\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 86, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE--2003/86\hfil Estimates for derivatives of the Green functions]
{Estimates for derivatives of the Green functions for the noncoercive
differential operators on homogeneous manifolds of negative curvature, II}
\author[Roman Urban\hfil EJDE--2003/86\hfilneg]
{Roman Urban}
\address{Institute of Mathematics, University of Wroclaw,
Pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland}
\email{urban@math.uni.wroc.pl}
\date{}
\thanks{Submitted February 18, 2003. Published August 15, 2003.}
\thanks{Partly supported by grant 5P03A02821 from KBN, and by
contract \hfill\break\indent HPRN-CT-2001-00273-HARP
from RTN Harmonic Analysis and Related Problems.}
\subjclass[2000]{11E25, 43A85, 53C30, 31B25}
\keywords{Green function, homogeneous manifolds of negative curvature, \hfill\break\indent
$NA$ groups, evolutions on nilpotent Lie groups}
\begin{abstract}
We consider the Green functions for second order
non-coercive differential operators on homogeneous manifolds of
negative curvature, being a semi-direct product of a nilpotent Lie
group $N$ and $A=\mathbb{R}^+$.
We obtain estimates for the mixed derivatives of the Green functions
that complements a previous work by the same author \cite{pota}.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newcommand{\e}{\text{\bf E}}
\newcommand{\p}{\text{\bf P}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\I}{I_{a,\eta}}
\section{Introduction}\label{sec. introduction}
Let $M$ be a connected and simply connected homogeneous manifold of
negative curvature.
Such a manifold is a solvable Lie group $S=NA$, a semi-direct
product of a nilpotent Lie group $N$ and an Abelian group
$A=\mathbb{R}^+$. Moreover, for an $H$ belonging to the Lie
algebra $\mathfrak{a}$ of $A$, the real parts of the eigenvalues
of $\mathop{\rm Ad}_{\exp H}|_{\mathfrak{n}}$, where $\mathfrak{n}$ is the Lie
algebra of $N$, are all greater than 0. Conversely, every such a
group equipped with a suitable left-invariant metric becomes a
homogeneous Riemannian manifold with negative curvature (see
\cite{H}).
On $S$ we consider a second order left-invariant operator
\begin{equation*}
\mathcal{L} =\sum_{j=0}^mY_j^2+Y.
\end{equation*}
We assume that $Y_0,Y_1,\ldots,Y_m$ generate the Lie algebra
$\mathfrak{s}$ of $S$ and $Y\in\mathfrak{s}$. We can always make
$Y_0,\ldots, Y_m$ linearly independent and moreover, we can choose
$Y_0,Y_1,\ldots,Y_m$ so that $Y_1(e),\ldots, Y_m(e)$ belong to
$\mathfrak{n}$. (Write $\mathcal{L}$ as
$\sum_{i,j=0}^{\text{dim}\mathfrak{n}}\alpha_{i,j}E_iE_j+\sum_{j=0}^{\text{dim}\mathfrak{n}}\beta_jE_j$, $E_0\in\mathfrak{a}$, $\{E_j\}$ is a basis of $\mathfrak{n}$, $\alpha_{i,j},\beta_j\in\R$ and then rewrite $\mathcal{L}$ as a
sum of squares). Let $\pi:S\to A=S\slash N$ be the canonical
homomorphism. Then the image of $\mathcal{L}$ under $\pi$ is a
second order left-invariant operator on $\mathbb{R}^+$,
\begin{equation*}
(a\partial_a)^2-\gamma_{\mathcal{L}} a\partial_a,
\end{equation*}
where $\gamma_{\mathcal{L}}\in\mathbb{R}$. We say that a second
order differential operator $\mathcal{L}$ on a Riemannian manifold
is \textit{noncoercive} (\textit{coercive} resp.) if there is no
$\varepsilon>0$ such that $\mathcal{L}+\varepsilon\text{Id}$
admits the Green function (if such an $\varepsilon$ exists resp.).
It is worth noting that our definition of coercivity is a little
bit different than that used e.g. in \cite{A}. Namely, for us,
$\mathcal{L}$ is coercive if it is weakly coercive in Ancona's
terminology. There is a relation between the notion of coercivity
property in the sense used in the theory of partial differential
equations (i.e., that an appropriate bilinear form is coercive,
\cite{Stamp}) and weak coercivity. For this the reader is
referred to \cite{A}.
