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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 85, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/85\hfil A topology on inequalities]
{A topology on inequalities}

\author[A. M. D'Aristotile, A. Fiorenza\hfil EJDE-2006/85\hfilneg]
{Anna Maria D'Aristotile, Alberto Fiorenza}  % in alphabetical order

\address{Anna Maria D'Aristotile
Dipartimento di Costruzioni e Metodi Matematici in Architettura, Universit\`a di
Napoli "Federico II", via Monteoliveto 3, 80134 Napoli, Italy}
\email{daristot@unina.it}

\address{Alberto Fiorenza \newline
 Dipartimento di Costruzioni e Metodi Matematici in
Architettura, Universit\`a di Napoli  "Federico II", via Monteoliveto 3, 80134
Napoli, Italy and Istituto per le Applicazioni del Calcolo "Mauro Picone", Consiglio Nazionale delle Ricerche, via Pietro Castellino 111, 80131 Napoli, Italy}
\email{fiorenza@unina.it}


\date{}
\thanks{Submitted July 12, 2006. Published August 2, 2006.}
\subjclass[2000]{46E99, 54C30, 54A20, 54B15}
\keywords{Real analysis; topology; inequalities;
 homogeneous operators; \hfill\break\indent
 Banach spaces;  Orlicz spaces; Sobolev spaces;
 norms; density}


\begin{abstract}
We consider sets of inequalities in Real Analysis and construct  a
topology such that inequalities usually called ``limit cases'' of
certain sequences of inequalities are in fact limits - in the
precise topological sense - of such sequences. To show
the generality of the results, several examples are given for the
notions introduced, and three main examples are considered:
Sequences of inequalities relating real numbers, sequences of
classical Hardy's inequalities, and sequences of embedding
inequalities for fractional Sobolev spaces. All examples are
considered along with their limit cases, and it is shown how they
can be considered as sequences of one ``big'' space of
inequalities. As a byproduct, we show how an abstract process to
derive inequalities among homogeneous operators can be a tool for
proving inequalities. Finally, we give some tools to compute
limits of sequences of inequalities in the topology introduced,
and we exhibit new applications.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}


\section{Introduction}


In Analysis it is frequent that authors consider inequalities that are
limiting cases of sequences of inequalities, or, more generally, of a
parametrized set of inequalities. The goal of this paper is to
construct a topology such that inequalities
usually called ``limit cases'' of certain sequences of inequalities
are in fact limits - in the precise topological sense - of such sequences of
inequalities. Such kind of problem can be studied from several points
of view, because, for instance, it is possible - in a quite general,
abstract setting - to speak about
inequalities in ordered sets; moreover, even confining ourselves
for instance to inequalities involving real functions, several notions
of convergence can be considered. Our point of view has to be
considered only as a first approach to the problem, which
seems new.


\section{The main question through examples}

To give an idea of the general setting of our results, we will examine
three examples of sequences of inequalities.
The first one is the case of elementary numerical inequalities. The second one
will be the classical
integral inequality, known as \sl Hardy's inequality \rm and,
finally, we conclude with a recent version of the Sobolev inequality
for fractional Sobolev spaces.

\subsection{Inequalities relating real numbers, part I}

Let $(a_{n})$, $(b_{n})$ be sequences of positive real numbers such that
$a_{n}\to a>0$, $b_{n}\to b>0$, and
\begin{equation}
a_{n}\le b_{n}\quad \forall n\in\mathbb{N}
\label{numerireali}
\end{equation}
From elementary Analysis we know that one can ``pass to the limit''
in \eqref{numerireali}, obtaining
\begin{equation}
a\le b
\label{limnumerireali}
\end{equation}
The question here is to identify the inequalities in \eqref{numerireali}
with a sequence of ``Inequalities'' in a suitable topological space,
let's call it ({$\mathcal{I}$},$\tau$), and to show that the limit of such sequence, in the space
({$\mathcal{I}$},$\tau$), is the ``Inequality'' identified with \eqref{limnumerireali}.

\subsection{Hardy's inequality, part I}

Let $p>1$, $f$ be a nonnegative (Lebesgue) measurable function on $(0,1)$.
The classical Hardy's integral inequality states that (\cite{HLP})
\begin{equation}
\int_{0}^{1} \Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)^{p}dx
\le \Big( \frac{p}{p-1}\Big)^{p} \int_{0}^{1} f^{p}(x)dx
\label{hardy}
\end{equation}
When $p\to 1+$ the constant (which is the best one such that
\eqref{hardy} holds) $\big( \frac{p}{p-1}\big)^{p}$ blows
up, and this leads immediately to conjecture that
it cannot exist a constant $c>0$ such that the inequality
$$
\int_{0}^{1} \Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)dx
\le c \int_{0}^{1} f(x)dx
\label{hardyp=1bad}
$$
holds for any $f$. This conjecture is in fact true: it is sufficient
to consider the sequence $f_n(x)=x^{-1+1/n}$ . Nevertheless,  the
``limiting'' case of \eqref{hardy}, when $p\to 1+$, can be expressed through
the norm of the Zygmund
space $LlogL(0,1)$:
\begin{equation}
\int_{0}^{1} \Big(\frac{1}{x} \int_{0}^{x}f(t)dt\Big) dx
\le C \| f \|_{LlogL(0,1)},
\label{hardyp=1good}
\end{equation}
where
\begin{equation}
  \| f \|_{LlogL(0,1)}=\inf \big\{ \lambda>0:\int_0^1
| \frac{f(x)}{\lambda}| \log \big(e+
 | \frac{f(x)}{\lambda}| \big)dx\leq 1\big\}.
\label{zyg}
\end{equation}
For a recent digression on equivalent norms in $LlogL$ see e.g. \cite{FK}.

