\documentclass[12pt, reqno]{amsart}
\usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color}
\usepackage[bookmarksnumbered, plainpages]{hyperref}
 \textheight 22.5truecm \textwidth 14.5truecm
\setlength{\oddsidemargin}{0.35in}\setlength{\evensidemargin}{0.35in}
 \setlength{\topmargin}{-.5cm}
\newtheorem{theorem}{Theorem}[section]
\newtheorem*{theorema}{Theorem A}
\newtheorem*{theoremb}{Theorem B}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}
\newtheorem{problem}[theorem]{Problem}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{remarks}[theorem]{Remarks}
\numberwithin{equation}{section}
\newcommand{\D}{\mathcal{D}}
\newcommand{\R}{\mathcal{R}}
\newcommand{\N}{\mathcal{N}}
\newcommand{\Hi}{\mathcal{H}}
\newcommand{\G}{\mathcal{G}}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\I}{\mathcal{I}}
\newcommand{\T}{\mathcal{T}}
\newcommand{\eps}{\epsilon}
\newcommand{\bal}{\begin{align}}
\newcommand{\eal}{\end{align}}
\newcommand{\benu}{\begin{enumerate}}
\newcommand{\eenu}{\end{enumerate}}

\begin{document}
\setcounter{page}{42}

\noindent\parbox{2.85cm}{\includegraphics*[keepaspectratio=true,scale=1.75]{BJMA.jpg}}
\noindent\parbox{4.85in}{\hspace{0.1mm}\\[1.5cm]\noindent
Banach J. Math. Anal. 2 (2008), no. 2, 42--58 \\
$\frac{\rule{4.55in}{0.05in}}{{}}$\\
{\footnotesize
\textcolor[rgb]{0.65,0.00,0.95}{\textsc{\textbf{\large{B}}anach
\textbf{\large{J}}ournal of \textbf{\large{M}}athematical
\textbf{\large{A}}nalysis}}\\
ISSN: 1735-8787 (electronic)\\
\textcolor[rgb]{0.00,0.00,0.84}{\textbf{http://www.math-analysis.org }}\\
$\frac{{}}{\rule{4.55in}{0.05in}}$}\\[.5in]}

\title
[Sum Inequalities in $\mathbf{L}^{\mathbf{p}}$ spaces]{Some weighted
sum and product inequalities in $\mathbf{L}^{\mathbf{p}}$ spaces and
their applications}
\author[R.C. Brown ]{R. C. Brown}

\address{Department of Mathematics, University
of Alabama-Tuscaloosa, AL 35487-0350, USA}
\email{\textcolor[rgb]{0.00,0.00,0.84}{dicbrown@bama.ua.edu}}
\dedicatory{This paper is dedicated to Professor Joseph E.
Pe\v{c}ari\'{c}\\ \vspace{.5cm} {\rm Submitted by Th. M. Rassias}}
 \subjclass[2000]{Primary: 26D10, 47A30, 34B24; Secondary
47E05}  \keywords{Weighted sum inequalities, weighted product
inequalities, Sturm Liouville operators, limit-point conditions,
relatively bounded perturbations}
\date{Received: 12 April 2008; Accepted 21 April 2008.}

\begin{abstract}
We survey some old and new results concerning weighted norm
inequalities of sum and product form and apply the theory to obtain
limit-point conditions for second order differential operators of
Sturm-Liouville form  defined in $L^p$ spaces. We also extend
results of Anderson and Hinton by giving necessary and sufficient
criteria that perturbations of such operators be relatively bounded.
Our work is in part a generalization of the classical Hilbert space
theory of Sturm-Liouville operators to a Banach space setting.
\end{abstract} \maketitle

\section{Introduction}
Let $w, v_0, v_1$ be positive a.e. measurable or ``weights'' on the
interval  $I_a=[a,\infty)$, $a>-\infty$. We are interested in
obtaining conditions which guarantee the validity of the weighted
``sum" inequality:
\begin{eqnarray}\label{Sumineq1}
 \int_{I_a} w| y^{(j)}| ^p \leq
K_1(\epsilon )\int_{I_a} v_0|y|^p +\epsilon \int_{I_a}
v_1|y^{(n)}|^p
\end{eqnarray}
for $0\le j<n$ where $1\le p\le \infty$ and $\epsilon\in
(0,\epsilon_0)$. The space of functions $\D^p(v_0,v_1;I_a)$ on which
\eqref{Sumineq1} holds is defined by
\begin{eqnarray*}
\D^p(v_0,v_1;I_a):=\left\{ y:y\in AC^{n-1}(I_a);\int_{I_a}
v_0|y|^p,\int_{I_a} v_1|y^{(n)}|^p<\infty \right\}\notag
\end{eqnarray*}
where $AC^j(I_a)$ denotes the class of functions whose $j$-th
derivative is locally absolutely continuous on $I_a$. We shall also
show the inequality \eqref{Sumineq1} often implies the ``product"
inequality
\begin{eqnarray*}
\int_{I_a} v_1|y^{(j)}|^p\le
K_2\left(\int_{I_a}v_1|y|^p\right)^\frac{n-j}{n}\left(
\int_{I_a}v_1|y^{(n)}|^p\right)^\frac{j}{n}.
\end{eqnarray*}
It will turn out that \eqref{Sumineq1} has a number of interesting
applications to problems in concerning second-order differential
operators determined by symmetric expressions of the form
$-(ry')'+qy$ and defined in $L^p$ spaces. The results generalize
both some aspects of the Hilbert theory presented in the book of
Naimark \cite{Nai} and criteria obtained by Anderson and Hinton in
\cite{AH} that perturbations of such operators in the $L^2$ setting
be relatively bounded.  We close  this section with  a few remarks
on notation.  Upper case letters such as $K$ or $C$ denote constants
whose value may change from line to line. We distinguish between
different constants by writing $K_1, K_2, C, C_1, \dots$, etc.
$K(\cdot)$ indicates dependence on a parameter, e.g.,
$K_1(\epsilon)$. $L^p(I_a)$ signifies the (complex) $L^p$ space on
$I_a$ having the norm
$$
||u||_{p,I_a}:= \left(\int_{I_a} |u|^p\right)^{1/p}.
$$
$C^\infty(I_a)$ and $C^j(I_a)$ respectively denote the infinitely or
$j$-fold differentiable functions having continuous $j$-th
derivative on $I_a$ and $C_0^\infty(I_a)$ or $C^j_0(I_a)$ consists
of the subspace of $C^\infty(I_a)$ or $C^j(I_a)$ having compact
support. A local property is indicated by the subscript ``loc",
e.g., $f\in L^p_{\text{loc}}(I)$, etc. Also, we write $AC^0(I_a)$ as
$AC(I_a)$ and $L^1(I_a)$ as $L(I_a)$, and when the context is clear
we abbreviate $\mathcal{D}^p(v_0, v_1;I_a)$ by ``$\D^p$." If $T:X\to
Y$ is an operator where $X$ and $Y$ are Banach spaces $\D(T)$,
$\R(T)$, $\G(T)$, and $\N(T)$ respectively denote the domain, range,
graph, and null space of $T$. Finally, if $f$ and $g$ are two
functions the notation $f\approx g$ means that there are constants
$C_1$ and $C_2$ such that $f\le C_1g$ and $g\le C_2f$.
\section{Some weighted norm inequalities of sum form}
Suppose that $f$ is a positive continuous function on $I_a$. Let
$J_{t,\epsilon}:=[t,t+\epsilon f(t)]$. For $1< p<\infty$ set
\begin{align}
S_1(t)&:= f^{-jp}\left\lbrack(\epsilon
f)^{-1}\int_{J_{t,\epsilon}}w\right\rbrack \left\lbrack(\epsilon
f)^{-1}\int_{J_{t,\epsilon}}v_0^{-p^{\prime}/p}\right\rbrack
^{p/p^\prime }\label{S1t}\\ S_2(t)&:=
f^{(n-j)p}\left\lbrack(\epsilon
f)^{-1}\int_{J_{t,\epsilon}}w\right\rbrack \left\lbrack(\epsilon
f)^{-1}\int_{J_{t,\epsilon}}v_1^{-{p^\prime}/p}\right\rbrack
^{p/{p^\prime}}\label{S2t}
\end{align}
where $p'=p/(p-1)$. In the case that  $p=1$ or $\infty$ some
modifications in these definitions are required.  If $p=1$ we
substitute the $L^\infty$ norm of $v^{-1}_i$, $i=0, 1$, on
$J_{t,\epsilon}$ for the integral term. For instance,
$$
S_1(t):= f^{-j}\left\lbrack(\epsilon
f)^{-1}\int_{J_{t,\epsilon}}w\right\rbrack \Vert
v^{-1}\Vert_{\infty, J_{t,\eps}}.
$$
And when $p=\infty$ we write
$$
S_1(t):= f^{-j}\Vert w\Vert_{\infty,J_{t,\eps}}\left\lbrack(\epsilon
f)^{-1}\int_{J_{t,\epsilon}}v_0^{-1}\right\rbrack.
$$
Similar changes  apply to $S_2(t)$.
\begin{theorema}\label{theoa} Suppose $1\le p\le\infty$.
If there exists $f$ and $\eps_0\in (0,\infty]$ such that
\begin{align}
S_1(\epsilon_0)&:=\sup _{t\in I_a,\, 0<\epsilon\le\epsilon_0}S_1(t)<\infty\label{S1}\\
S_2(\epsilon_0)&:=\sup _{t\in I,\, 0<\epsilon \leq
\epsilon_0}S_2(t)<\infty,
 \label{S2}
\end{align}
then the inequality
\begin{eqnarray}\label{Sumineq2}
\int_{I_a} w| y^{(j)}| ^p\leq K_1\left\{ \epsilon
^{-j\eta}\int_{I_a} v_0|y|^p +\epsilon^{(n-j)\eta}\int_{I_a}
v_1|y^{(n)}|^p\right\}
\end{eqnarray}
where $\eta=p$ if $1\le p<\infty$, $\eta=1$ if $p=\infty$ holds on
$\D^p(v_0, v_1; I_a)$ for $\epsilon\le \epsilon_0$, and $K_1\approx
\max\{S_1(\epsilon_0), S_2(\epsilon_0)\}.$
\end{theorema}
\begin{proof} For $1\le p<\infty$ this was shown in \cite{BH1}.
The main idea in the proof of Theorem A is to partition $I_a$ in the
following way. Let $t_0:=a$ and set $t_{j+1}=t_j+\epsilon f(t_j)$.
On each interval $J_{t_j, \epsilon}$ we start with the basic
interpolation inequality (see \cite[Lemma 2.1]{BH1})
\begin{eqnarray}\label{Interpeq}
|y^{(j)}(t)|\le K_1\left\{(\epsilon f)^{-(j+1)}\int_{J_{t_j,
\epsilon}}|y|+ (\epsilon f)^{n-(j+1)}\int_{J_{t_j,
\epsilon}}|y^{(n)}|\right\}.
\end{eqnarray}
We then raise both sides to the $p$-th power, apply the  inequality
$(A+B)^p\le 2^{p-1}(A^p+ B^p)$ to the right hand side, and use
H\"{o}lder's inequality to introduce the weights $v_0$, $v_1$. Next,
we multiply both sides by $w$ and integrate over $J_{t,\epsilon}$.
The functions $S_1(t)$ and $S_2(t)$ will naturally appear. We bound
them by $S_1$ and $S_2$ and add the resulting inequalities over all
the intervals to obtain \eqref{Sumineq2}. A requirement of this
argument is that the sequence $\{t_j\}$ have no finite limit point.
This is guaranteed by the continuity and positivity of $f$. For
$p=\infty$ H\"{o}lder's inequality, multiplication by $w$ in
\eqref{Interpeq}, and an easy estimate gives
\begin{align}
 w|y^{(j)}(t)|&\le K_1\left\{(\epsilon
f)^{-j}\Vert w\Vert_{\infty,J_{t_j, \epsilon}}\left[(\epsilon
f)^{-1}\int_{J_{t_j, \epsilon}}v_0^{-1}\right] \Vert
v_0y\Vert_{\infty, J_{t_j, \epsilon}}\right.\notag\\ &\quad+
\left.(\epsilon f)^{(n-j)}\Vert w\Vert_{\infty,J_{t_j, \epsilon}}
\left[(\epsilon f)^{-1}\int_{J_{t_j, \epsilon}}v_1^{-1}\right]\Vert
v_1y^{(n)}\Vert_{\infty, J_{t_j, \epsilon}}\right\}\notag\\ &\le
K_1\max\{S_1, S_2\}\left(\Vert v_0 y\Vert_{\infty, I_a}+ \Vert v_0
y^{(n)}u\Vert_{\infty, I_a}\right).\notag
\end{align}
Taking the $L^\infty$ norm of the left side completes the argument.
\end{proof}