In this paper we shall study the operators $\mathcal{L}$ that are
noncoercive ($\gamma_\mathcal{L}=0$). In this case $\mathcal{L}$
can be written as
\begin{equation}\label{operator}
\mathcal{L}=\sum_{j=1}^m\Phi_a(X_j)^2+\Phi_a(X)+a^2\partial_a^2
+\partial_a,
\end{equation}
and $X,X_1,\ldots,X_m$ are left-invariant vector fields on $N$,
moreover, the vector fields $X_1,\ldots,X_m$ are linearly
independent and generate $\mathfrak{n}$,
\begin{equation*}
\Phi_a={\mathop{\rm Ad}}_{\exp(\log a)Y_0}=e^{(\log a) \mathop{\rm
ad}_{Y_0}}=e^{(\log a) D},
\end{equation*}
where $D=\mathop{\rm ad}_{Y_0}$ is a derivation of the Lie algebra
$\mathfrak{n}$ of the Lie group $N$ such that the real parts $d_j$
of the eigenvalues $\lambda_j$ of $D$ are positive. By multiplying
$\mathcal{L}$ by a constant, i.e., changing $Y_0$, we can make
$d_j$ arbitrarily large (see \cite{DHU}).
Let $\mathcal{G}(xa,yb)$ be the {\it Green function} for
$\mathcal{L}$. The Green function $\mathcal{G}$ is (uniquely)
defined by two conditions:
\begin{itemize}
\item [i)] $\mathcal{L}\mathcal{G}(\cdot,yb)=-\delta_{yb}$ as distributions
(functions are identified with distributions via the right Haar
measure),
\item [ii)] for every $yb\in S$, $\mathcal{G}(\cdot,yb)$ is a potential for
$\mathcal{L}$, i.e., is a positive superharmonic function such
that its largest harmonic minorant is equal to zero (cf. \cite{pt}).
\end{itemize}
Let
\begin{equation}\label{gf}
\mathcal G(x,a):=\mathcal G(xa;e),
\end{equation}
where $e$ is the identity element of the group $S$. It is easy to
show that\begin{equation*} \mathcal G(xa;yb)=\mathcal
G((yb)^{-1}xa;e)=\mathcal G((yb)^{-1}xa).
\end{equation*}
In this paper we refer to $\mathcal G(x,a)$ defined in \eqref{gf}
as the Green function for $\mathcal L$.
The main goal of this paper is to prove the following estimates
for derivatives of the Green function \eqref{gf} for the
noncoercive operator $\mathcal{L}$, i.e., with
$\gamma_{\mathcal{L}}= 0$. For every neighborhood $\mathcal U$ of
the identity $e$ of the group $NA$ there is a
constant $C=C(\mathcal{U})$ such that
we have
\begin{equation}\label{ge}
|\partial_a^l\mathcal{X}^I\mathcal{G}(x,a)|\leq
\begin{cases}
C(|x|+a)^{-\|I\|-Q}a^{-l}&\\
\times (1+|\log(|x|+a)^{-1}|)^{\|I\|_0},&\text{$(x,a)\in(\mathcal Q\cup\mathcal U)^c$,}\\
Ca^{-l},&\text{$(x,a)\in \mathcal Q\setminus\mathcal U$.}
\end{cases}
\end{equation}
where $|\cdot|$ stands for a ``homogeneous norm" on $N$, $\mathcal
Q=\{|x|\leq 1, a\leq 1\}$, $\|I\|$ is a suitably defined length of
the multi-index $I$ and $\|I\|_0$ is a certain number depending on
$I$ and the nilpotent part of the derivation $D$. In particular,
$\|I\|_0$ is equal to 0 if the action of $A=\R^+$ on $N$, given by
$\Phi_a$, is diagonal or, if $I=0$.
$\mathcal{X}_1,\ldots,\mathcal{X}_n$ is an appropriately chosen
basis of $\mathfrak{n}$. For the precise definitions of all the
notions that have appeared here see Sect. \ref{sec.
preliminaries}.