As in the previous example, setting e.g. $p=1+1/n$ in \eqref{hardy}, it is natural
to ask whether in some topological space it is really true that the sequence of
Hardy's inequalities \sl converges \rm to the inequality \eqref{hardyp=1good}.




\subsection{Sobolev inequalities for fractional Sobolev spaces, part I}

Let $\Omega\subset\mathbb{R}^N$ $(N\ge 1)$ be a bounded smooth open set,
and let $p\ge 1$. Consider the classical Sobolev space $W_0^{1,p}(\Omega)$,
defined as the completion of $\mathcal{C}_0^\infty(\Omega)$ - the set of all functions
defined on $\Omega$ which have derivatives of any order on $\Omega$, whose
supports are compact sets - in the norm
$\| \nabla f\|_{L^p(\Omega)}$.  Let $0<s<1$ and consider the fractional
Sobolev space $W_0^{s,p}(\Omega)$,
defined (see e.g. \cite{K, MS}) as the completion of $\mathcal{C}_0^\infty(\Omega)$ in the norm
$$
\|f\|_{W^{s,p}(\Omega)}=\Big( \int_\Omega
\int_\Omega \frac{ |f(x)-f(y)|^p}{|x-y|^{N+sp}}\,dx\,dy\Big)^{1/p}
$$
The following version of Sobolev inequality for fractional Sobolev spaces holds
(see e.g. \cite{A, T, BBM}):
\begin{equation}
\|f\|_{L^q(\Omega)}\le c(s,p,N) \|f\|_{W_0^{s,p}(\Omega)}
\label{sobembed}
\end{equation}
where $sp<N$, $q=Np/(N-sp)$.
Putting formally $s=1$ in \eqref{sobembed} we get
\begin{equation}
\|f\|_{L^{Np/(N-p)}(\Omega)}\le c(p,N) \|f\|_{W_0^{1,p}(\Omega)}
\label{sobembedclas}
\end{equation}
It would be natural to think the inequality \eqref{sobembedclas}
as limit, in the topological sense, of \eqref{sobembed}, as
$s\to 1$ (notice that, differently from the case of Hardy's
inequality previously discussed, inequality \eqref{sobembedclas} is true!).
The main point here is that
when $s\to 1$ the norm in $W_0^{s,p}(\Omega)$ blows up.
This problem has been studied in
\cite{BBM, BBM2, MS}, and solved by proving a version
of \eqref{sobembed} where the right dependence of the constant
$c(s,p,N)$ with respect to $s$ has been found. We will examine
the problem of the convergence (in the topological sense) of
\eqref{sobembed} to \eqref{sobembedclas} in Section 5.3.


\section{Preliminary considerations for the well-posedness
of the problem}

Some preliminary considerations are due
because the risk is that
the problem is trivial or not well-posed.

Let us analyze, for the moment, the first example (Example 2.1). Let
$c_{n}$ be a sequence of positive real numbers such that
$c_{n}\to 0$, say, $c_{n}=1/n$. Then for each $n\in N$ the inequality
$a_{n}\le b_{n}$ is \emph{equivalent} to $a_{n}c_{n}\le b_{n}c_{n}$, and
our topology should be chosen in such a way that both sequences of
inequalities converge to the same inequality. But while it is natural that
the first one converges to $a\le b$,
the second one (for analogous reason)
should converge to the trivial $0\le 0$. This conclusion
shows a first difficulty for our construction.

To avoid phenomena like the previous one, we
should look carefully to the phenomenon described before.
Suppose that $c_n\to c$, where $c$ is \emph{positive}. In this case
no contradiction arises, because the limit of ``$a_{n}\le b_{n}$'',
i.e. ``$a\le b$'', is \emph{equivalent}
to the limit of ``$a_{n}c_{n}\le b_{n}c_{n}$'', i.e. ``$ac\le bc$''.
Therefore, in order to overcome the difficulty,
we need always to consider inequalities in which both the
left hand side and the right hand side are not zero, but
positive (we will give in the sequel a precise meaning to this
sentence). Of course, starting from $a_{n}\le b_{n}$, one can
always consider the equivalent inequality $a_{n}c_{n}\le b_{n}c_{n}$,
but the limit inequality, in order to be called \emph{limit inequality},
must have each side different from zero. In the most general
setting, this means that we have to introduce a class of
possible left hand sides and  right hand sides (the \emph{admissible
operators}) in which the trivial zero
must be excluded.