Since the integrals or $L^\infty$ norms in the definitions of
$S_1(t)$ and $S_2(t)$ may be difficult to handle  we can replace
$S_1(t)$ and $S_2(t)$ by simpler expressions  provided the weights
satisfy a certain condition.
\begin{theorem}\label{Hdymax}
Let $f$ and $J_{t,\epsilon}$ be as above. Suppose that there is a
constant $K$ not depending on $t$ (but possibly on $\epsilon$) such
that
\begin{eqnarray}\label{wtcond}
\frac{v_0(s)}{v_0(t)}, \frac{v_1(s)}{v_1(t)}\ge K
\end{eqnarray}
a.e. on $J_{t,\epsilon}$. If $p=1, \infty$ assume also that
\begin{eqnarray}\label{wtcondw}
\frac{w(s)}{w(t)}\le K_1
\end{eqnarray}
a.e. on $J_{t,\epsilon}$.
  Then the sum
inequality \eqref{Sumineq2} holds on $\D^p$ for $1\le p<\infty$ if
\begin{align}
T_1(\eps_0)&:=\sup_{t\in I_a, 0<\epsilon\le\epsilon_0}f^{-jp}wv_0^{-1}<\infty\label{T1}\\
T_2(\eps_0)&:=\sup_{t\in
I_a,0<\epsilon\le\epsilon_0}f^{(n-j)p}wv_1^{-1}<\infty.\label{T2}
\end{align}
\end{theorem}
\begin{proof}
To prove this for $1<p<\infty$ we proceed as in the proof of Theorem
A beginning with the basic interpolation inequality
\eqref{Interpeq}. We then raise both sides of this inequality to the
$p$-th power, etc., and multiply by $w$. Next, using (\ref{wtcond})
to move $v_0^{-1/p}$ and $v_1^{-1/p}$ out of the integrals we get
that
\begin{align}
w|y^{(j)}|^p&\le K_2\max\{T_1(\epsilon_0),
T_2(\epsilon_0)\}2^{p-1}\left\{(\epsilon ^{-jp}\left((\epsilon
f)^{-1}\int_{J_{t,\epsilon}}v_0^{1/p}|y|\right)^p \right.\notag\\
&\quad\quad \left.+ \epsilon ^{(n-j)p}\left((\epsilon
f)^{-1}\int_{J_{t,\epsilon}}v_1^{1/p}|y^{(n)}|\right)^p\right\}.\notag
\end{align}
Finally, we integrate both sides over $I_a$ and apply the
Hardy-Littlewood Maximal Theorem (cf. \cite[Theorem 21.76]{HS} to
the two integral terms on the right-hand side. This gives
\eqref{Sumineq2}. The cases $p=1, \infty$ amount to  special cases
of Theorem A, where we use \eqref{wtcond} and \eqref{wtcondw} to
replace $S_1(\eps_0)$ and $S_2(\eps_0)$ by $T_1(\eps_0)$ and
$T_2(\eps_0)$.
\end{proof}
\begin{remark}\label{rem2.15}
Using a different argument it was shown in \cite{BH1} that
\eqref{Sumineq2} also remains true if $f$ is nondecreasing  and the
``semi-pointwise" averages
\begin{align}
R_1(\epsilon_0)&:=\sup_{t\in I_a,\, 0<\epsilon\le\epsilon_0}
f(t)^{-pj}w(t) \left[(\epsilon
f)^{-1}\int_{J_{t,\epsilon}}v_0^{-p^\prime /p}\right]^{p/p^\prime}
\label{R1}\\ R_2(\epsilon_0)&:=\sup_{t\in I_a,\, 0<\epsilon\le
\eps_0} f(t)^{p(n-j)}w(t) \left[(\epsilon
f)^{-1}\int_{J_{t,\epsilon}}v_1^{-p^\prime
/p}\right]^{p/p^\prime}\label{R2}
\end{align}
are finite.
\end{remark}
\begin{remark} Another possibility lies in the application of the
Besicovitch covering theorem. Let $I$ be some finite or infinite
interval. Suppose that each $t\in I$ is the center of an interval
$\Delta_{t,\eps}:=[t-\eps f(t)/2,t+\eps f(t)/2]$ contained in $I_a$
where $f$ is bounded if $I$ is infinite. Let ${\frak J}_\epsilon$
denote the collection of these intervals. It is then possible (see
\cite[Theorem 1.1, p.2]{Guz}) to extract from ${\frak J}_\epsilon$
finitely many families $\Gamma_1,\dots,\Gamma_l$ of disjoint
intervals in ${\frak J}_\epsilon$ whose union covers $I$. If we
redefine $S_1(t)$ and $S_2(t)$ by replacing the intervals
$J_{t,\eps}$ by $\Delta_{t,\eps}$, then \eqref{Sumineq2} is readily
seen to hold on each $\Gamma_j$, $1\le j\le l$. Hence,
$$
\int_{\Gamma_j} w| y^{(j)}| ^p\leq K_1\left\{ \epsilon
^{-j\eta}\int_I v_0|y|^p +\epsilon^{(n-j)\eta}\int_I
v_1|y^{(n)}|^p\right\},
$$
so that
$$
\int_I w| y^{(j)}|^p\leq lK_1\left\{ \epsilon ^{-j\eta}\int_I
v_0|y|^p +\epsilon^{(n-j)\eta}\int_I v_1|y^{(n)}|^p\right\}.
$$
\end{remark}

