\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 44, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/44\hfil Integration by parts]
{Integration by parts for the $L^r$ Henstock-Kurzweil integral}

\author[P. Musial, F. Tulone \hfil EJDE-2015/44\hfilneg]
{Paul Musial, Francesco Tulone}

\address{Paul Musial \newline
Department of Mathematics and Computer Science,
Chicago State University \newline
9501 South King Drive, Chicago, Illinois 60628, USA}
\email{pmusial@csu.edu}

\address{Francesco Tulone \newline
Department of Mathematics and Computer Science \newline
University of Palermo,
Via Archirafi, 34, 90132 Palermo, Italy}
\email{francesco.tulone@unipa.it}

\thanks{Submitted December 8, 2014. Published February 16, 2015.}
\subjclass[2000]{26A39}
\keywords{Henstock-Kurzweil; integration by parts}

\begin{abstract}
 Musial and Sagher \cite{Musial Sagher} described a
 Henstock-Kurzweil type integral that integrates $L^r$-derivatives.
 In this article, we develop a product rule for the $L^r$-derivative and
 then an integration by parts formula.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks


\section{Introduction}

\begin{definition}[\cite{Musial Sagher}]\rm 
A real-valued function $f$ defined on $[a,b]$ is said to be 
$L^r$ Henstock-Kurzweil integrable ($f\in HK_r[ a,b] $) if there exists a function
$F\in L^r[ a,b] $ so that for any $\varepsilon >0$ there exists a
gauge function $\delta ( x) >0$ so that whenever 
$\{ ( x_i,[c_i,d_i] ) \} $ is a $\delta $-fine tagged partition
of $[ a,b] $ we have
\[
\sum_{i=1}^{n}\Big( \frac{1}{d_i-c_i}( L)
\int_{c_i}^{d_i}| F( y) -F( x_i)
-f( x_i) ( y-x_i) | ^rdy\Big) ^{1/r}<\varepsilon .
\]
\end{definition}

In the sequel, if an integral is not specified, it is a Lebesgue integral.
 It is shown in \cite{Musial Sagher} that if $f$ is $HK_r$-integrable
 on $[ a,b] $, the following function is well-defined for all $x\in [ a,b] $:
\begin{equation}
F( x) =( HK_r) \int_{a}^{x}f( t) \,dt
\label{HKrprim}
\end{equation}

Here the function $F$ is called the {\it indefinite} $HK_r$ {\it integral of} $f$.
Our aim is to establish an integration by parts formula for the $
HK_r $ integral.  In a manner similar to L. Gordon \cite{LGordon} we state the following

\begin{theorem}\label{thm-int-by-parts}
Suppose that $f$ is $HK_r$-integrable on $[ a,b] $, and $G$ is
absolutely continuous on $[ a,b] $ with $G'\in
L^{r'}( [ a,b] ) $, where $1\leq r<\infty ,r'=r/(r-1)$ if $r>1$, and
$r'=\infty$ if $r=1$.  Then $fG$ is $HK_r$-integrable on $[ a,b] $ and if
$F$ is the indefinite $HK_r$ integral of $f$, then
\[
( HK_r) \int_{a}^{b}f( t) G( t)\,dt
=F( b) G( b) -\int_{a}^{b}F(t) G'( t) \,dt.
\]
\end{theorem}

We note that if $r=1$ so that $r'=\infty$, the condition on $G$ is that 
it is a Lipschitz function of order 1 on $[a,b]$.

In the classical case where $f$ is Henstock-Kurzweil integrable 
($r=\infty,r' =1$), Theorem \ref{thm-int-by-parts} holds, but it is 
enough to assume that $G$ is of bounded variation on $[a,b]$. 
 In that case the integral on the right is the Riemann-Stieltjes 
 integral $\int_{a}^{b}F dG$. See \cite{R Gordon} for a proof of this statement.