Some comments should be made at this moment. First of all, it
should be said that the estimates for the Green function (i.e.,
when $I$=0) with $\gamma_{\mathcal{L}}>0$, also from below, was
proved by E. Damek in \cite{D} and then by the author for
$\gamma_{\mathcal{L}}=0$ in \cite{CM} but at that time it was
impossible to prove analogous estimate for derivatives. The reason
was that we did not have sufficient estimates for the derivatives
of the transition probabilities of the evolution on $N$ generated
by an appropriate operator which appears as the ``horizontal"
component of the diffusion on $N\times\R^+$ generated by
$a^{-2}\mathcal L_{-\gamma}$ (cf. \cite{DHU}). These estimates
have been obtained by the author in \cite{MathZ} and eventually
led up to the estimates for derivatives of the Green function
$\mathcal{G}$ with respect to the $N$ and $A$-variables (see
\cite{pota}). Here we are going to present how to use the results
from \cite{pota} in order to get the estimates for the mixed
derivatives, i.e., not only for $N$ and $A=\R^+$-variables
separately.
The proof of \eqref{ge} requires both analytic and probabilistic
techniques. Some of them have been introduced in \cite{DHZ,DHU}
and \cite{CM}. This paper also heavily depends on some results
from \cite{pota}. \smallskip
\noindent\textbf{Guide to the paper.} The structure of the paper
is as follows. In Sect. \ref{sec. preliminaries} we set the
notation and give all the necessary definitions. In particular, we
recall a definition of the Bessel process which appears as the
``vertical" component of the diffusion generated by
$a^{-2}\mathcal{L}_{-\gamma}$ on $N\times\R^+$ as well as the
notion of the evolution on $N$ generated by an appropriate
operator which appears as the ``horizontal" component of the
diffusion on $N\times\R^+$ mentioned in the Introduction above
(cf. \cite{DHU, pota}).
The main tool in the proof of \eqref{ge} is Proposition
\ref{mainproposition}. Its proof is given in Sect.~\ref{pl}.
Finally, in Sect. \ref{sec. proof of theorem} we state the
estimates \eqref{ge} precisely (see Theorem \eqref{main}) and we
give a sketch of its proof.
\section{Preliminaries}\label{sec. preliminaries}
\subsection{$NA$ groups} Good references for this topic are \cite{DHZ, DHU} and
\cite{FS}. Let $N$ be a connected and simply connected nilpotent
Lie group. Let $D$ be a derivation of the Lie algebra
$\mathfrak{n}$ of $N$. For every $a\in\R^+$ we define an
automorphism $\Phi_a$ of $\mathfrak{n}$ by the formula
\begin{equation*}
\Phi_a=e^{(\log a)D}.
\end{equation*}
Writing $x=\exp X$ we put
\begin{equation*}
\Phi_a(x):=\exp\Phi_a(X).
\end{equation*}
It is clear that $\Phi_a:N\to N$ defines an automorphism. Let
$\mathfrak{n}^\C$ be the complexification of $\mathfrak{n}$. Define
\begin{equation*}
\mathfrak{n}_\lambda^\C=\{X\in\mathfrak{n}^\C:\exists k>0\text{
such that }(D-\lambda I)^k=0\}.
\end{equation*}
Then
\begin{equation}\label{gradation}
\mathfrak{n}=\bigoplus_{\mathop{\rm Im}\lambda\geq 0}V_\lambda,
\end{equation}
where
\begin{equation*}
V_\lambda=
\begin{cases}
V_{\bar\lambda}=(\mathfrak{n}^\C\oplus\mathfrak{n}_{\bar\lambda}^\C)\cap\mathfrak{n}
&\text{if }\mathop{\rm Im}\lambda\not=0,\\
\mathfrak{n}_\lambda^\C\cap\mathfrak{n}&\text{if }\mathop{\rm Im}\lambda=0.
\end{cases}
\end{equation*}
We assume that the real parts $d_j$ of the eigenvalues $\lambda_j$
of the matrix $D$ are strictly greater than 0. We define the
number
\begin{equation}\label{qdef}
Q=\sum_j\text{Re }\lambda_j=\sum_j d_j
\end{equation}
and we refer to this as a \textit{``homogeneous dimension"} of $N$.