How can  conditions like (\ref{S1}), (\ref{S2}), \eqref{T1},
\eqref{T2}, or \eqref{R1}, \eqref{R2}  be verified? Essentially, as
we have already in part done in Theorem \ref{Hdymax},  we will want
to choose $f$ so that $v_i(s)\approx v_i(t)$, $i=0,1$, and
$w(s)\approx w(t)$ on $J_{t,\epsilon}$.  In a very general case this
can always be done as we now demonstrate.
\begin{proposition}\label{prop1} Suppose that $w, v_0$, and $v_1$
are continuous on $I_a$. Then there exists a positive function
$f^\ast$ depending on $t$ and possibly $\epsilon$ such
\begin{eqnarray}\label{wtcond3}
\frac12\le \frac{w(s)}{w(t)}, \frac{v_0(s)}{v_0(t)},
\frac{v_1(s)}{v_1(t)}\le \frac32
\end{eqnarray}
on $J_{t,\epsilon}$. Moreover, the sequence $\{t_j\}$ defined as in
Theorem A using $f^\ast$ has no finite limit point.
\end{proposition}
\begin{proof}  Given $t\ge a$, $\epsilon>0$, and for $i=0,1$ let
$f_i(t, \eps):=(s_i(t)-t)/\epsilon$ where
\begin{eqnarray}\label{f}
s_i(t)=\min\{t+\eps, \sup\{z>t: 3v_i(t)/2 \ge v_i(u)\ge v_i(t)/2
\text{\ for\ } u\in (t, z]\}.
\end{eqnarray}
Define $s_2(t)$ and $f_2(t, \eps)$ similarly for $w$ and set
$f^\ast(t, \eps)=\min\{f_i(t, \epsilon)\}$, $i=0, 1, 2$. With this
construction of $f^\ast$ \eqref{wtcond3} follows. To prove the
second assertion  Set
$$
s_{\ast,0}:=a <\dots< s_{\ast,j+1}:=s_{\ast,j}+\epsilon
f_\ast(s_{\ast,j},\eps)\equiv s_0(s_{\ast,j})
$$
 and suppose that $\{s_{\ast,j}\}$ converges to $\bar{s}_\ast<\infty$.
 We show that for all sufficiently large $j$ and for
$u\in (s_{\ast,j}, \bar{s}_\ast]$ that $3v_i(s_{\ast,j})/2 \ge
v_i(u)\ge v_i(s_{\ast,j})/2$. If this is not so then for every $j$
there is a $k>j$ and a $u^\ast \in [s_{\ast,k}, \bar{s}_\ast]$ such
that for one of the weights, say, $v_0$ either (i)
$v_0(s_{\ast,k})/2>v_0(u^\ast)$ or (ii)
$3v_0(s_{\ast,k})/2<v_0(u^\ast)$. But from the continuity and
positivity of $v_0$, given, say, $1/10>\mu>0$ there is a $j$ such
that for any $k>j$ and all $u\in [s_{\ast,k},\bar{s}_\ast]$ we have
that
$$
(1-\mu)v_0(\bar{s}_\ast)<v_0(u)<(1+\mu)v_0(\bar{s}_\ast).
$$
If (i) is true then
$$
(1-\mu)v_0(\bar{s}_\ast)<v_0(u^\ast)<v_0(s_{\ast,k})/2<(1+\mu)v_0(\bar{s}_\ast)/2
$$
so that $9/10<(1/2)(11/10)$ which is false. Similarly, if (ii) holds
we have
$$
(3/2)(1-\mu)v_0(\bar{s}_\ast)<(3/2)v_0(s_{\ast,k})<v_0(u^\ast)<(1+\mu)v_0)(\bar{s}_\ast)
$$
so that $27/20< 11/10$ which is also false. This argument shows that
$$
\bar{s}_\ast\le
s_\ast(s_{\ast,k})=s_{\ast,k+1}<s_\ast(s_{\ast,k+1}),
$$
and so $\bar{s}_\ast$ cannot be a limit point of the sequence
$\{s_{\ast,j}\}$.
\end{proof}

\begin{remark} With this definition of $f^\ast$ we see that
\begin{align}
S_1(t)&\approx f^{\ast -jp}wv_0^{-1}\label{ptwise1}\\
S_2(t)&\approx f^{\ast (n-j)p}wv_1^{-1}.\label{ptwise2}
\end{align}
It is even simpler to define $f^\ast$ so that \eqref{wtcond} holds
in Theorem \ref{Hdymax} since we can omit $w$ and only consider the
lower bounds in \eqref{f}. We omit the details. In particular, this
means that if  the weights are continuous then the integral
expressions \eqref{S1t} and \eqref{S2t} in Theorem A can in theory
always be replaced by the point evaluation expressions
\eqref{ptwise1} and \eqref{ptwise2}. Also, in Theorem \ref{Hdymax}
if the weights are continuous then the conditions \eqref{wtcond}
will be satisfied if $f^\ast$ is chosen. But while Proposition
\ref{prop1} is  of some theoretical interest it is usually not of
much practical use since  it is difficult to characterize $f^\ast$
in a convenient fashion from its definition. Fortunately, a
satisfactory substitute for $f^\ast$ is often suggested by the
particular weights $w, v_0,$ and $v_1$.
\end{remark}
\begin{example}\label{Ex1} Let $a=1$,
 $w(t)=t^\beta $, $v_0(t)=t^\gamma $, $v_1(t)=t^\alpha$ and
$f(t)=t^\delta$ where $\delta \leq 1$. Then
$$
1\leq \sup _{s\in J_{t,\epsilon}}st^{-1}\leq 1+\epsilon t^{\delta
-1}\leq 1+\epsilon .
$$
A calculation shows that $ S_1(\epsilon_0)$, $S_2(\epsilon_0)$ are
finite if
\begin{eqnarray}\label{eq2.5}
\beta \leq \min \left\{ \delta pj + \gamma ,-\delta (n-j)p+\alpha
\right\},
\end{eqnarray}
and any fixed $\epsilon_0$  (say $\epsilon_0=1$). In (\ref{eq2.5})
$\beta$ will be as large as possible relative to $\alpha $ and
$\gamma $ if $\delta $ is chosen by ``equality'', i.e.,
\begin{eqnarray}\label{eq2.6}
 \delta
=(\alpha -\gamma)/np\leq 1.
\end{eqnarray}
With this choice of $\delta$
$$
 \beta\le   \gamma \left(\frac{n-j}{n}\right)+
\alpha \left( \frac {j}{n}\right).
$$
\end{example}
\begin{example}\label{ex2.1} Let $w(t)=v_0(t)=v_1(t)=\text{e}^t$, $a\ge 0$, and $f(t)=1$.
Then $1\le \text{e}^s/\text{e}^t\le \epsilon$ on $J_{t,\eps}$.
\eqref{Sumineq2} follows.
\end{example}
It was demonstrated in \cite[Theorem 3.2]{BHCAN} that either of the
conditions \eqref{S1} or \eqref{S2} is necessary as well as
sufficient for \eqref{Sumineq2}  provided the weights are chosen so
that $S_1(t)\approx S_2(t)$. The choice of $\delta$ according to
\eqref{eq2.6} forces this in Example \ref{Ex1}.
\begin{example}\label{ex3} Let $v_0=v_1=w$ and take $f(t)=1$. Then
\eqref{Sumineq2} is true if and only if
$$
\sup_{t\in I_a, 0<\eps\le
\eps_0}\left(\eps^{-1}\int_t^{t+\eps}w\right)\left(\eps^{-1}\int_t^{t+\eps}w^{-p'/p}\right)^{p/p'}<\infty.
$$
\end{example}
A necessary and sufficient condition for \eqref{Sumineq2} can also
be stated in a more general setting if the weights satisfy certain
growth conditions.
\begin{theorem}\label{theorem1} Let $w, v_0, v_1$ be weights such that $w, v_0^{-p'/p},
v_1^{-p'/p}\in L_{\text{loc}}(I_a)$  and  the following conditions
are satisfied:
\begin{align}\label{2.0}
|v_1'|&\le Kv_1^{1-1/np}v_0^{1/np},\notag\\
|v_0'|&\le npv_1^{-1/np}v_0^{1+1/np}
\end{align}
on $I_a$. Then the sum inequality \eqref{Sumineq2} holds for
$1<p<\infty$ and $j=0,\dots,n-1$ if and only if
\begin{eqnarray}\label{S}
 S(\epsilon_0):=\sup_{t\in I_a, \epsilon\in
(0,\epsilon_0)}\frac{1}{\epsilon f(t)}\int_t^{t+\epsilon f(t)}
wv_1^{-j/n}v_0^{(j-n)/n}<\infty
\end{eqnarray}
 where
$f(t)=(v_1/v_0)^{1/np}$.
\end{theorem}