To prove Theorem \ref{thm-int-by-parts} we will need a product rule 
for the $L^r$-derivative.
 We will also utilize a characterization of the space of 
$HK_r$-integrable functions that involves generalized absolute
continuity in $L^r $ sense ($ACG_r( [ a,b] ) )$.

\section{Product rule for the $L^r$-derivative}

\begin{definition}[\cite{Calderon Zygmund}]\rm 
For $1\leq r<\infty $, a function $F\in L^r( [ a,b]) $ is said to be
$L^r$-differentiable at $x\in [ a,b] $
if there exists $a\in \mathbb{R}$ such that
\[
\int_{-h}^{h}| F( x+t) -F( x) -at|^rdt=o( h^{r+1}).
\]
\end{definition}

It is clear that if such a number $a$ exists, then it is unique. 
 We say that $a$ is the {\it{$L^r$-derivative}} of $F$ at $x$, and denote
the value $a$ by $F_r'( x)$.


\begin{theorem}\label{Thm:Lr-product-rule}
For $1\leq r<\infty $, let $x\in \mathbb{R}$ and suppose $F\in L^r( I) $
 where $I$ is an interval having $x$ in its interior, and suppose $F$ is
 $L^r$-differentiable at $x$. Suppose
also that $G\in L^{\infty }( I) $ and that $G$ is $L^r$-differentiable at $x$.
Then $FG$ is $L^r$-differentiable at $x$ and
$( FG) _r'( x) =F_r'(x) G( x) +F( x) G_r'( x) $.
\end{theorem}

\begin{proof} 
Let $\varepsilon >0$.  We need to choose $\gamma $ so that for 
$0<h<\gamma $
\begin{equation} \label{E:Product Rule}
\int_{-h}^{h}| F( x+t) G( x+t) -F(x) G( x) - H(x)\\
t| ^rdt < \varepsilon h^{r+1}
\end{equation}
where $H(x)= F_r'( x) G(x)+F( x) G_r'( x)$. We add and subtract
the terms $F( x) G( x+t) $ and $F_r'( x) G( x+t)t $ to the part of the
integrand inside the absolute value signs. 
 We also note that if $a,b$ and $c$ are non-negative numbers then
\[
( a+b+c) ^r\leq C( a^r+b^r+c^r)
\]
where $C$ is a positive constant that depends on $r$.

Choose $\gamma _{0}>0$ and $N>0$ so that 
$F\in L^r( [ x-\gamma_{0},x+\gamma _{0}] ) $ and that
\[
\operatorname{esssup}_{[ x-\gamma _{0},x+\gamma _{0}] }G<N.
\]
We then have that if $0<h<\gamma _{0}$ then the integral in 
\eqref{E:Product Rule} is less than
or equal to
\begin{align}
&C\int_{-h}^{h}| G( x+t) | ^r|
F( x+t) -F( x) -F_r'( x)
t| ^rdt  \label{First line of inequality} \\
&+C\int_{-h}^{h}| F( x) | ^r|
G( x+t) -G( x) -G_r'( x)
t| ^rdt  \label{Second line of inequality} \\
&+C\int_{-h}^{h}| F_r'( x) |
^r| ( G( x+t) -G( x) )
t| ^rdt.  \label{Third line of inequality}
\end{align}
For \eqref{First line of inequality}, choose $\gamma _{1}<$ $\gamma _{0}$ so
that if $0<h<\gamma _{1}$ we have
\[
\int_{-h}^{h}| F( x+t)-F( x) -F_r'( x) t| ^rdt
<\frac{\varepsilon h^{r+1}}{4CN^r}
\]
so that
\[
C\int_{-h}^{h}| G( x+t) | ^r|
F( x+t) -F( x) -F_r'( x)
t| ^rdt<\frac{\varepsilon h^{r+1}}{4}.
\]

