from
ours as it does not rest on the regularity of solutions
to degenerate  elliptic  quasilinear equations.

Since the commutators of step two commute with the horizontal vector fields,
it should be  clear that the equations satisfied by the derivatives of 
the solution
along the commutators must be considerably simpler than the ones
satisfied by the derivatives of the solution along the horizontal directions.
This observation would induce one to think that the nonhorizontal derivatives
of weak solutions should enjoy extra regularity properties.
Indeed, this is the case.
We present an argument based on a  weak reverse H\"{o}lder
inequality (analogous to the one in \cite{G2})
 that will lead to the following  higher integrability result.
 
 \begin{theorem8} Let $u$ be a weak solution to 
(2)
 in an open set  $\OXYZii \subset G$. 
If $f=0$ and $A_{i}(p,\xi )$ does not depend on $V^{2}$ (when expressed
in exponential coordinates), then there exists $\e >0$ such that
\begin{equation*}Tu\in S^{1,2+\e }_{\loc }(\OXYZii ),\end{equation*}
for any $T\in V^{2}$.
\end{theorem8}


Let us observe that, in the case of lowest dimension, $Q=4$,
 Theorem F  provides
enough regularity in the coefficients
of one of the pde's solved by $X_{i}^{1} u$ so that the non-Euclidean  version of
the De Giorgi, Nash, Moser, Stampacchia theorem  in \cite{CDG1} can be used.
This would give a proof of Theorem B that does not involve the
use of Morrey's ideas.

 \section*{Sketch of the proofs}The classical  argument for the proof of the optimal  regularity of
weak solutions to (1) can be divided into two parts: the first deals with the
higher differentiability (in the weak $L^{2}$ sense) of the solutions; the
second consists in proving regularity results for second order elliptic
linear partial differential equations with nonsmooth coefficients.
The first part of the program  is carried  by using 
 difference quotients.

Since we are interested in horizontal derivatives, 
 the approach in the stratified nilpotent Lie groups  setting should,
arguably, start by considering difference quotients of the solutions
along the horizontal directions $X^{1}_{i}$, $i=1,\dots,m$.
This is not possible. In fact, in contrast with the Euclidean case,
the difference quotient along any nonzero vector $X\in V^{1}$ of
a function $u\in S^{1,2}_{\loc }(\OXYZii )$,
that is, 
\begin{equation*}u_{(1,X)}(p)=\bigg 
(\frac{u(pe^{sX})-u(p)}{|s|}\bigg )\end{equation*}
 in general is not in $S^{1,2}_{\loc }(\OXYZii )$, for any value of $s$.
This is due to the noncommutative nature of the group, and
can be easily seen by using exponential coordinates
and the Baker-Campbell-Hausdorff formula
\begin{equation*}e^{X}e^{Y}=e^{X+Y+\frac{1}{2}[X,Y]}, 
\end{equation*}
where $X,Y\in g$.
 In fact one has,
for any $\theta \in G$, that
\begin{equation*}
\begin{split}
X_{i}^{1}u(p\theta )&=\frac{d}{dt}u(pe^{tX^{1}_{i}}\theta)
=\frac{d}{dt} u(p\theta e^{tX_{i}^{1}+t
\sum _{l=1}^{m^{1}} \theta _{l}^{1}[X^{1}_{i},X^{1}_{l}]})\\
&=\left(X^{1}_{i}u + \sum _{l=1}^{m^{1}} 
\theta _{l}^{1}[X^{1}_{i},X^{1}_{l}]u\right)(p\theta ),
\end{split}\tag{3}
\end{equation*}
for any $p\in G$.
Equation (3) determines, with its structure, the strategy to be followed
 in the proof of the  sharp regularity theorem:
show that the weak solutions are differentiable (in the weak sense)
along the commutators direction, and then deal with the horizontal derivatives.