The comment above suggests also that
if one inequality in a sequence of inequalities is changed
by another \emph{equivalent} inequality, the limit
should no change. This means that, for a given inequality,
we must consider all the equivalent inequalities,
and treat all of them in the same way. It will be natural, therefore,
to consider classes of equivalence of inequalities,
that we will call ``Inequalities'' with capital ``I'', and speak about limits
of ``Inequalities''.

Consider now the second example (Example 2.2). If we start our limiting process
from inequality \eqref{hardy}
$$
\int_{0}^{1} \Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)^{p}dx
\le \Big( \frac{p}{p-1}\Big)^{p} \int_{0}^{1} f^{p}(x)dx
$$
or from the same inequality raised to $1/p$:
\begin{equation}
\Big(\int_{0}^{1} \Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)^{p}dx\Big)^{1/p}
\le \Big( \Big( \frac{p}{p-1}\Big)^{p} \int_{0}^{1} f^{p}(x)dx\Big)^{1/p}
\label{goood}
\end{equation}
or, say, from the same inequality raised to $2$:
\begin{equation}
\Big(\int_{0}^{1} \Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)^{p}dx\Big)^2
\le \Big( \Big( \frac{p}{p-1}\Big)^{p} \int_{0}^{1} f^{p}(x)dx\Big)^2
\label{baaad}
\end{equation}
we should expect to get the same limit, in the sense that the
inequalities (possibly different) obtained as limits (respectively) of \eqref{hardy},
\eqref{goood}, \eqref{baaad} should be equivalent each other.
This gives an idea of the properties that we must require, to
a couple of inequalities, in order to be defined ``equivalent''.

The plan of the rest of the paper is the following:
in the next section  we define the
admissible operators and we
introduce a method to \emph{homogenize} an inequality. Besides
being of interest in itself, this method will play a key role for one of our main
examples (Hardy's inequality), which will be discussed later. In Section 5
we introduce a topology in inequalities, and describe the notion of convergence.
Such notion is applied to three main examples:
sequences of inequalities relating real numbers, sequences of classical Hardy's inequalities, and
sequences of embedding inequalities for fractional Sobolev spaces.
In Section 6 we introduce a notion of equivalent inequalities, and we construct
an abstract setting, starting from the notions
already introduced, which seems more suitable for
inequalities relating admissible operators. Finally, in Section 7,
we compute explicitly some limits and we see how the notion of convergence
introduced in the paper can be a tool to derive new results.

\section{Homogenizing inequalities}\label{homog}

We begin by introducing some notation. If not differently specified,
 $\Omega\subset\mathbb{R}^N$ $(N\ge 1)$ will denote a bounded smooth open set.
 $\mathcal{C}_{0,+}^\infty(\Omega)$ will be the set of all nonnegative functions
defined on $\Omega$ which have derivatives of any order, and whose
supports are compact sets. We will denote by ${\bf 0}$ the function
 whose value is zero on all $\Omega$.

 Let  $\mathcal{O}$ be the set of all operators (we will call them
admissible)
$$
T: \mathcal{C}_{0,+}^\infty(\Omega) \to [0,+\infty[
$$
 such that, setting
 $$
 F_{T,f}: \lambda\in [0,+\infty[ \; \to \; F_{T,f}(\lambda)
=T(\lambda f)\in [0,+\infty[ \quad \forall f\in \mathcal{C}_{0,+}^\infty(\Omega)
 $$
 it is
 \begin{gather}
 F_{T,f} \text{ is continuous for all }f\in \mathcal{C}_{0,+}^\infty(\Omega)
  \label{cont}
\\
 F_{T,f} \text{ is strictly  increasing for all
$f\in \mathcal{C}_{0,+}^\infty(\Omega)$, $f\neq {\bf 0}$}
  \label{incr}
\\
 \lim_{\lambda\to\infty}F_{T,f} (\lambda)=+\infty \quad
\forall f\in \mathcal{C}_{0,+}^\infty(\Omega), \; f\neq {\bf 0}
  \label{infinity}
\\
 \inf_{\lambda>0}T\big( \frac{f}{\lambda}\big)
=T({\bf 0}):=m_T\in [0,+\infty[\quad \forall
f\in \mathcal{C}_{0,+}^\infty(\Omega)
 \label{inzero}
\end{gather}
For $T$, $S$ in $\mathcal{O}$
we shall often write
 $$
d=d(T,S),$$
   instead of
$$
Tf\le Sf \quad \forall f\in \mathcal{C}_{0,+}^\infty(\Omega)
$$
  We observe that the operators that we consider are very common
  in Analysis, because many known inequalities have both sides
  enjoying the properties listed above.

  Before listing some examples,
  let us recall some definitions which will be useful in the sequel.