For \eqref{Second line of inequality}, choose $\gamma _{2}<$ $\gamma _{1}$
so that if $0<h<\gamma _{2}$ we have
\[
\int_{-h}^{h}| G( x+t) -G( x) -G_r'( x) t| ^rdt<\frac{\varepsilon h^{r+1}}{4C(
| F( x) | ^r+1) }
\]
so that
\[
C\int_{-h}^{h}| F( x) | ^r| G(
x+t) -G( x) -G_r'( x) t|
^rdt<\frac{\varepsilon h^{r+1}}{4}
\]
For \eqref{Third line of inequality}, we note that
\begin{align*}
&C\int_{-h}^{h}| F_r'( x) |^r| ( G( x+t) -G( x) )t| ^rdt \\
&= C| F_r'( x) |^r\int_{-h}^{h}| ( G( x+t) -G( x)
-G_r'( x) t+G_r'( x) t) t| ^rdt \\
&\leq  C^2| F_r'( x) |^rh^r\Big( \int_{-h}^{h}| ( G( x+t) -G(x) -G_r'( x) t) | ^rdt
\\
&\quad +\int_{-h}^{h}| G_r'( x) t| ^rdt\Big) \\
&\leq C^2| F_r'( x) |^rh^r\Big( \int_{-h}^{h}| ( G( x+t) -G(
x) -G_r'( x) t) | ^rdt\Big)
\\
&\quad +2C^2| F_r'( x) |^rh^{2r+1}| G_r'( x) | ^r.
\end{align*}
Now we note that we can choose
\[
0<\gamma <\min \Big( 1,\gamma _{2},\big(\varepsilon /
\big( 8C^2(| G_r'( x) | +1) (| F_r'( x) | +1) \big)\big)^{1/r}\Big)
\]
so that if $0<h<\gamma $ we have
\[
\Big( \int_{-h}^{h}| ( G( x+t) -G( x)
-G_r'( x) t) | ^rdt\Big) 
<\frac{\varepsilon h^{r+1}}{4C^2( | F_r'( x)| ^r+1) }
\]

We then have that if $0<h<\gamma $, then
\begin{align*}
&C^2| F_r'( x) | ^rh^r\Big(\int_{-h}^{h}| ( G( x+t) -G( x)
-G_r'( x) t) | ^rdt\Big) \\
&<( C^2| F_r'( x) |
^rh^r) \Big( \frac{\varepsilon h^{r+1}}{4C^2( |F_r'( x) | ^r+1) }\Big) \\
&\leq \frac{\varepsilon h^{2r+1}}{4}<\frac{\varepsilon h^{r+1}}{4}
\end{align*}
and that
\begin{align*}
&2C^2| F_r'( x) |
^rh^{2r+1}| G_r'( x) | ^r \\
&\leq 2C^2| F_r'( x) |^rh^{r+1}| G_r'( x) | ^r
\Big(\frac{\varepsilon }{8C^2( | F_r'( x)| +1) ( | G_r'( x) | +1) }\Big) \\
&\leq \frac{\varepsilon h^{r+1}}{4}.
\end{align*}
We can then conclude that \eqref{E:Product Rule} holds and 
the theorem is therefore proved.
\end{proof}

In \cite{Musial Sagher} we find sufficient conditions for $HK_r$-integrability. 
 We will need the following definitions.

\begin{definition}[\cite{Musial Sagher}]\rm 
 We say that $F\in AC_r( E) $ if for all $\varepsilon >0$ there exist $\eta >0$ 
and a gauge function $\delta (x) $ defined on $E$ so that if 
$\mathcal{P}=\{ (x_i,[c_i,d_i]) \} $ is a finite collection of non-overlapping 
$\delta $-fine tagged intervals having tags in $E$ and satisfying
\[
\sum_{i=1}^{q}( d_i-c_i) <\eta
\]
then
\[
\sum_{i=1}^{q}\Big( \frac{1}{d_i-c_i}\int_{c_i}^{d_i}|
F( y) -F( x_i) | ^rdy\Big) ^{1/r}<\varepsilon .
\]
\end{definition}