Before proving this we need a lemma showing that our choice of $f$
works like $f^\ast$ in Proposition \ref{prop1}.
\begin{lemma}\label{lemma0} There are constants $K_2, K_3>0$ possibly depending on $\epsilon$ so that
\begin{eqnarray}\label{2.2}
K_2\le \frac{f(s)}{f(t)}, \frac{v_1(s)}{v_1(t)},
\frac{v_0(s)}{v_0(t)}\le K_3
\end{eqnarray}

 on the intervals $J_{t,\epsilon}$ for sufficiently small $\epsilon>0$.
\end{lemma}
\noindent
\begin{proof} First, we note that
\begin{align}
|f'|&= \left|\left((v_1/v_0)^{1/np}\right)'\right|= \left|(1/np)(v_1/v_0)^{1/np-1}v_0^{-2}(v_0v_1'-v_1v_0')\right|\notag\\
   &\le (1/np)v_1^{1/np-1}v_0^{-1/np}|v_1'|+ (1/np)(v_1/v_0)^{1/np-1}v_0^{-2}v_1|v_0'|\notag\\
     &\le K/np+1.\notag
\end{align}
Hence for $t<s\le t+\epsilon f(t)$,
$$
|f(s)-f(t)|=\left|\int_t^sf'\right|\le \int_t^s|f'|\le \epsilon
f(t)(K/np+1),
$$
so that $f(s)/f(t)\le 1+\epsilon(K/np+1)$ and $f(s)/f(t)\ge
1-\epsilon(K/np+1).$  Next, consider $v_0$. Since
\begin{align}
|fv_0'|&\le (v_1/v_0)^{1/np}npv_0^{1+1/np}v_1^{-1/np},\notag\\
           &=npv_0\notag
\end{align}
and by the previous result we obtain that
$$
 v_0'/v_0\le |v_0'/v_0|\le np(1-\epsilon(K/np+1))f(t))^{-1}.
$$
Integrating this over $J_{t,\epsilon}$ implies that
\begin{align}
v_0(s)/v_0(t)&\le {\rm exp}(K_4(\epsilon)(s-t)/f(t))\notag\\
         &\le {\rm exp}(K_4(\epsilon)).\label{q(s)/q(t)}
\end{align}
But also since
$$
-v_0'/v_0\le |v_0'/v_0|\le np(1-\epsilon(K/np+1))f(t))^{-1},
$$
we get by integration that
\begin{eqnarray}
v_0(s)/v_0(t)\ge ({\rm exp}(K_4(\epsilon)))^{-1}.\label{q(t)/q(s)}
\end{eqnarray}
(\ref{2.2}) for $v_0$ follows from (\ref{q(s)/q(t)}) and
(\ref{q(t)/q(s)}). That (\ref{2.2}) holds also for $v_1$ is a
consequence of the identity
$$
v_1(s)/v_1(t)=(v_0(s)/v_0(t)(f(s)/f(t))^{np}
$$
and (\ref{2.2}) for $v_0$ and $f$.
\end{proof}

\noindent {\sl Proof of Theorem \ref{theorem1}.}$\quad$ We know by
Theorem A that the sum inequality (\ref{Sumineq2}) holds if
\eqref{S1} and \eqref{S2} hold.
 However because of the conditions (\ref{2.2})
allowing $f, v_0,$ and $v_1$ to be taken in and out of integrals
over the interval $J_{t,\epsilon}$ both conditions are found to be
equivalent to \eqref{S}.  Since the assumptions on the weights
guarantee that $S_1(t)=S_2(t)$ we could apply \cite[Theorem
3.2]{BHCAN} to conclude that \eqref{S} is a necessary condition; but
we choose to give an explicit argument. Let $\phi\ge 0$ be a
$C^\infty_0$ function such that $\phi(t)=1$ on $[0,1]$ and $\phi$
has support on $(-2,2)$. Define
$$
H_j(t)=\phi(t)(t^j/j!).
$$
Then $H_j^{(j)}(t)=1$ on $[0,1]$ and $H_j(t)$ has support on
$(-2,2)$. Set $u=(s-t)/\epsilon f(t)$ for $t-2\epsilon f(t)\le s\le
t+2\epsilon f(t)$ and define
$$
H_{j,t}(s)=(\epsilon f(t))^j H_j(u).
$$
Note that
$$
H_{j,t}^{(k)}(s)=H^{(k)}_j(u)(\epsilon f(t))^{j-k}
$$
so that in particular $H_{j,t}^{(j)}(s)=1$. Next, choose $t$ and
sufficiently small $\epsilon$ such that $t-2\epsilon f(t)>a$ and
consider the function
\begin{align}
S(t)&:=(\epsilon f(t))^{-1}\int_t^{t+\epsilon f(t)}wv_1^{-j/n}v_0^{j/n-1}\notag\\
    &=(\epsilon f(t))^{-1}\int_t^{t+\epsilon f(t)}wv_1^{-j/n}
    v_0^{j/n-1}|H^{(j)}_{j,t}|^p\notag\\
    &\approx (\epsilon f(t))^{-1}(v_1^{-j/n}v_0^{j/n-1})(t)\notag\\
    &\quad \times\int_t^{t+\epsilon f(t)}w|H^{(j)}_{j,t}|^p.\notag
\end{align}
Therefore if \eqref{Sumineq2}
 holds we have

\begin{eqnarray*}
S(t)&=&\text{O}\Big\{(\epsilon
f(t))^{-1}(v_1^{-j/n}v_0^{j/n-1})(t)\Big[\eps^{-jp}\int_{t-2\epsilon
f(t)}^{t+2\epsilon
f(t)}v_0|H_{j,t}|^p\\&&+\eps^{(n-j)p}\int_{t-2\epsilon
f(t)}^{t+2\epsilon f(t)}v_1|H_{j,t}^{(n)}|^p\Big]\Big\}.
\end{eqnarray*}
However
\begin{align}
&(\epsilon
f(t))^{-1}(v_1^{-j/n}v_0(t)^{j/n-1})(t)\left(\int_{t-2\epsilon
f(t)}^{t+2\epsilon f(t)}v_0|H_{j,t}|^p\right)\notag\\ &\quad \approx
f(t)^{-jp}(\epsilon f(t))^{-1} \left(\int_{t-2\epsilon
f(t)}^{t+2\epsilon f(t)}|H_{j,t}|^p\right)\notag\\
&\quad=\int_{-2}^2|H_j(u)|^p,\notag
\end{align}
and
\begin{align}
&(\epsilon
f(t))^{-1}(v_1^{-j/n}v_0^{j/n-1})(t)\left(\int_{t-2\epsilon
f(t)}^{t+2\epsilon f(t)}v_1|H^{(n)}_{j,t}|^p\right)\notag\\
&\quad \approx f(t)^{-jp}(v_1/v_0)(\epsilon f(t))^{-1})
\left(\int_{t-2\epsilon f(t)}^{t+2\epsilon
f(t)}|H_{j,t}^{(n)}|^p\right)\notag\\ &\quad
=\int_{-2}^2|H_j^{(n)}(u)|^p.\notag
\end{align}
Putting these two estimates together shows that $S(t)$ is uniformly
bounded for $t\in I_a$ and $\epsilon\in (0, \epsilon_0]$  which is
equivalent to \eqref{S} as was to be proved. \qed
\medskip

Our next result  extends an inequality of Anderson and Hinton
\cite[Theorem 3.1]{AH} from $L^2(I_a)$ to the  $L^p$ case.
\begin{corollary} \label{col1} Suppose $v_0, v_1\in L^p_{\text{loc}}(I_a)$,
$v_1^{1/2}|v_0'|\le 2v_0^{3/2}$, and $|v_1'|\le (K/p)\sqrt{v_0v_1}$
then the sum inequality
\begin{eqnarray}\label{cor}
\Vert v_1'y'\Vert_{p, I_a}\le K(\epsilon)\Vert v_0y\Vert_{p,
I_a}+\epsilon \Vert v_1y''\Vert_{p, I_a}
\end{eqnarray}
holds for all $1<p<\infty$ and $\epsilon>0$.
\end{corollary}
\noindent
\begin{proof} Here the weights $(v_1')^p, v_0^p$, and $v_1^p$ replace
the weights $w, v_0$, and $v_1$. The conditions on $v_1'$ and $v_0'$
are equivalent to (\ref{2.0}) for this choice of weights. $f(t)$ is
given by $(v_1^p/v_0^p)^{1/2p}\equiv \sqrt{v_1/v_0}$. We have also
that
$$
S_1(t)\equiv
S_2(t)=\eps^{-1}\sqrt{\frac{v_0}{v_1}}\int_t^{t+\epsilon
\sqrt{v_1/v_0}} |v_1'|^p(v_1v_0)^{-p/2}\le (K/p)^p.
$$
(\ref{cor}) follows by Theorem \ref{theorem1}.
\end{proof}

\begin{example}\label{ex2}
Suppose $w=v_0=v_1$ and $|v_1'|\le npv_1$. Then $f=1$ and
$S_1(\infty)=S_2(\infty)=1$. By Theorem \ref{theorem1} we have the
inequality \begin{eqnarray} \label{v1}
 \int_{I_a} w|y^{(j)}|^p\le
K_1\left(\epsilon^{-jp}\int_{I_a}w|y|^p+
\epsilon^{(n-j)p}\int_{I_a}w|y^{(n)}|^p\right).
\end{eqnarray}
In this example unlike Examples \ref{ex2.1} and \ref{ex3} since the
inequality holds for all $\eps>0$.
\end{example}