  A function $A:[0,+\infty[\to[0,+\infty[$ is called \emph{N-function} if
  it is continuous, strictly increasing, convex and such that
  $$
  \lim_{t\to0}\frac{A(t)}t=0,\quad  \lim_{t\to\infty}\frac{A(t)}t=\infty.
  $$
  Typical examples of N-functions are powers, with exponent greater than $1$.
  Starting from the notion of N-function it is possible to consider the norm
  $$
  \|f\|_A=\| f \|_{L^A(0,1)}=\inf \big\{ \lambda>0:\int_0^1 A
\big(\big| \frac{f(x)}{\lambda}\big| \big)dx\leq 1\big\}
  $$
  which defines the \emph{Orlicz} space $L^A(0,1)$. If $A(t)=t\log (e+t)$,
  we get the norm considered
  in \eqref{zyg}; in this case the Orlicz space is called \emph{Zygmund} space.
  We refer to \cite{FK}  for expressions for the norm in such spaces.
  For properties and
  further examples of N-functions and Orlicz spaces see e.g. \cite{A}.

  Let $f$ be a (Lebesgue) measurable function defined on $(0,1)$, a.e. finite,
and for any   Lebesgue measurable set
  $E\subset (0,1)$ let $|E|$ be its measure. The \emph{decreasing
rearrangement} of $f$ is the function, denoted by $f^*$, defined by
  $$
  f^*(t)=\inf\{ \lambda>0 : | \{ x\in (0,1) : |f(x)|>\lambda \} |\le t \} \quad t\in (0,1)
  $$
  This definition is usually given for a much more general class of functions, but
  it is not in our purposes to give details here. For interested readers we refer to \cite{BS}.

  \begin{example}  \label{exa4.1}\rm
  Let $X$ be a Banach space whose elements are measurable functions. Suppose that
  $\mathcal{C}_0^\infty(\Omega)\subset X$. An example of operator
  in $\mathcal{O}$ is
  $$
  T_1f=\|f\|_X \quad \forall f\in \mathcal{C}_{0,+}^\infty(\Omega).
  $$
  \end{example}

  \begin{example}  \label{exa4.2} \rm
Let $p$ be a measurable function on $\Omega$, whose values are in
$[1,\infty]$, and set $\Omega_\infty=\{ x\in\Omega : p(x)=\infty\}$. Then an
  example of operator
  in $\mathcal{O}$ is
$$
T_2f=\inf\big\{ \lambda>0:
\int_{\Omega\backslash\Omega_\infty}\big| \frac{f(x)}\lambda
\big|^{p(x)}dx+\mathop{\rm ess\,sup}_{\Omega_\infty}
\big| \frac{f(x)}\lambda \big|\le1 \big\}
$$
This operator is a special case of the previous example, in fact it is the norm in
 the space $L^{p(\cdot)}(\Omega)$. For details see \cite{KR}.
\end{example}

  \begin{example} \label{exa4.3}\rm
  Let $A:[0,+\infty[\to[0,+\infty[$ be an N-function. Then an
  example of operator
  in $\mathcal{O}$ is
  $$
  T_3f=\int_\Omega A(f)dx \quad \forall f\in \mathcal{C}_{0,+}^\infty(\Omega).
  $$
  \end{example}

    \begin{example} \label{exa4.4} \rm
  Let $A_1, A_2, A_3:[0,+\infty[\to[0,+\infty[$ be  continuous and strictly
increasing functions such that
  $A_i(+\infty)=+\infty$, $i=1,2,3$, and let
  $w_1, w_2$ be nonnegative, locally integrable functions defined respectively in $\Omega\times\Omega$
  and $\Omega$, such that
  $$
  w_1(x, \cdot)\not\equiv 0\quad \forall x\in\Omega, \quad
 w_2>0 \quad\text{a.e. in }\Omega.
  $$
  Then an   example of operator
  in $\mathcal{O}$ is
  $$
  T_4f=A_1\Big( \int_\Omega A_2\Big(   \int_\Omega A_3(f(y))w_1(x,y)dy
  \Big)w_2(x)dx\Big) \quad \forall f\in \mathcal{C}_{0,+}^\infty(\Omega)
  $$
 Next we consider the particular case: $\Omega=]0,1[\subset\mathbb{R}$,
  $A_1(t)=A_2(t)=A_3(t)=t$, $w_1(x,y)=\chi_{(0,x)}(y)$, $w_2(x)=1/x$,
 which gives the operator
  $$
  T_5=\int_0^1 \Big( \frac1x \int_0^x f(t)dt\Big) dx \quad \forall
f\in \mathcal{C}_{0,+}^\infty(\Omega)
  $$
  \end{example}

\begin{example} \label{exa4.5} \rm
  Let $A_1, A_2:[0,+\infty[\to[0,+\infty[$ be  N-functions and
  let $X$ be a Banach Function Space. Then an
  example of operator
  in $\mathcal{O}$ is
  $$
  T_6f=A_1( \|A_2(f)\|_X) \quad \forall f\in \mathcal{C}_{0,+}^\infty(\Omega).
  $$
  \end{example}