This approach presents some  immediate advantages but also
new difficulties. On the one hand, 
the difference quotient along any vector $T\in V^{2}$ of a function
$u\in S^{1,2}_{\loc }(\OXYZii )$  
 is always in $S^{1,2}_{\loc }(\OXYZii )$
(this is a simple consequence of (3)); on the other hand,
however, the fact that a priori a weak solution is only differentiable
along horizontal vector fields is an obstruction to the use
of such difference quotients. 

In order to remove this considerable obstruction we introduce difference 
quotients of fractional order along the directions $X^{2}_{1},\dots,
X^{2}_{m^{2}}$.
In exponential coordinates, for $1\le i_{0} \le m^{2}$,  
we write these difference quotients
as 
\begin{equation*}u_{(\a , X^{2}_{i_{0}})}(p)\!=\!\bigg (\! \frac{
u(p^{1}_{1},...,p^{1}_{m},p^{2}_{1},...,p^{2}_{i_{0}}\!+\!s^{2},...,
p^{2}_{m^{2}})\!-\!
u(p^{1}_{1},...,p^{1}_{m},p^{2}_{1},...,p^{2}_{i_{0}},...,p^{2}_{m^{2}})
}{|s|^{2\a }}\!\bigg ),\end{equation*}
where $0<\a <1$, and $s\neq 0$.
The estimates on the fractional derivatives of the solution will be
carried through by using the  following two norms on the 
space $C^{\infty }_{0}(g)$.
 The first is 
\begin{equation*}|w|_{\frac{\p }{\p p^{2}_{i_{0}}},\a } =\sup _{\e >s>0}\int _{\br ^{\text{dim}(g)}} |w_{(\a ,X^{2}_{i_{0}})}|^{2} dp.\tag{4}\end{equation*}
The second is the usual Sobolev norm
\begin{equation*}\begin{split}
 \Big\|\frac{\p ^{\a }w }{\p p^{2}_{\nu }}\Big\|_{L^{2}(g)} &=
\int _{\br ^{\text{dim}(g)}}
 |h|^{2\a } |\hat {w}(p_{1}^{1},\dots,p_{\nu -1}^{2},h,p^{2}_{\nu +1},
\dots,p^{2}_{m^{2}})|^{2}\\
 &\qquad\qquad \times dp^{1}_{1}\cdots dp^{2}_{\nu -1}dh\, 
dp^{2}_{\nu +1}\cdots dp^{2}_{m^{2}},
\end{split}\tag{5}
\end{equation*}
where we have denoted by $\hat {w}$ the partial Fourier transform in the
variable $p^{2}_{\nu }$. 
The two norms (4) and (5) are related by the following theorem of Peetre
\cite{Pe}
(see also \cite{S}):

\begin{theorem9} Let $0<\b <\a <1$, and $w\in C^{\infty }_{0}(G)$.
There exists a positive constant depending only on $\alpha $ and $\b $
such that
\begin{equation*}C\Big{\|}\frac{\p ^{\b }w}{\p p^{2}_{\nu }}
\Big{\|}_{L^{2}(g)} \le |w|_{\frac{\p }{\p p^{2}_{\nu }},\a } 
\le C^{-1}\Big{\|}\frac{\p ^{\a }w}{\p p^{2}_{\nu }}\Big{\|}_{L^{2}(g)}.
\end{equation*}
\end{theorem9}

 
The differentiability of the weak solutions to (2) along the commutator
directions is achieved by an iteration argument, where the order
of differentiability is increased step by step, from one half to one.
The first step in this iteration is given by the following proposition, that in turn is just a consequence of the Baker-Campbell-Hausdorff formula.

\begin{theorem10}
Let $\OXYZii \subset G$ be an open set, and $B(x_{0},r)\subset \OXYZii $
a homogeneous gauge ball. If $\eta \in C^{\infty }_{0}(B(x_{0},r))$ and
$w\in C^{\infty }(\OXYZii )$, then there exists a positive
constant $C$ such that
\begin{equation*}|w\eta |_{\frac{\p }{\p p^{2}_{\nu }},\frac{1}{2}} \le C \sum _{l=1}^{m^{1}} ||X(w\eta )||_{L^{2}(\OXYZii )}
.
\end{equation*}
\end{theorem10}