\begin{definition}[\cite{Musial Sagher}] \rm  
We say that $F\in ACG_r( E) $ if $E$ can be written
\[
E=\cup_{i=1}^{\infty }E_i
\]
and $F\in AC_r( E_i)$ for all $i$.
\end{definition}

\begin{lemma} \label{T:ACG products} 
Suppose that $F$ and $G$ are in $ACG_r( [ a,b] ) $, 
and that $G\in L^{\infty }( [ a,b]) $. 
 Then $FG\in ACG_r( [ a,b] ) $.
\end{lemma}

\begin{proof} 
The function $F\in ACG_r( [ a,b] ) $ and so
we can find a sequence of sets $\{ A_{n}\} _{n=1}^{\infty }$ so
that $[ a,b] =\cup _{n=1}^{\infty }A_{n}$ and 
$F\in AC_r( A_{n}) $ for all $n$. 
Since $G$ belongs to $ACG_r( [ a,b]) $, we can also find a sequence of sets 
$\{ B_{m}\} _{m=1}^{\infty }$ so that $[ a,b] =\cup _{m=1}^{\infty }B_{m}$
and $G\in AC_r( B_{m}) $ for all $m$.  We can then write
\[
[ a,b] =\cup_{n=1}^{\infty }\cup_{m=1}^{\infty }( A_{n}\cap B_{m}) .
\]

We will rewrite the sequence $\{ A_{n}\cap B_{m}\} _{n,m\geq 1}$
as $\{ E_{k}\} _{k\geq 1}$.  We then have that both $F$ and $G$
are in $AC_r( E_{k}) $ for all $k\geq 1$.  We will show that 
$FG\in ACG_r(  E_{k} ) $ for all $k$.

Let $N=1+\left\Vert G\right\Vert _{\infty }$ and fix $k$. \ For $j\geq 1$ let
\[
U_{j}=\{ x\in E_{k}:j-1\leq | F( x) |
<j\}
\]
We then have
\[
E_{k}=\cup_{j=1}^{\infty }U_{j}.
\]
We will show that $FG\in AC_r( U_{j}) $ for all $j$.

Let $\varepsilon >0$. There exist $\eta >0$ and a gauge function 
$\delta ( x) $ defined on $U_{j}$ so that if
$\mathcal{P}=\{ x_i,[ c_i,d_i] \} $ is a finite
collection of non-overlapping $\delta $-fine tagged intervals having 
tags in $U_{j}$ and satisfying
\[
\sum_{i=1}^{q}( d_i-c_i) <\eta
\]
then
\begin{gather*}
\sum_{i=1}^{q}\Big( \frac{1}{d_i-c_i}\int_{c_i}^{d_i}|
F( y) -F( x_i) | ^rdy\Big) ^{1/r}<\frac{\varepsilon }{2N}, 
\\
\sum_{i=1}^{q}\Big( \frac{1}{d_i-c_i}\int_{c_i}^{d_i}|
G( y) -G( x_i) | ^rdy\Big) ^{1/r}<\frac{\varepsilon }{2j}.
\end{gather*}
Then for such $\mathcal{P}$,
\begin{align*}
&\sum_{i=1}^{q}\Big( \frac{1}{d_i-c_i}\int_{c_i}^{d_i}|
F( y) G( y) -F( x_i) G( x_i) | ^rdy\Big) ^{1/r} 
\\
&\leq \sum_{i=1}^{q}\Big( \frac{1}{d_i-c_i}\int_{c_i}^{d_i}
| F( y) G( y) -F( x_i) G(y) | ^rdy\Big) ^{1/r} 
\\
&\quad +\sum_{i=1}^{q}\Big( \frac{1}{d_i-c_i}\int_{c_i}^{d_i}|
F( x_i) G( y) -F( x_i) G(x_i) | ^rdy\Big) ^{1/r}. 
\\
&\leq N\Big( \sum_{i=1}^{q}\Big( \frac{1}{d_i-c_i}
\int_{c_i}^{d_i}| F( y) -F( x_i)| ^rdy\Big) ^{1/r}\Big) 
\\
&\quad +| F( x_i) | \Big( \sum_{i=1}^{q}(
\frac{1}{d_i-c_i}\int_{c_i}^{d_i}| G( y)
-G( x_i) | ^rdy) ^{1/r}\Big) 
\\
&\leq N\big( \frac{\varepsilon }{2N}\big) + j( \frac{\varepsilon }{2j})=\varepsilon.
\end{align*}
Now we can conclude that for $\mathcal{P}$,
\[
\sum_{i=1}^{q}\Big( \frac{1}{d_i-c_i}\int_{c_i}^{d_i}|
F( y) G( y) -F( x_i) G( x_i)
| ^rdy\Big) ^{1/r}<\varepsilon
\]
and so that $FG\in ACG_r( [ a,b] ) .$\bigskip
\end{proof}