   \begin{example} \label{exa4.6}\rm
  Let us denote by $f^*$ the decreasing rearrangement of $f$, defined in
  the interval $]0, |\Omega|]$. An example of operator
  in $\mathcal{O}$ is
  $$
  T_7f=f^*\big(\frac{|\mathop{\rm supp}(f)|}{2}\big)\quad \forall
f\in \mathcal{C}_{0,+}^\infty(\Omega)
  $$
  where $|\mathop{\rm supp}(f)|$ denotes the Lebesgue measure of the
support of $f$.
    \end{example}

   \begin{example}  \label{exa4.7} \rm
   Let us denote by $f^*$ the decreasing rearrangement of $f$, defined in
  the interval $]0, |\Omega|]$, let $A_1$ be a strictly increasing, continuous
  function on $[0, |\Omega|]$ such that $A_1(0)=0$, $A_1(+\infty)=+\infty$,
  let $A_2:[0,+\infty[\to[0,+\infty[$ be an  N-function, and let
  $w_1, w_2$ be positive, locally integrable functions defined in $]0,|\Omega|[$.
  Then an
  example of operator
  in $\mathcal{O}$ is
  $$
  T_8f=\int_0^{|\Omega|}A_1\Big( \int_0^t A_2 (f^*(s))w_1(s)ds\Big)w_2(t)dt
\quad \forall f\in \mathcal{C}_{0,+}^\infty(\Omega)
  $$
  Operators of this type occur in Function Space Theory, see e.g. \cite{FKa}.
  \end{example}

Let us now introduce a notion which will play a key role in the sequel.
For each $T\in \mathcal{O}$ we define the
\emph{associate family $\{T^{(\mu)}\}$ of
homogeneous operators}  as the class of operators
$T^{(\mu)}: \mathcal{C}_{0,+}^\infty(\Omega) \to [0,+\infty[$
defined by
$$
T^{(\mu)}f=\inf \big\{ \lambda>0: T\big( \frac{f}{\lambda}\big)\le
\mu\big\}\quad\forall \mu>m_T\quad
\forall f\in \mathcal{C}_{0,+}^\infty(\Omega)
$$
We remark that since $T\in \mathcal{O}$ the set
$$
\big\{ \lambda>0: T\big( \frac{f}{\lambda}\big)\le \mu\big\}
$$
is nonempty for all $f\in \mathcal{C}_{0,+}^\infty(\Omega)$ and all
$\mu>m_T$, so that the definition of $T^{(\mu)}f$
 is well posed for all $\mu>m_T$.
The homogeneity of the operator
$T^{(\mu)}$ is proved by the following result.

\begin{proposition}
If $T$ is admissible, then for all $\mu>m_T$ the operator $T^{(\mu)}$
is admissible and has the further property to be homogeneous (of degree $1$):
$$
T^{(\mu)}(kf)=kT^{(\mu)}f\quad \forall f\in \mathcal{C}_{0,+}^\infty(\Omega)
\; \forall k\ge 0.
$$
\label{homo}
\end{proposition}

\begin{proof} First we prove the homogeneity. If $k=0$ it is
sufficient to see that
\begin{equation}
T^{(\mu)}{\bf 0}=0\quad \forall \mu>m_T,
\label{T0}
\end{equation}
and this is trivial, because for all $\lambda>0$ it is
$$
T\big( \frac{{\bf 0}}{\lambda}\big)=T{\bf 0}\le \mu\quad \forall \mu>m_T.
$$
For $k>0$ we have
\begin{align*}
T^{(\mu)}(kf)
&=\inf \big\{ \lambda>0: T\big(\frac{kf}{\lambda}\big)\le \mu\big\}\\
&=\inf \big\{ k\lambda>0: T\big( \frac{f}{\lambda}\big)\le \mu\big\} \\
&=k\inf \big\{ \lambda>0: T\big( \frac{f}{\lambda}\big)\le \mu\big\} \\
&=kT^{(\mu)}(f)
\end{align*}
Now we show that the operator $T^{(\mu)}$
is admissible. Property \eqref{cont} is trivial, because
the homogeneity of $T^{(\mu)}$ implies that
$$
\lambda\in [0,+\infty[ \to T^{(\mu)}(\lambda f)\in [0,+\infty[
$$
is linear. In order to show properties \eqref{incr} and \eqref{infinity},
because of the homogeneity of $T^{(\mu)}$, it is sufficient to
see that
$$
f\neq {\bf 0} \Longrightarrow T^{(\mu)}(f)>0
$$
Since $T$
is admissible, property  \eqref{infinity} is true for
$T$, therefore there exists $\lambda>0$ such that
$ T\big( \frac{f}{\lambda}\big)>\mu$. The conclusion is
that $T^{(\mu)}(f)>0$.