In this paper $D=\mathop{\rm ad}_{Y_0}$ (see Introduction). Under the
assumption on positivity of $d_j$, \eqref{gradation} is a
gradation of $\mathfrak{n}$.
We consider a group $S$ which is a \textit{semi-direct} product of
$N$ and the multiplicative group
$A=\mathbb{R}^+= \{\exp tY_0:t\in\R\}$:
\begin{equation*}
S=NA=\{xa:x\in N,a\in A\}
\end{equation*}
with multiplication given by the formula
\begin{equation*}
(xa)(yb)=(x\Phi_a(y)\ ab).
\end{equation*}
On $N$ we define a \textit{``homogeneous norm"}, $|\cdot|$ (cf.
\cite{DHZ,DHU}) as follows. Let $(\cdot,\cdot)$ be a fixed inner
product in $\mathfrak{n}$. We define a new inner product
\begin{equation}\label{product}
\langle X,Y\rangle =\int_0^1\Big(\Phi_a(X),\Phi_a(Y)\Big)\frac{d
a}{a}
\end{equation}
and the corresponding norm
\begin{equation*}
\|X\|=\langle X,X\rangle^{1\slash 2}.
\end{equation*}
We put
\begin{equation*}
|X|=\left(\inf\{a>0:\|\Phi_a(X)\|\geq 1\}\right)^{-1}.
\end{equation*}
One can easily show that for every $Y\ne 0$ there exists precisely
one $a>0$ such that $Y=\Phi_a(X)$ with $|X|=1$. Then we have
$|Y|=a$.
Finally, we define the homogeneous norm on $N$. For $x=\exp X$ we
put
\begin{equation*}
|x|=|X|.
\end{equation*}
Note that if the action of $A=\mathbb{R}^+$ on $N$ (given by
$\Phi_a$) is diagonal the norm we have just defined is the usual
homogeneous norm on $N$ and the number $Q$ in \eqref{qdef} is
simply the homogeneous dimension of $N$ (see \cite{FS}).
Having all that in mind we define appropriate derivatives (see
also \cite{DHZ}). We fix an inner product \eqref{product} in
$\mathfrak{n}$ so that $V_{\lambda_j}$, $j=1,\ldots,k$ are
mutually orthogonal and an orthonormal basis
$\mathcal{X}_1,\ldots,\mathcal{X}_n$ of $\mathfrak{n}$. The
enveloping algebra $\mathfrak{U}(\mathfrak{n})$ of $\mathfrak{n}$
is identified with the polynomials in
$\mathcal{X}_1,\ldots,\mathcal{X}_n$. In
$\mathfrak{U}(\mathfrak{n})$ we define $\langle
\mathcal{X}_1\otimes\ldots\otimes\mathcal{X}_r,\mathcal{Y}_1\otimes\ldots\otimes\mathcal{Y}_r\rangle
= \prod_{j=1}^r\langle \mathcal{X}_j,\mathcal{Y}_j\rangle$. Let
$V_j^r$ be the symmetric tensor product of $r$ copies of
$V_{\lambda_j}$. For $I=(i_1,\ldots,i_k)\in(\N\cup\{0\})^k$ let
\begin{equation*}
\mathcal{X}^I=\mathcal{X}_1^{(i_1)}\ldots\mathcal{X}_k^{(i_k)},\quad
\text{where }\mathcal{X}_j^{(i_j)}\in V_j^{i_j}.
\end{equation*}
Then for $\mathcal{X}\in V_{\lambda_j}$, \begin{equation*}
\|\Phi_a(\mathcal{X})\|\leq c\exp(d_j\log a+D_j\log(1+|\log a|)),
\end{equation*}
where $d_j=\text{Re}\lambda_j$ and $D_j=\dim V_{\lambda_j}-1$, and
so
\begin{equation}\label{den}
\|\Phi_a(\mathcal{X}^I)\|\leq\exp\Big(\sum_{j=1}^ki_j(d_j\log
a+D_j\log(1+|\log a|))\Big)\prod_{j=1}^k\|\mathcal{X}_j^{(i_j)}\|.