Let us describe the iteration without going into the details.
 From Proposition H one has
that the one-half derivative of $u$ along any  commutator direction exists in 
the
$L^{2}$-sense.
Through the use of fractional difference quotients one can prove a Caccioppoli
type inequality for derivatives of solutions to (2); namely, for $\eta \in C^{\infty }_{0}(B(0,2R))$ we have
\begin{equation*} \int _{\OXYZii } |(\eta Xu)_{(\X ,\frac{1}{2})}|^{2}dp
\le C\bigg [\int _{B(0,2R)} |Xu|^{2} +|f|^{2} dp\bigg ]. 
\end{equation*}
 This inequality
(almost)  proves that the $L^{2}$ norm of the horizontal gradient of
the weak derivative of order one half along the commutators of the solution is
controlled by the $L^{2}$ norm of the horizontal gradient of $u$.
Using Proposition H once more (with $w=\frac{\p ^{\frac{1}{2}}}{\p p^{2}_{i_{0}}}u$) one obtains that the full derivative along
the commutator is in $L^{2}$. 

This is the rough idea behind the proof of the first part of  Theorem C: an interplay
between the Baker-Campbell-Hausdorff formula and a  Caccioppoli type inequality. 
However, the  argument we have just described is not precise.
Things are more complicated because, in view of Theorem G, the two fractional 
norms (4) and (5)  are not equivalent,
so a longer, and more difficult,  iteration has to be carried through
in order to reach the (nonfractional) differentiability.

Once one has differentiability of the weak solution along the commutators,
the difference quotients along horizontal directions can be used
and the classical approach may be pursued, yielding $u\in S^{2,2}_{\loc }(\OXYZii )$. 
There are more difficulties,
of course, due to the noncommutative structure of the vector fields, but
these can be overcome since a  certain amount of regularity
of the derivatives along the commutators directions is now known.

\begin{remark1} It seems  worthwhile to observe that the iteration argument works also for
any nilpotent stratified Lie group, without the restriction to the
step two case. It is always true that the derivatives  of weak
solutions to (2) along the 
directions of the highest component of $g$, namely $V^{r}$,
are in $S^{1,2}_{\loc }(\OXYZii )$. 
In this case the iteration on the order of differentiability
goes from $\frac{1}{r}$ to one with step $\frac{1}{r}$.
 However, the next level 
offers some new difficulties. In fact one needs  to 
estimate the commutators between horizontal
vector fields and fractional derivatives in the direction $V^{s}$, with $1s$. This problem is  the object of work in progress, and is the only obstruction to the extension of Theorems B and C to any nilpotent stratified Lie group.
\end{remark1}
Let us go back to the step two case and describe briefly
the ideas behind the proof of Theorem B.

Theorem C allows us to differentiate the equation (2) along the horizontal 
and the $V^{2}$ directions. In this way one obtains two sets of equations.
Since the group is of step two, it is clear that the equations obtained
by differentiating along the $V^{2}$ directions
must be considerably simpler than the other one
(all the commutators are missing). 
Following the classical approach we would like to treat such
equations as linear equations. The presence of terms with
mixed derivatives, horizontal and nonhorizontal, creates a problem:
these terms are not regular enough to allow the use
of the various regularity theorems for subelliptic linear equations
with nonsmooth coefficients \cite{CDG1}, \cite{X1}.
This problem can be solved by recurring to the same strategy used in Theorem
C. First, one deals with the equations obtained by differentiating
(2) along the commutator directions. These are simpler and
 can be dealt with by developing, to some extent, a 
non-Euclidean  analogue
of some of Morrey's techniques (Chapter 5 in \cite{M}).
Once one has obtained the sharp regularity for the "mixed" derivatives,
the other set of equations may be studied, the terms corresponding
to the commutators being now regular enough.

\section*{Acknowledgement}The results announced  here are part of my Ph.D. dissertation
at Purdue University.
I am very grateful to my thesis advisor, Professor Nicola Garofalo,
for having introduced me to the problem addressed in this paper.

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\end{document}