Finally, we observe that both sides of  \eqref{inzero}
are equal to zero: the left hand side because of the homogeneity
of $T^{(\mu)}$, the right hand side because of \eqref{T0}.
\end{proof}

\begin{remark} \rm
If $T$ is homogeneous, then it is easy to show that $T^{(\mu)}=(1/\mu)T$
 for all $\mu>0$.
\label{Thomo}
\end{remark}

The main result of this Section is the following.

\begin{theorem}
Let $T,S\in  \mathcal{O}$. The following equivalence holds:
$$
Tf\le Sf\quad\forall f\in \mathcal{C}_{0,+}^\infty(\Omega)
$$
if and only if
$$
T^{(\mu)}f\le S^{(\mu)}f \quad \forall f\in \mathcal{C}_{0,+}^\infty(\Omega)
\quad \forall \mu>\max(m_T, m_S)
$$
\label{homogen}
\end{theorem}

\begin{proof} Let us assume first that
\begin{equation}
Tf\le Sf\quad\forall f\in \mathcal{C}_{0,+}^\infty(\Omega)
\label{lhs}
\end{equation}
and let $\mu>\max(m_T, m_S)$. Fix $f\in \mathcal{C}_{0,+}^\infty(\Omega)$
and let $\lambda>0$ be such that $S\big(\frac{f}{\lambda}\big)\le \mu$.
 By \eqref{lhs} the number $\lambda$ is also such that
$T\big(\frac{f}{\lambda}\big)\le \mu$; therefore,
$$
\big\{ \lambda>0: T\big( \frac{f}{\lambda}\big)\le
\mu\big\}\supseteq \big\{ \lambda>0: S\big( \frac{f}{\lambda}\big)\le
\mu\big\}
$$
from which the first part of the assertion follows.
\par
On the other hand, by contradiction, let us assume that there exists
$ \overline{f}$ such that
$$
T\overline{f}>S\overline{f}
$$
We observe that $\overline{f}$ can be always chosen different from
${\bf 0}$. In fact, if
$$
T{\bf 0}>S{\bf 0}
$$
 then, fixing any $f\neq {\bf 0}$, by
property \eqref{cont} for $S$ in $\lambda=0$, there exists $\lambda$ sufficiently small
such that
$$
T{\bf 0}>S(\lambda f)>S{\bf 0}
$$
and therefore, by property \eqref{incr} for $T$,
$$
T(\lambda f)>T{\bf 0}>S(\lambda f).
$$
Setting $\overline{f}=\lambda f$ we get the existence of
$\mu>\max(m_T,m_S)$ such that
\begin{equation}
T\overline{f}>\mu>S\overline{f}
\label{4contr}
\end{equation}
Now from the inequality $S\overline{f}<\mu$, applying
property \eqref{cont} for $S$ in $\lambda=1$, we can consider
$\epsilon>0$ such that
$$
1-\epsilon\in \{ \lambda>0: S\big( \frac{\overline{f}}{\lambda}\big)\le
\mu\}
$$
By our assumption
$$
\inf \big\{ \lambda>0: T\big( \frac{\overline{f}}{\lambda}\big)\le
\mu\big\}= T^{(\mu)}{\overline{f}}\le S^{(\mu)}{\overline{f}}\le 1-\epsilon
$$
from which
$T\overline{f}\le \mu$; this conclusion is in contrast with \eqref{4contr}.
\end{proof}

In the sequel we will use the first implication proved above, which, starting
from a generic inequality, leads to a family of inequalities relating homogeneous
operators. We will say that such family of inequalities is obtained
\emph{homogenizing} the original inequality.


\subsection*{Application 1.}
 Let $\varphi:[0,+\infty[\to[0,+\infty[$ be increasing and such that
$\varphi(0)=0$, and
suppose to know that
\begin{equation}
\varphi(\| f\|_X)\le \| f\|_Y \quad  \forall
f\in \mathcal{C}_{0,+}^\infty(\Omega),
\label{appl1}
\end{equation}
where $X$ and $Y$ are Banach Function Spaces (see \cite[Def 1.3 p. 3]{BS}),
 such that $\mathcal{C}_0^\infty(\Omega)$
functions are dense in $X$ and $Y$. Applying  \cite[Theorem 1.8 p. 7]{BS},
one can deduce that $Y$ is continuously embedded into $X$; therefore,
there exists a constant $k>0$ such that
$$
\| f\|_X\le k \|f\|_Y
$$
We observe that it is possible to get the same conclusion homogenizing
the inequality \eqref{appl1}, getting also an estimate of the constant $k$.