\end{equation}
\subsection{Bessel process}
Let $b_t$ denote the Bessel process with a parameter $\alpha\geq
0$ (cf. \cite{RY}), i.e., a continuous Markov process with the
state space $[0,+\infty)$ generated by
$\partial_a^2+\frac{2\alpha+1}{a}\partial_a$. The transition
function with respect to the measure $y^{2\alpha+1}dy$ is given
(cf. \cite{Borodin, RY}) by:
\begin{equation}\label{tf}
p_t(x,y)=\begin{cases}
\frac{1}{2t}\exp\left(\frac{-x^2-y^2}{4t}\right) I_\alpha\left(
\frac{xy}{2t}\right) \frac{1}{(xy)^\alpha}
&\text{for $x,y>0,$}\\
\frac{1}{2^\alpha (2t)^{\alpha+1}\Gamma(\alpha+1)}
\exp\left(\frac{-y^2}{4t}\right) &\text{for $x=0$, $y>0,$}
\end{cases}
\end{equation}
where
\begin{equation}\label{ab}
I_\alpha(x)= \sum_{k=0}^\infty\frac{(x\slash
2)^{2k+\alpha}}{k!\Gamma(k+\alpha+1)}
\end{equation}
is \textit{the Bessel function} (see \cite{L}). Therefore for
$x\geq 0$ and a measurable set $B\subset(0,\infty)$:
\begin{equation*}
\p_x(b_t\in B)=\int_Bp_t(x,y)y^{2\alpha+1}dy.
\end{equation*}
If $b_t$ is the Bessel process with a parameter $\alpha$ starting
from $x$, i.e., $b_0=x$, then we will write that
$b_t\in\mathop{\rm BESS}_x(\alpha)$ or simply $b_t\in\mathop{\rm BESS}(\alpha)$ if the
starting point is not important or is clear from the context.
Properties of the Bessel process are very well known and their
proofs are rather standard. They can be found e.g. in \cite{RY,
DHU, pams, Phd}. However, in our paper we will not explicitly make
use of any particular property of the Bessel process. What we only
need is the possibility of generalization of some lemmas from
Section 5 in \cite{pota}.
\subsection{Evolutions}
Let $X,X_1,\ldots, X_m$ be as in \eqref{operator}. Let $\sigma:
[0,\infty)\longrightarrow [0,\infty)$ be a continuous function
such that $\sigma(t)>0$ for every $t>0$. We consider the family of
evolutions operators $L_{\sigma(t)}-\partial_t$, where
\begin{equation}\label{operatorsigma}
L_{\sigma(t)}=\sigma(t)^{-2}\Big(\sum_j\Phi_{\sigma(t)}(X_j)^2+
\Phi_{\sigma(t)}(X)\Big).
\end{equation}
Since we can assume that $X_1,\ldots,X_m$ are linearly independent,
we can select $X_{m+1},\ldots, X_n$ so that $X_1,\ldots ,X_n$ form a
basis of $\mathfrak{n}$. For a multi-index $I=(i_1,\ldots,i_n)$, $i_j\in\Z^+$ and the basis $X_1,\ldots,X_n$ of the Lie algebra
$\mathfrak{n}$ of $N$ we write: $X^I=X_1^{i_1}\ldots X_n^{i_n}$
and $|I|=i_1+\ldots+i_n$. For $k=0,1,\ldots,\infty$ we define:
\begin{equation*}
C^k=\{f:X^If\in C(N),\text{ for }|I|r>s$. The
existence of the family $U^\sigma(s,t)$ follows from \cite{T}.
\section{Probabilistic Lemma}\label{pl}
For the rest of this article, as in \cite{pota},
we consider a Bessel process $\sigma_t$ with a
parameter 0, i.e., $\sigma_t\in\text{BESS}(0)$. In other words
$\sigma_t=\|w_t\|$, where $w_t$ is a standard Brownian motion on
$\R^2$.
In the whole section we use the following notation. For every
$\eta>0$ and $a>0$, $\I=[a-\eta,a+\eta]$. The kernel $p^\sigma$ is
the evolution kernel defined in the previous section and
``directed" by the Bessel process $\sigma_t\in\text{BESS}(0)$ and
is fixed for the whole section. Moreover, for every measurable
subset $B$ of $\R^+$, $m(B)=\int_Bydy$.