\section{Linearity of $ACG_r(E)$}

We now show that $ACG_r(E)$ is a linear space.

\begin{theorem} \label{T:Linearity of ACGr} 
Suppose $F$ and $G$ are in $ACG_r( E) $.  Then
for any constants $a$ and $b$ we have that $aF+bG\in ACG_r( E)$.
\end{theorem}

\begin{proof} 
Write $E$ as $\cup _{n=1}^{\infty }E_{n}$.  We will show that $
aF+bG\in AC_r( E_{n}) $ for every $n$.

First we  show that $aF\in AC_r( E_{n}) $.  Let $\varepsilon >0$ and 
choose $\eta >0$ and a gauge function $\delta (x) $ defined on $E_{n}$ 
so that if $\mathcal{P}=\{ x_i,[c_i,d_i] \} $ is a finite collection 
of non-overlapping $\delta $-fine tagged intervals having tags in $E$ 
and satisfying
\[
\sum_{i=1}^{q}( d_i-c_i) <\eta
\]
then
\[
\sum_{i=1}^{q}\Big( \frac{1}{d_i-c_i}\int_{c_i}^{d_i}|
F( y) -F( x_i) | ^rdy\Big) ^{1/r
}<\frac{\varepsilon }{| a| +1}.
\]
Then
\begin{align*}
&\sum_{i=1}^{q}\Big( \frac{1}{d_i-c_i}\int_{c_i}^{d_i}|
aF( y) -aF( x_i) | ^rdy\Big) ^{1/r} \\
&=| a| \Big( \sum_{i=1}^{q}\Big( \frac{1}{d_i-c_i}
\int_{c_i}^{d_i}| F( y) -F( x_i)
| ^rdy\Big) ^{1/r}\Big) \\
&< | a| \big( \frac{\varepsilon }{|
a| +1}\big) <\varepsilon .
\end{align*}

Now we  show that $F+G\in ACG_r( E) $.  Let $\varepsilon
>0 $ and choose $\eta >0$ and a gauge function $\delta ( x) $
defined on $E_{n}$ so that if $\mathcal{P}=\{ x_i,[ c_i,d_i
] \} $ is a finite collection of non-overlapping $\delta $-fine
tagged intervals having tags in $E$ and satisfying
\[
\sum_{i=1}^{q}( d_i-c_i) <\eta\,,
\]
then 
\begin{gather*}
\sum_{i=1}^{q}\Big( \frac{1}{d_i-c_i}\int_{c_i}^{d_i}|
F( y) -F( x_i) | ^rdy\Big) ^{1/r} <\frac{\varepsilon }{2},\\
\sum_{i=1}^{q}\Big( \frac{1}{d_i-c_i}\int_{c_i}^{d_i}|
G( y) -G( x_i) | ^rdy\Big) ^{1/r}<\frac{\varepsilon }{2}.
\end{gather*}
Then we have for this $\mathcal{P}$, using Minkowski's inequality,
\begin{align*}
&\sum_{i=1}^{q}\Big( \frac{1}{d_i-c_i}\int_{c_i}^{d_i}|
F( y) +G( y) -(F( x_i) +G(
x_i))| ^rdy\Big) ^{1/r} \\
&\leq \sum_{i=1}^{q}\Big( \frac{1}{d_i-c_i}\int_{c_i}^{d_i}
| F( y) +F( x_i) | ^rdy\Big) ^{1/r} \\
&\quad +\sum_{i=1}^{q}\Big( \frac{1}{d_i-c_i}\int_{c_i}^{d_i}|
G( y) -G( x_i) | ^rdy\Big) ^{1/r} \\
&< \frac{\varepsilon }{2}+\frac{\varepsilon }{2}=\varepsilon .
\end{align*}
\end{proof}