\begin{remark}
If a sum inequality of the form \eqref{Sumineq2} holds for arbitrary
$\epsilon>0$ we can minimize the right-hand side of the inequality
as a function of $\epsilon$ provided $j\ne 0$ and
$\int_{I_a}v_1|y^{(n)}|^p\ne 0$. This procedure applied to
\eqref{v1} in the previous example will yield the product inequality
\begin{eqnarray*}
\int_{I_a} w|y^{(j)}|^p\le
K_2\left(\int_{I_a}w|y|^p\right)^\frac{n-j}{n}\left(
\int_{I_a}w|y^{(n)}|^p\right)^\frac{j}{n}.
\end{eqnarray*}
Note that $v_1$ can be taken as e$^{\pm bt}$ where $0<b\le np$.
\end{remark}
\begin{remark}
So far we have supposed because of the applications we have in mind
that each of the terms in our weighted sum or product inequalities
have a common $L^p$ norm. However, versions of these inequalities
exist when the three norms are different. One can have, for
instance, an inequality of the form
$$ \int_{I_a} w| y^{(j)}| ^p\leq
K_1\left\{ \epsilon ^{-\mu_1}\left[\int_{I_a} v_0|y|^s\right]^{p/s}
+\epsilon^{\mu_2}\left[\int_{I_a} v_1|y^{(m)}|^t\right]^{p/t}\right)
$$
 where $1\le p, s, t<\infty$,
\begin{align}
\mu_1&=p(j+s^{-1}-p^{-1})\notag\\
\mu_2&=p(n-j-t^{-1}+p^{-1}),\notag
\end{align}
and $n, j, p, s, t$ satisfy various relationships. Also
generalizations  exist in $\mathbb{R}^n$, $n>1$. For information on
these more general cases see \cite{BHP}, \cite{BHCAN},
\cite{BHLONDON},  \cite{M}, and \cite{GI8}. There are additionally
other approaches to weighted norm inequalities similar to
\eqref{Sumineq2}. See for example Wojteczek-Laszczak \cite{Woj} and
Kwong and Zettl \cite{KZ3}.
\end{remark}





\section{Some Applications to Relative Boundedness and Limit-point conditions for
differential operators in $L^p$ spaces} In \cite{GI8} we gave
applications of sum and product inequalities to various spectral
theoretic problems involving Sturm-Liouville operators in
$L^2(I_a)$. In this section we look at  applications to operators
determined by expressions of Sturm-Liouville form but defined in
$L^p$ spaces.  We first require some preliminary definitions and
abstract results. In what follows $\Vert (\cdot)\Vert$ will denote
the norm in an arbitrary Banach space.
\begin{definition} Suppose $A$ and $T$ are
operators from a Banach space $X$ to a Banach space $Y$. Then $A$ is
said to be $T$ bounded if the domain of $T$ is contained in the
domain of $A$ and the inequality \begin{eqnarray} \label{TBd}
 \Vert A(x)\Vert\le
K(\Vert x\Vert+\Vert T(x)\Vert)
\end{eqnarray} holds for all $x$ in the domain of
$T$. Furthermore $A$ is said to have $T$ bound 0 if $A$ is $T$
bounded and the inequality \eqref{TBd} has the form
$$
\Vert A(x)\Vert\le K(\epsilon)\Vert x\Vert+\epsilon \Vert T(x)\Vert)
$$
for all $\epsilon\in (0, \epsilon_0)$ for some $\epsilon_0 \in
(0,\infty)$.
\end{definition}
\begin{lemma}\label{lemma2} Suppose $A, B, C$, and $L$ are
operators from a Banach space $X$ to a Banach space $Y$. Suppose
that $A$ is $L$-bounded and $B, C$ are $L$-bounded with $L$ bound 0.
Then $A$ is $L+B+C$ bounded.
\end{lemma}
\begin{proof}
By the triangle inequality \begin{eqnarray}\label{L+1}
 \Vert x\Vert+ \Vert L(x)\Vert\le
\Vert x\Vert+ \Vert(L+B+C)(x)\Vert +\Vert B(x)\Vert+\Vert C(x)\Vert.
\end{eqnarray}
>From the hypotheses on $B$ and $C$ we also have the estimates
\begin{align}
\Vert B(x)\Vert&\le K_1(\epsilon/2)\Vert x\Vert+ (\epsilon/2) \Vert L(x)\Vert\label{B}\\
\Vert C(x)\Vert&\le K_2(\epsilon/2)\Vert x\Vert+ (\epsilon/2) \Vert
L(x)\Vert\label{C}.
\end{align}
Substituting (\ref{B}) and (\ref{C}) into (\ref{L+1} gives that
\begin{eqnarray}\label{L+B+C}
(1-\epsilon)(\Vert x\Vert + \Vert L(x)\Vert)\le (1+
K_1(\epsilon/2)+K_2(\epsilon/2))\Vert x\Vert+ \Vert(L+B+C)(x)\Vert.
\end{eqnarray}
But since $A$ is $L$ bounded $\Vert A(x)\Vert\le K(\Vert
x\Vert+\Vert L(x)\Vert)$. Combining this with (\ref{L+B+C}) yields
that
$$
\Vert A(x)\Vert\le K(1-\epsilon)^{-1}[(1+
K_1(\epsilon/2)+K_2(\epsilon/2))\Vert x\Vert+ \Vert(L+B+C)(x)\Vert].
$$
\end{proof}

\begin{lemma}\label{lemma4} Let $A$, $B$, $C$ and $L$ be operators from a
Banach space $X$ to a Banach space $Y$. Suppose the inequalities
\begin{align}
\Vert A(y)\Vert&\le K(\epsilon)\Vert B(y)\Vert+ \epsilon\Vert L(y)\Vert)\label{2.8}\\
\Vert B(y)\Vert&\le K(\epsilon)\Vert C(y)\Vert+\epsilon \Vert
L(y)\Vert\label{2.9}\\ \Vert C(y)\Vert&\le K(\Vert y\Vert+\Vert
T(y)\Vert)\label{2.10}
\end{align}
where $T(y)=(A+B+L)(y)$. Then $A$ is $T$ bounded with relative bound
$0$.
\end{lemma}
\noindent
\begin{proof} By \eqref{2.9} and the triangle inequality
$$
\Vert B(y)\Vert\le K(\epsilon)\Vert
C(y)\Vert+\epsilon\Vert(L+B)(y)\Vert+\epsilon\Vert B(y)\Vert.
$$
Hence,
$$ \Vert B(y)\Vert\le K(\epsilon)(1-\epsilon)^{-1}\Vert
C(y)\Vert+\epsilon(1-\epsilon)^{-1}\Vert(L+B)(y)\Vert.
$$
Substituting this into (\ref{2.8}) after  noting again that $\Vert
L(y)\Vert\le \Vert(L+B)(y)\Vert+\Vert B(y)\Vert$ gives the
inequality
\begin{eqnarray}\label{2.11}
\Vert A(y)\Vert\le K(\epsilon)(1+\epsilon(1-\epsilon)^{-1})\Vert
C(y)\Vert+\epsilon^2(1-\epsilon)^{-1}\Vert(L+B)(y)\Vert.
\end{eqnarray}
 Finally,
\begin{align}
 \Vert (L+B)(y)\Vert&\le \Vert T(y)+\Vert C(y)\Vert\notag\\
                    &\le K\Vert y\Vert+ (K+1)\Vert T(y)\notag\Vert.
\end{align}
Substitution into (\ref{2.11}) now gives the desired conclusion.
\end{proof}