In fact, by Theorem \ref{homogen}, from  \eqref{appl1} we get, for all
$\mu>0$ and $f\in \mathcal{C}_{0,+}^\infty(\Omega)$,
\begin{gather*}
\inf \big\{ \lambda>0: \varphi\big( \big\|\frac{f}{\lambda}\big\|_X\big)\le
\mu\big\} \le
\inf \big\{ \lambda>0:  \big\|\frac{f}{\lambda}\big\|_Y\le
\mu\big\},
\\
\inf \big\{ \lambda>0: \frac1\lambda \|f\|_X\le \sup\{ \xi:\varphi(\xi)\le \mu\}
\big\} \le  \frac1\mu \|f\|_Y,
\\
\|f\|_X\le \frac{\sup\{ \xi:\varphi(\xi)\le \mu\}}\mu  \|f\|_Y\,.
\end{gather*}
In conclusion:
$$
\|f\|_X\le \inf_{\mu>0} \frac{\sup\{ \xi:\varphi(\xi)\le \mu\}}\mu
 \|f\|_Y\quad  \forall f\in \mathcal{C}_{0,+}^\infty(\Omega)
$$
and therefore the same inequality is true for all $f$, due to the assumed
 density
of the $\mathcal{C}_0^\infty(\Omega)$ functions.


\subsection*{Application 2.}  Let $\Phi$ be an N-function.
  By the well-known Jensen inequality it is
  $$
  \Phi\big( \int_0^1 f(x)dx\big)\le  \int_0^1 \Phi(f)dx
\quad\forall f\in \mathcal{C}_{0,+}^\infty(\Omega)
  $$
  Let us homogenize this inequality.
For all $\mu>0$ and $f\in \mathcal{C}_{0,+}^\infty(\Omega)$ we get
\begin{gather*}
\inf\{ \lambda>0:  \Phi\big( \int_0^1 \frac{f(x)}\lambda dx\big)
\le\mu \big\}
\le \inf \big\{ \lambda>0:  \int_0^1\Phi\big( \frac{f(x)}\lambda \big) dx
\le\mu \big\},
\\
\inf \big\{ \lambda>0:   \int_0^1 \frac{f(x)}\lambda dx\le\Phi^{-1}(\mu)
 \big\}
\le
\inf \big\{ \lambda>0:  \int_0^1\big(\frac\Phi\mu\big)\big( \frac{f(x)}\lambda
 \big) dx\le1 \big\},
\\
\frac1{\Phi^{-1}(\mu)}\int_0^1 f(x)dx\le \|f\|_{\Phi/\mu},
\\
\int_0^1 f(x)dx\le \Phi^{-1}(\mu) \|f\|_{\Phi/\mu}
\end{gather*}
For $\mu=1$ such inequality reduces to
$$
\int_0^1 f(x)dx\le \Phi^{-1}(1) \|f\|_{\Phi}
$$
which is the inequality which shows that the Orlicz space $L^\Phi(0,1)$
is embedded in  $L^1(0,1)$. Notice that the
constant $\Phi^{-1}(1)$ on the right hand side is optimal
(the inequality becomes equality for $f\equiv 1$).

\subsection*{Application 3.}
 Let us consider - as usual, we will consider functions
  $f$ in $\mathcal{C}_{0,+}^\infty$ - a well known inequality
  by Hardy and Littlewood (see \cite[241 (i) page 169]{HLP}, or
\cite[ vol.1, p.32, Theorem 13.15(iii)]{Z}):
  \begin{equation}
  \int_0^1 \Big( \frac1x \int_0^x f(t)dt\Big) dx
\le c_1 \int_0^1 f(x)\log^+f(x)dx+c_2
  \label{appl3}
  \end{equation}
  which is true for some $c_1, c_2>0$. Here $\log^+t=\max (\log t, 0)$. We now compute
  the family of the homogenized inequalities. Since the left hand side is already
  homogeneous, by Remark \ref{Thomo} it is immediate to compute the associated
  family of homogeneous operators. Let us consider the right hand side:
 $$
 Tf=c_1 \int_0^1 f(x)\log^+f(x)dx+c_2=\int_0^1 [c_1f(x)\log^+f(x)+c_2] dx
 $$
 Let $c_3>0$ be such that
 $$
 c_1t\log^+t+c_2\ge c_3t\log(e+t)\quad\forall t>0\,.
 $$
For all $\mu>0$ we have,
\begin{align*}
T^{(\mu)}(f)
&=\inf\big\{\lambda>0: T\big( \frac{f}{\lambda}\big)\le
\mu\big\}\\
&=\inf\big\{\lambda>0: \int_0^1 \big[c_1\frac{f(x)}{\lambda}\log^+
\frac{f(x)}{\lambda}+c_2\big] dx\le \mu\big\}\\
&\ge \inf\big\{\lambda>0: \int_0^1 c_3\frac{f(x)}{\lambda}\log
\big( e+\frac{f(x)}{\lambda}\big)dx\le \mu\big\} \\
&=\inf\big\{\lambda>0: \int_0^1 \frac{c_3}\mu\frac{f(x)}{\lambda}\log
\big( e+\frac{f(x)}{\lambda}\big)dx\le 1\big\} \\
&\ge \min \big(1, \frac{c_3}\mu\big)\|f\|_{LlogL(0,1)}
\end{align*}
where $\|f\|_{LlogL(0,1)}$ is defined in \eqref{zyg}. Last inequality is
easily obtained considering separately the cases $c_3\ge \mu$, $c_3<\mu$.
 On the other hand, for all $\mu>c_2$ we have
\begin{align*}
T^{(\mu)}(f)
&=\inf\big\{\lambda>0: T\big( \frac{f}{\lambda}\big)\le \mu\big\} \\
&=\inf\big\{\lambda>0: \int_0^1 \big[c_1\frac{f(x)}{\lambda}\log^+
\frac{f(x)}{\lambda}+c_2\big] dx\le \mu\big\} \\
&=\inf\big\{\lambda>0: \int_0^1c_1\frac{f(x)}{\lambda}\log^+
 \frac{f(x)}{\lambda} dx\le \mu-c_2\big\} \\
&\le \inf\big\{\lambda>0: \int_0^1 \frac{c_1}{\mu-c_2}\frac{f(x)}{\lambda}
\log\big( e+\frac{f(x)}{\lambda}\big) dx\le 1\big\} \\
&\le \max \big(1, \frac{c_1}{\mu-c_2}\big)\|f\|_{LlogL(0,1)}
\end{align*}
In conclusion, the homogenized family of inequalities of \eqref{appl3}
can be written as
$$
 \int_0^1 \big( \frac1x \int_0^x f(t)dt\big) dx\le C(\mu) \|f\|_{LlogL(0,1)}
\quad \forall\mu>c_2.
 $$
We observe that such inequalities are of the type \eqref{hardyp=1good}.
\medskip