For a positive $\delta<1\slash 2$, let
\begin{gather*}
T_\delta=\{(x,a)\in N\times\R^+:1-\delta\delta>1\slash 2$, for every
$0<\chi_0\leq 1$, $00$ there exists a constant $C$ such that for every
$(x,a)\in\mathcal Q\setminus T_\delta$ and for every $0\leq l\leq
k-1$,
\begin{equation*}
\sup_{0<\eta<\delta\slash 2}\big|\int_1^\infty\e_\chi
X^Ip^\sigma(t,0)(x)\partial_a^lm(\I)^{-1}\partial_a^{k-l}1_{\I}(\sigma_t)dt\big|
\leq Ca^{-k}.
\end{equation*}
\end{proposition}
\begin{proof}[Sketch of the proof]
To prove Proposition \ref{mainproposition}, we divide the set
$\mathcal Q\setminus T_\delta$ into five subsets as in Lemmas
5.1-5.5 in \cite{pota} (in the case $\chi=1$). The case
$0<\chi\leq\chi_0$ is done similarly. What we have to notice is
the fact that the only difference between the above proposition
and corresponding Lemmas 5.1-5.5 in \cite{pota} is the appearance
of derivatives with respect to $a$ variable and in some sense
uniform estimate with respect to the starting point
$\chi\leq\chi_0$. To overcame this difficulty we notice the
following.
For every $\eta0$,
\begin{equation}\label{da1e}
\begin{split}
\e_\chi\partial_a1_{\I}(\sigma_t)
=&\lim_{h\to 0}h^{-1}(\p_\chi(a+\eta\leq\sigma_t\leq a+\eta+h)\\
&-\p_\chi(a-\eta+h\leq\sigma_t\leq a-\eta))\\
=&p_t(\chi,a+\eta)(a+\eta)-p_t(\chi,a-\eta)(a-\eta),
\end{split}
\end{equation}
where $p_t$ is the transition function \eqref{tf}. Formula
\eqref{da1e} together with \eqref{tf} allow us to calculate
$\e_\chi\partial_a^l1_{\I}(\sigma_t)$ for $l\geq 2$, \begin{equation}\label{dale}
\begin{split}
\e_\chi\partial_a^l1_{\I}(\sigma_t)=&(2t)^{-1}e^{-\chi^2\slash
4t}\partial_a^{l-1}(e^{-(a+\eta)^2\slash 4t}I_{0}(\chi
(a+\eta)\slash 2t)
(a+\eta)\\
&-e^{-(a-\eta)^2\slash 4t}I_{1}(\chi (a-\eta)\slash 2t)(a-\eta)).
\end{split}
\end{equation}
Since we get $\lim_{\eta\to 0}\frac{(C\eta)^l}{m
(\I)^l}=\partial_a^lm(\I)$. Using \eqref{da1e}, \eqref{dale} and
the following formulae (cf. \cite{L}):
\begin{equation*}
\frac{d}{dx}I_\alpha(x)=I_{\alpha-1}(x)-\frac{\alpha}{x}I_\alpha(x)=I_{\alpha+1}(x)+\frac{\alpha}{x}I_\alpha(x)
\end{equation*}
we get, after not difficult but a little tedious computation,
that
\begin{multline}\label{*}
\lim_{\eta\to 0}|\e_{\chi}m
(\I)^{-1}\partial_a^{k-l}1_{\I}(b_{t\slash 2})|\leq
Ct^{-k+l-2}a^{-k+l}e^{-\chi^2\slash 4t}e^{-a^2\slash 4t}\\
\times\sum_{(w_1,w_2,w_3,w_4)\in W}
c_{w_1,w_2,w_3,w_4}\chi^{w_1}a^{w_2}t^{w_3}I_{w_4}(\chi a\slash
2t),
\end{multline}
where
\begin{multline*}
W=\big\{(w_1,w_2,w_3,w_4):0\leq w_1\leq k-l+1,0\leq w_2\leq 2(k-l)+1,\\
0\leq w_3\leq k-l,w_4\in\{0,1\}\text{ and }w_1\slash 2+w_3