Given a  Banach space $X$ with dual $X^\ast$, $[x, x^\ast]$
signifies the complex conjugate $\overline{{x^\ast}(x)}$ for $x\in
X$ and $x^\ast\in X^\ast$. If $T$ is an operator on $X$ we consider
the set of pairs $G(T^\ast):=(z, z')\in X\times X^\ast$ such that
\begin{eqnarray}\label{Green}
[T(y),z]=[y, z'].
\end{eqnarray}
 The density of $\mathcal{D}(T)$ implies that
\eqref{Green}  determines an operator $T^\ast$ called the
\emph{adjoint} of $T$ such that $T^\ast(z)=z'$. If $T: X^\ast\to
X^\ast$ has a domain $\mathcal{D}(T)$ which is {\it total} over $X$
(i.e., $[x, x^\ast]=0$ for a fixed  $x\in X$ and all $x^\ast\in
\mathcal{D}(T) \Rightarrow x=0$) the set $G(^\ast T)$ of pairs
$(y',y)\in X\times X$ satisfying
$$
[y', z]=[y, T(z)]
$$
also determines an operator which we denote by $^\ast T$ and call
the adjoint of $T$ in $X$.\footnote{Goldberg calls this operator the
preconjugate of $T$. Its properties are studied in
\cite[VI.I]{Gold}.} Both $T^\ast$ and $^\ast T$ are closed and
$[T(y), z]=[y, T^\ast(z)]$ or $[^\ast T(y), z]=[y, T(z)]$ for all
$y\in \mathcal{D}(T)$ or $\D(^\ast T)\subset X$ and $z\in
\mathcal{D}(T^\ast)$ or $\D(T) \subset X^\ast$ . In the particular
case $X=L^p(I_a)$ and $X^\ast=L^{p'}(I_a)$ where $1\le p\le\infty$
the pairing $[(\cdot), (\cdot)]$ on $L^p(J)\times L^{p'}(J)$ for
some interval $J$ is given by $[y,z]_J:=\int_Jy\bar{z}$. Consider
now the differential expression $M[y]:=-(ry')'+qy$. Assume that $r>
0$, $r\in C^1(I_a)$, and $q\in C(I_a)$. Define
$$
\{y,z\}(t):=r(t)(y(t)\bar{z}'(t)-y'(t)\bar{z}(t)).
$$
for $y,z\in AC_{\text{loc}}(I_a)$ and the following operators and
domains in $L^p(I_a)$.
\begin{definition}\label{def2} For $p\in [1,\infty]$ let
let $T'_{0,p}$, $T_p$, $T_{0,p}$, and be the operators with domain
and range in $L^p(I_a)$ determined by $M$ on
\begin{align}
\mathcal{D}'_{0,p}&:=\{y\in C^\infty_0(I_a)\},\notag\\
\mathcal{D}_p&:=\{y\in L^p(I_a): y'\in AC_{\text{loc}}(I_a); M[y]\in L^p(I_a)\},\notag\\
\mathcal{D}_{0,p}&:=\{y\in \mathcal{D}_p: y(a)=y'(a)=0;
\lim_{t\to\infty}\{y,z\}(t)=0, \forall z\in
\mathcal{D}_{p'}\}.\notag
\end{align}
\end{definition}
\noindent We call $T_{0,p}'$, $T_{0,p}$ respectively the
``preminimal" and ``minimal" operators, and $T_p$ the ``maximal"
operator determined by $M$.   \noindent
\begin{theoremb} If $1\le p<\infty$ the operators $T'_{0,p}$, $T_{0,p}$, and $T_p$ have
the following properties:
\begin{enumerate}
\item[(i)] $T_{0,p}$ and $T_p$ are closed densely defined operators.
\item[(ii)] $[T_p(y), z]_{[a,t]}=\{y,z\}(t)-\{y,z\}(a)+ [y,
T_{p'}(z)]_{[a,t]}$.
\item[(iii)]  $T_p^\ast=T_{0, p'}$ and $T_{0,p}^\ast= T_{p'}$.
\item[(iv)] $T'_{0,p}$ is closable and $\overline{T'_{0,p}}=T_{0,p}$.
\end{enumerate}
Moreover,  $T_{0,\infty}$ and $T_\infty$ are closed, the closure of
$T_{0,\infty}'$ is $T_{0,\infty}$, $^\ast T_{0,\infty}=T_1$, and
$^\ast T_\infty= T_{0,1}$.
\end{theoremb}
\noindent Proofs of (i)--(iii), the last statement, as well as more
general results may be found in one of \cite[Chapter VI]{Gold},
\cite{Rota}, or   \cite{BC}). The $L^2$ theory is thoroughly treated
in Naimark \cite[~\S 17]{Nai}. It is almost certain that by
extending the procedure of Naimark given in the $L^2$ case that $q,
r$ and $r^{-p'/p}$ need only be locally integrable for Theorem B to
hold. However, the verification of this is technically complicated
and will be omitted here.
\begin{definition} The operators
$T_{0,p}$ or $T_p$ are separated if on $\D_{0,p}$ or $\D_p$
$(ry')'\in L^p(I_a)$
\end{definition}
\begin{remark} By the triangle inequality the separation of
$T_{0,p}$ or $T_p$ is equivalent to $qy\in L^p(I_a)$ on the domains
of these operators. The closed graph theorem in turn implies that
separation is equivalent to the existence of an inequality of the
form
$$
\Vert(ry')'\Vert_{p, I_a}+ \Vert qy\Vert_{p, I_a}\le K\{\Vert
y\Vert_{p, I_a}+ \Vert M[y]\Vert_{p, I_a}\}.
$$
Necessary and sufficient conditions for separation in $L^p$ when
$M[y]=-y''+qy$ have been given by Chernyavskaya and Shuster
\cite{CS}. However their conditions can be difficult to verify. For
$p=2$ various sufficient conditions may be found in \cite{ABH},
\cite{BHsep}, \cite{Hg02}, \cite{EG}. The simplest condition
guaranteeing separation for all $p$ is to require that $q$ be
essentially bounded.
\end{remark}



\noindent {\scshape Limit-point Results in $L^p$ spaces.}
\begin{definition} We say that $T_p$ is $p$-limit-point ($p$-LP) at $\infty$
if $\lim_{t\to \infty} \{y,z\}(t)=0$ for all $y\in \mathcal{D}_p$
and $z\in \mathcal{D}_{p'}$
\end{definition}

\begin{theorem}\label{Limpt} Consider the following conditions:

\begin{enumerate}
\item[(i)] $T_p$ is $p$-LP.
\item[(ii)] There do not exist linearly independent solutions
$y_p\in L^p(I_a)$ and $z_{p'}\in L^{p'}(I_a)$ of $M[y]=0$.
\item[(iii)] dim\,$\mathcal{D}_p/\mathcal{D}_{0,p}=2$.
\end{enumerate}
Then (i) and (iii) are equivalent conditions and (i) $\Rightarrow$
(ii).
\end{theorem}
\noindent
\begin{proof} Suppose that (i) is true and that there were linearly independent solutions $y_p\in
L^p(I_a)$ and $z_{p'}\in L^{p'}(I_a)$. Let $t\in I_a$ By (ii) of
Theorem B  $\{y,z\}(t)=\{y,z\}(a)$ and the fact that $T_p$ is $p$-LP
we have
$$
0=\lim_{t\to\infty}\{y_p,z_{p'}\}(t)= \{y_p,z_{p'}\}(a)=
r^{-1}(a)\{y_p,z_{p'}\}(a).
$$
But this is impossible since $r^{-1}\{y,z\}$  is just the Wronskian
of of the solutions $y_p, z_{p'}$ and its zero value at $a$ or $t$
contradicts their assumed linear independence.  Turning now to
(iii), let $\phi_1$ and $\phi_2$ be $C^\infty_0(I_a)$ functions such
that $\phi_1(a)=1$, $\phi_1'(a)=0$ and $\phi_2(a)=0$,
$\phi_2'(a)=1$. Since $\phi_1, \phi_2\in \mathcal{D}_p$ and are
linearly independent, we see that
dim\,$\mathcal{D}_p/\mathcal{D}_{0,p}\ge 2$. Suppose there exists
$u\in \mathcal{D}_p$ such that $\{\phi_1, \phi_2, u\}$ is linearly
independent mod $\mathcal{D}_{0,p}$.  Let $h$ be a linear
combination of these functions such that $h(a)=h'(a)=0$. Let $G_p$
and $G_{0,p}$ respectively be $\mathcal{D}_p$ or $\mathcal{D}_{0,p}$
endowed with the graph norm. Now the dual $G_p^\ast$ of $G_p$ can be
identified with the space of pairs $\xi=(z, z^\ast)$ in
$L^{p'}(I_a)\times L^{p'}(I_a)$ such that $\xi(y)$, $y\in G_p$, is
given by
$$
\int_{I_a} y\bar{z}+\int_{I_a}M[y]\bar{z}^\ast.
$$
Since $h\notin G_{0,p}$ and $G_{0,p}$ is closed in $G_p$ there
exists an element $\xi\in G_p^\ast$  such that $\xi(h)=1$ and
$\xi(y)=0$ for all $y\in G_{0,p}$, implying that $\xi=(-M[z^\ast],
z^\ast)$ for some $z\in \mathcal{D}_p$. It follows that
\begin{align}
1&=\xi(h)=\int_{I_a}h(-\overline{M[z]} + \int_{I_a} M[h]\bar{z}
\notag\\ &=\{h,z\}(\infty),\notag
\end{align}
contradicting (i). We conclude that
dim\,$\mathcal{D}/\mathcal{D}_0=2.$  In the above argument we have
shown that (i)$\Rightarrow$ (ii) and that (i)$\Rightarrow$ (iii). It
is clear that (iii) $\Rightarrow$ (i). For if $\D_p$ is a two
dimensional extension of $\D_{0,p}$ then span$\{\phi_1, \phi_2,
\D_{0,p}\}=\D_p$. Since $\phi_1$ and $\phi_2$ have compact support
and by the definition of $\D_{0,p}$, (i) will hold.
\end{proof}