We will use the following characterization of $HK_r$-integrable functions.

\begin{theorem}[\cite{Musial Sagher}] \label{T:Sufficient Conditions HKr} 
 Let $1\leq r<\infty $.  A function $f$ is $HK_r$-integrable
on $[ a,b] $ if and only if there exists a function $F\in ACG_r( [ a,b] ) $ 
so that $F_r^{'}=f$ a.e.
\end{theorem}

\section{Integration by Parts}

We are now ready to give the proof of Theorem \ref{thm-int-by-parts}.

\begin{proof} Define
\begin{gather*}
V( x) =f( x) G( x),\\
J( x) =F( x) G( x) -\int_{a}^{x}F( t) G'( t) \,dt.
\end{gather*}
We note that $FG' $ is integrable by H\"{o}lder's inequality 
\cite{Wheedan and Zygmund}.  Our task is to show that $J$ is the $HK_r$-integral 
of $V$.  By Theorem \ref{T:Sufficient Conditions HKr}, we see that it is 
sufficient to demonstrate that $J\in ACG_r([a,b])$ and that $J_r^{'}=V$ a.e.

We note that the function
\[
\int_{a}^{x}F( t) G'( t) \,dt
\]
is absolutely continuous on $[ a,b] $ and therefore is in 
$ACG_r( [ a,b] )$ \cite{Musial Sagher}. 
 Its derivative, and therefore its $L^r$-derivative, is equal to 
$F( x) G'(x) $ a.e. in $[ a,b] $.

Using Theorem \ref{Thm:Lr-product-rule} we can see that $FG$ has 
an $L^r$-derivative equal to $F_r'G+FG'$ a.e. in $[ a,b] $.  Using the
linearity of the $L^r$-derivative, we have that $J_r^{'}=V$
a.e.  Thus all that remains is to show that $J\in ACG_r( [ a,b]) $. 
By Theorem \ref{T:Linearity of ACGr} it is sufficient to show that 
$FG\in ACG_r( [ a,b] ) $.

The function $F\in ACG_r( [ a,b] ) $.  Since $G\in AC([a,b])$, it is also
 in $ACG_r( [ a,b] ) $ and $G$ is also in $L^{\infty }$
so by Lemma \ref{T:ACG products}, $FG\in ACG_r( [ a,b] )$ and 
Theorem \ref{thm-int-by-parts} is proved.
\end{proof}

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Calderon, A. P.; Zygmund, A.;
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\bibitem{LGordon} Gordon, L.;
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Studia Math. \textbf{28} (1966/1967), pp. 295-316.

\bibitem{R Gordon} Gordon, R. A.;
\emph{The Integrals of Lebesgue, Denjoy, Perron, and Henstock},
 Grad. Stud. Math. 4, Amer. Math. Soc., 1994.

\bibitem{Musial Sagher} Musial, P.; Sagher, Y.;
\emph{The $L^r$ Henstock-Kurzweil integral}, 
Stud. Math. \textbf{160} (1) (2004), pp. 53-81.

\bibitem{Wheedan and Zygmund} Wheedan, R.; Zygmund, A.;
\emph{Measure and Integral}, Marcel Dekker, Inc., New York, 1977.

\end{thebibliography}

\end{document}