\noindent
\begin{remark}\label{plimitpt} For $p\ne 2$
Theorem  \ref{Limpt} represents an extension of the well-known limit
point concept for differential operators in $L^2(I_a)$. In
particular (ii) is a generalization of the fact that if $M$ is
$2$-LP at $\infty$ then $M$ has at most one $L^2$ solution. In the
limiting case $p=1$, $p'=\infty$, if there is a solution $y$ in
$\mathcal{D}_1$ in $L(I_a)$,  (ii) says that that there cannot be
another solution independent of $y$ which is bounded. As in the
Hilbert space case we have also shown that $T_p$ is $p$-limit-point
if and only if $\D_p/\D_{0,p}=2$.
\end{remark}
\begin{remark}
If any one of $T_{0,p}$, $T_p$, $T_{0,p'}$ and $T_{p'}$ has closed
range then  more can be said. Specifically all the other minimal and
maximal operators also have closed range. In particular, the minimal
operators are one-to-one and have bounded inverses while the maximal
operators are surjective  and
$$
\text{dim}\,(\D_p/\D_{0,p})=
\text{dim}\,\N(T_p)+\text{dim}\,\N(T_{p'}).
$$
For the proof in a considerably more general setting see Goldberg
\cite{Gold} Theorems VI.2.7 and VI.2.11. Furthermore, since the
dimensions of both null spaces do not exceed 2 and since as we have
seen  $\D_p$ is at least a two dimensional extension of $\D_{0,p}$
we have that
$$
2 \le\text{dim}\,(\D_p/\D_{0,p})\le 4.
$$
Brown and Cook \cite[Corollary 2.9]{BC} have shown that $T_{0,p}$
defined on $I_a$ has a bounded inverse and thus closed range for
$1\le p\le \infty$ if both $\int_{I_a} r^{-1}<\infty$ and
$\int_{I_a} q^{-1}<\infty$
\end{remark}

\begin{remark}
If $q\ge 0$ then $M$ is disconjugate and by  by Corollary 6.4 and
Theorem 6.4 of Hartman \cite{Hart} there is a fundamental set of of
positive linearly independent
 solutions
$y_1$ and $y_2$ of $M[y]=0$, called respectively the principal and
nonprincipal solutions, such that  $y_1'\le 0$ and  $y'>0$ on $I_a$.
Additionally, $\lim_{t\to\infty} y_1/y_2=0$. It follows at once in
our setting that dim\,$\N(T_p)= \text{dim}\,\N(T_{p'})\le 1$. In
\cite{SD} it is furthermore shown that if $r=1$, $q\ge 0$, and there
exists $b\in (0,\infty)$ such that
\begin{eqnarray}\label{closedrange}
\inf_{x\in I_a, x-b>a}\int_{x-b}^{x+b}q>0,
\end{eqnarray}
then
\begin{enumerate}
\item[(i)] $T_p$ has closed range.
\item[(ii)] dim\,$(\mathcal{D}_p/\mathcal{D}_{0,p})=2$ and
dim\,$(R(T_p)/R(T_{0,p})=1$,
\item[(iii)]  dim\,$\N(T_p)=1$,
\item[(iv)] If $y_1$ denotes the principal or ``small" solution of $M[y]=0$
then $y_1\in L^p(I_a)$ for all $p\in [1,\infty]$.
\end{enumerate}
The condition \eqref{closedrange} was shown by Chernyavskya and
Shuster \cite{CS2} to be necessary and sufficient for $T_p$ defined
on $\mathbb{R}$ to have a bounded inverse. In the case when $q\ge
k>0$ one can show that $M[y]=0$ has exponentially growing and
exponentially decaying solutions. Read \cite{Read} has extended this
by showing that the same is true if
$$
 \liminf_{x\to\infty}\int_x^{x+a}
q^{1/2}\,dt>0
$$
 for some $a>0$.



\end{remark}

\noindent {\scshape Relative Boundedness for perturbations of
Differential Operators in $L^p$ spaces.}
\smallskip

\noindent The following two results are  generalizations to $L^p$ of
relative boundedness criteria given by Anderson and Hinton \cite{AH}
in the $L^2$ setting.
 \noindent
\begin{theorem}\label{theorem5} Let $A_j: L^p\to L^p$ be given by $\int_{I_a}
a_jy^{(j)}$, $j=0,1$, on $\D_p$  where the $a_j$ are locally
integrable functions. Let $T_{0,p}$  be the minimal  operator in
$L^p(I_a)$ determined by $M[y]=-(ry')'+ qy$ and Assume that $|r'|\le
(K/p)\sqrt{r}$ and that
\begin{eqnarray}\label{qcond}
\sup_{t\in
I_a}\frac{1}{\sqrt{r}}\int_t^{t+\epsilon\sqrt{r}}|q|^p<\infty.
\end{eqnarray}
Then the $A_j$ are $T_{0,p}$-bounded if and only if
\begin{eqnarray}\label{rcond}
\frac{1}{\sqrt{r}}\int_t^{t+\epsilon \sqrt{r}} |a_j|^p
r^{-jp/2}<\infty,\quad j=0,1,
\end{eqnarray}
is bounded on $I_a$. If $T_p$ is $p$-LP at $\infty$ then the $A_j$
are also $T_p$ bounded.
\end{theorem}
\noindent
\begin{proof} This is an application of Theorem \ref{theorem1} and  Lemma \ref{lemma2}.  Define the operators $L, B, C: L^p(I_a)\to L^p(I_a)$
on $C^\infty_0(I_a)$ by $L(y)=ry''$, $B(y)=r'y'$, and $C(y)=qy$
where $y\in \mathcal{D}_{0,p}$. Then  $T'_{0,r}= L+B+C$. Let
$f=\sqrt{r}$, and take $v_0=1$, The condition on $r'$ is just the
first condition in (\ref{2.0}) with $r^p$ replacing $v_1$. By
Theorem \ref{theorem1} the $A_j$ are $L$-bounded or equivalently the
sum inequalities
$$
\Vert A_j(y)\Vert_{p, I_a}\le K(\Vert y\Vert_{p, I_a}+\Vert
ry''\Vert_{p, I_a}),\quad j=0,1,
$$
hold if and only if (\ref{rcond}) is true. By Corollary \ref{col1}
(with $v_0=1$) $B$ is $L$-bounded with relative bound $0$. Finally,
another application of Theorem \ref{theorem1} using \eqref{qcond}
gives that $C$ is $L$-bounded with relative bound $0$. The fact that
the $A_j$ are $T'_{0,p}$ bounded now  now follows from Lemma
\ref{lemma2}. A closure argument shows that the $A_j$ are $T_{0,p}$
bounded. Finally, if $T_p$ is $p$-LP at $\infty$, then $T_p$ is a
two dimensional extension of $T_{0,p}$ via the functions $\phi_1$
and $\phi_2$. Hence, if  $y\in \D_p$, $y=y_0 + z$ where
$z=c_1\phi_1+ c_2\phi_2$. Since $A_j$ is $T_0$ bounded and by an
elementary inequality we have
$$
\Vert A_j(y_0)\Vert_{p, I_a}^p\le K^p2^{p-1}\{\Vert y_0\Vert_{p,
I_a}^p+ \Vert T(y_0)\Vert_{p, I_a}^p\}.
$$
The same inequality is true for $z$ since the $p$-th root of each
side defines two norms on $Z:=\text{span}\,\{\phi_1, \phi_2\}$ and
the mapping $j:Z\to Z$ given by $j(z)=z$ is continuous with respect
to these norms  since $Z$ is finite (2!) dimensional. Hence,
\begin{align}
\Vert A_j(y_0+z)\Vert_{p,I_a}^p&\le 2^{p-1}[\Vert
A_j(y_0)\Vert_{p,I_a}^p+ A_j(z)\Vert_{p,I_a}^p\notag\\ &\le
K_1[\Vert y_0\Vert_{p,I_a}^p+ \Vert z\Vert_{p,I_a}^p+ \Vert
T(y_0)\Vert_{p,I_a}^p+ \Vert T(z)\Vert_{p,I_a}^p]\notag\\ &\le
K_1[\Vert y_0+z\Vert_{p,I_a}^p+ \Vert
T(y_0+z)\Vert_{p,I_a}^p]\notag\\ &=K_1\{\Vert y\Vert_{p,
I_a}^p+\Vert T(y)\Vert_{p,I_a}^p\}.\notag
\end{align}
\end{proof}

\begin{theorem}\label{theorem3} Suppose that $T_{0,p}$ is separated
 Additionally suppose that
\begin{align}
|r'|&\le (K/p)\sqrt{r|q|}\notag\\
|q'|&\le 2|q|^{3/2}r^{-1/2}.\notag
\end{align}
 Let $A_j$, $j=0,1$,  be as
in Theorem \ref{theorem5}. Then $A_j$ is $T_{0,p}$-bounded with
$T_p$-bound $0$ if and only if
\begin{eqnarray}\label{2.12}
 S(t)= \sqrt{\frac{|q|}{r}}\int_t^{t+\epsilon
 \sqrt{r/q}}|a_j|^pr^{-jp/2}|q|^{p(j-2)/2}
 \end{eqnarray}
is bounded on $I_a$.
\end{theorem}
\noindent
\begin{proof} As before we begin with the $C^\infty_0$ functions. Let $C(y)=qy$, $B(y)= r'y'$ and $L(y)=ry''$. By
the hypothesis of separation  (\ref{2.10}) holds. By Theorem
\ref{theorem1} where $f(t)=(r^p/q^p)^{1/2p}=\sqrt{r/q}$ (\ref{2.8})
holds if and only if (\ref{2.12}) is true. By Corollary \ref{col1}
\eqref{2.9} is true. The conclusion that $A_j$ is $T'_{0,p}$ bounded
follows from Lemma \ref{lemma4}.
\end{proof}



\bibliographystyle{amsplain}
\begin{thebibliography}{99}

\bibitem{AH}
T. G. Anderson and D. B. Hinton, \textit{Relative boundedness and
compactness for second order differential operators}, J. Ineq. and
Appl. \textbf{1} (4) (1997), 375--400.
\bibitem{Hg02}
R. C. Brown, \textit{Separation and discongugacy}, JPAM \textbf{4}
(3), Article 56, 2002 (Electronic).
\bibitem{SD}
\bysame, \textit{A limit-point criterion for a class of
Sturm-Liouville operators defined in $L^p$ spaces}, Proc. Am. Math.
Soc \textbf{132} (2004), 2273--2280.
\bibitem{BH1}
R. C. Brown and D. B. Hinton, \textit{Sufficient conditions for
weighted inequalities of sum form}, J. Math. Anal. Appl.
\textbf{112} (1985), 563--578.
\bibitem{BHP}
\bysame, \textit{Sufficient conditions for weighted Gabushin
inequalities}, Casopis P\v est. Mat. \textbf{111} (1986), 113--122.

\bibitem{BHLONDON}
\bysame, \emph{Weighted interpolation inequalities of sum and
product form in  R$^n$}, J. London Math Soc. (3) \textbf{56} (1988),
261--280.
\bibitem{BHCAN}
\bysame, \textit{Weighted interpolation inequalities and embeddings
in R$^n$},   Canad. J. Math. \textbf{47} (1990), 959--980.
\bibitem{BHHardy}
 \bysame, \textit{A Weighted Hardy's inequality and nonoscillatory differential
equations},   Quaes. Mathematicae  \textbf{115} (1992), 197--212.

\bibitem{M}
\bysame, \textit{An interpolation inequality and applications},
Inequalities and Applications (R. P. Agarwal, ed.), World
Scientific, Singapore-New Jersey-London-Hong Kong, 1994, 87--101.

\bibitem{BHsep}
 \bysame,
\textit{Two separation criteria for second order ordinary or partial
differential operators}, Math. Bohem. \textbf{124} (1999), 273--292.

\bibitem{GI8}
\bysame, \textit{Some One Variable Weighted Norm Inequalities and
Their Applications to Sturm-Liouville and other Differential
Operators}, to appear in Inequalities and Applications, (C. Bandle,
A. Gil\'{a}nyi, L. Losonczi, Z. Pales, M. Plum, eds.), Vol. 157,
International Series of Numerical Mathematics, Birkh\"{a}user,
Boston, Basel, Stuttgart, 2008.
\bibitem{ABH}
 R. C. Brown, D. B. Hinton, and M. F. Shaw,
\textit{Some separation criteria and inequalities associated with
linear second order differential operators}, in Function Spaces and
Aplications (D. E. Edmunds, et al., eds.),
 Narosa Publishing House, New Delhi, 2000, 7--35.

\bibitem{BC}
R. C. Brown and J. Cook, \textit{Continuous invertibility of minimal
Sturm-Liouville operators in Lebesgue spaces}, Proc. Roy. Soc.
Edinburgh. A \textbf{136} (01) (2006), 53--70.
\bibitem{CS}
 N. Chernyavskaya and L. Schuster,
\textit{Weight summability of solutions of the Sturm-Liouville
equation}, J. Diff. Eqs. \textbf{151} (1999), 456--473.
\bibitem{CS2}
 \bysame, \textit{A criterion for correct solvability of the Sturm-Liouville
equation in the space $L_p(R)$}, Proc. Amer. Math. Soc.
\textbf{130}(4) (2001), 1043--1054.
\bibitem{EG}
W. N. Everitt and M. Giertz, \textit{Some properties of the domains
of certain differential operators},
 Proc. London Math. Soc.\,(3) \textbf{23} (1971), 301--324.



\bibitem{EverGWeid}
 \bysame, M. Giertz, and J. Weidmann, \textit{Some
remarks on a separation and limit-point criterion of second-order
ordinary differential expressions},  Math. Ann. \textbf{200} (1973),
335--346.
\bibitem{Gold}
 S. Goldberg,
\textit{Unbounded Linear Operators Theory and Applications},
McGraw-Hill Series in Higher Mathematics (E. H. Spanier, ed.),
McGraw-Hill Book Company, New York, St. Louis, San Francisco,
Toronto, London, Sydney, 1966.
\bibitem{Guz}
M. De Guzman, ``Differentiation of Integrals in ${\Bbb R}^n$"
(Lecture Notes in Mathematics 481), Springer-Verlag, Berlin, 1975.

\bibitem{Hart}
 P. Hartman,
\textit{Ordinary Differential Equations}, Second Edition,
Birkh\"auser,
 Boston, Basel, Stuttgart, 1982.


\bibitem{HS}
E. Hewitt and K. Stromberg, \textit{Real and Abstract Analysis},
Springer-Verlag, New York Heidelberg, Berlin, 1969.
\bibitem{KZ3}
M. K. Kwong and A. Zettl, \textit{Norm Inequalities for Derivatives
and Differences}, Lecture Notes in Mathematics 1536,
Springer-Verlag, Berlin, 1992.
\bibitem{Nai}
 M. A. Naimark,
\textit{Linear Differential Operators, Part II},  Frederick Ungar,
New York, 1968.
\bibitem{Read}
T. T. Read, \textit{Exponential solutions of $y''+(r-q)y=0$ and the
least eigenvalues of Hill's equation} Proc. Amer. Math. Soc.
\textbf{50} (1975), 273--280.
\bibitem{Rota}
G. Rota, \textit{Extension theory of differential operators I},
Comm. in Pure and Applied Math. \textbf{13} (1958), 23--65.
\bibitem{Woj}
K. Wojteczek-Laszczak, \textit{On quadratic integral inequalities of
the second order}, J. Math. Anal. Appl. \textbf{342} (2) (2008),
1356--1352.
\end{thebibliography}
\end{document}
</XMP></FONT><FONT  COLOR="#0f0f0f" BACK="#fffffe"
style="BACKGROUND-COLOR: #fffffe" SIZE=2 PTSIZE=10
FAMILY="SANSSERIF" FACE="Arial" LANG="0">
----------------------- Headers --------------------------
------
Return-Path: <pecaric@element.hr> Received: from rly-dd07.mx.aol.com
(rly-dd07.mail.aol.com [172.19.141.154]) by air-dd10.mail.aol.com
(v121.4) with ESMTP id MAILINDD101-b89480f3d633d4; Wed, 23 Apr 2008
09:45:33 -0400 Received: from ls405.t-com.hr (ls405.t-com.hr
[195.29.150.135]) by rly-dd07.mx.aol.com (v121.4) with ESMTP id
MAILRELAYINDD073-b89480f3d633d4; Wed, 23 Apr 2008 09:45:08 -0400
Received: from ls242.t-com.hr (ls242.t-com.hr [195.29.150.134])
    by ls405.t-com.hr (Postfix) with ESMTP id 798CE6F810E
    for <RBrown4163@aol.com>; Wed, 23 Apr 2008 13:17:15 +0200 (CEST)
Received: from ls242.t-com.hr (localhost.localdomain [127.0.0.1])
    by ls242.t-com.hr (Qmlai) with ESMTP id 6B844B08384
    for <RBrown4163@aol.com>; Wed, 23 Apr 2008 13:17:15 +0200 (CEST)
X-Envelope-Sender: pecaric@element.hr Received: from [192.168.2.101]
(89-172-55-198.adsl.net.t-com.hr
    [89.172.55.198])by ls242.t-com.hr (Qmali) with ESMTP id 9D75D5BC258for
    <RBrown4163@aol.com>; Wed, 23 Apr 2008 13:17:12 +0200 (CEST)
Message-ID: <480F1AB4.5020102@element.hr> Date: Wed, 23 Apr 2008
13:17:08 +0200 From: Pecaric <pecaric@element.hr> User-Agent:
Thunderbird 1.5.0.14 (Windows/20071210) MIME-Version: 1.0 To:
RBrown4163@aol.com Subject: [Fwd: Text file of Brown_galley]
Content-Type: multipart/mixed;
    boundary=------------060209010106060905020000
X-imss-version: 2.050 X-imss-result: Passed X-imss-scanInfo: M:P L:N
SM:0 X-imss-tmaseResult: TT:0 TS:0.0000 TC:00 TRN:0
TV:5.0.1023(15810.001) X-imss-scores: Clean:62.61307 C:2 M:3 S:5 R:5
X-imss-settings: Baseline:1 C:1 M:1 S:1 R:1 (0.0000 0.0000)
X-AOL-IP: 195.29.150.135 X-AOL-SCOLL-SCORE:0:2:486576288:9395240
X-AOL-SCOLL-URL_COUNT: X-AOL-SCOLL-AUTHENTICATION: listenair ;
SPF_helo : n X-AOL-SCOLL-AUTHENTICATION: listenair ; SPF_822_from :
n  </FONT></HTML>

.