We conclude this Section answering to a natural question: What happens
if we homogenize one of the inequalities obtained after a homogenization?

Let $T,S\in  \mathcal{O}$ and fix a number $\mu>\max(m_T, m_S)$.
If
$$
Tf\le Sf\quad\forall f\in \mathcal{C}_{0,+}^\infty(\Omega)
$$
by Theorem \ref{homogen} we know that
$$
T^{(\mu)}f\le S^{(\mu)}f
\quad \forall f\in \mathcal{C}_{0,+}^\infty(\Omega)
$$
Let us now apply again Theorem \ref{homogen}
to this inequality, and consider
$$
T^{(\mu)(\sigma)}f\le S^{(\mu)(\sigma)}f
\quad \forall f\in \mathcal{C}_{0,+}^\infty(\Omega)\quad \forall\sigma>0
$$
By Remark \ref{Thomo} we get
$$
\frac1\sigma T^{(\mu)}f\le \frac1\sigma S^{(\mu)}f
\quad \forall f\in \mathcal{C}_{0,+}^\infty(\Omega)\quad\forall\sigma>0
$$
i.e.
$$
T^{(\mu)}f\le S^{(\mu)}f \quad \forall f\in \mathcal{C}_{0,+}^\infty(\Omega)
$$
The conclusion is that for each fixed $\mu>0$ the inequality
$T^{(\mu)}f\le S^{(\mu)}f $
has a trivial family of associated homogeneous inequalities,
pairwise identical to the original one.


\section{A topology on inequalities}

Let us consider the set of inequalities
$$
{\mathcal{I}}_{0}=\{ d(T,S) : T,S\in {\mathcal{O}}\}
$$
We  construct a topology in ${\mathcal{I}}_{0}$, taking inspiration
from the classical procedure used to define the topology of pointwise
convergence.

Fix $d=d(T,S)\in {\mathcal{I}}_{0}$. For any $n\in \mathbb{N}$, for any finite
subset
$F\subset \mathcal{C}_{0,+}^\infty(\Omega)$, let us set
$$
{\mathcal{U}}_{n,F}(d)=\{ d'=d'(T',S'): |T'f-Tf|<\frac1n \; \forall f\in F,
\; |S'f-Sf|<\frac1n \; \forall f\in F\}
$$
Let ${\mathcal{N}}(d)$ be the family of subsets of ${\mathcal{I}}_{0}$
whose elements contain some set of the type ${\mathcal{U}}_{n,F}(d)$:
$$
A\in {\mathcal{N}}(d) \Leftrightarrow \exists n\in\mathbb{N} , \; \exists
F\subset \mathcal{C}_{0,+}^\infty(\Omega)   \text{ finite: }A
\supseteq {\mathcal{U}}_{n,F}(d).
$$
We will prove the following result.

\begin{proposition}
The family
$\{ {\mathcal{N}}(d) \}_{d\in {\mathcal{I}}_{0}}$
satisfies the following Hausdorff's axioms:
\begin{itemize}
\item[(i)] $d\in A$ for all $A\in {\mathcal{N}}(d)$

\item[(ii)]$ A\in {\mathcal{N}}(d)$ and $B\in {\mathcal{N}}(d)$
implies $A\bigcap B\in {\mathcal{N}}(d)$

\item[(iii)] $A\in {\mathcal{N}}(d)$ and $B\supseteq A$ implies
$B\in {\mathcal{N}}(d)$

\item[(iv)] for all $A\in {\mathcal{N}}(d)$ there exists
$B\in {\mathcal{N}}(d)$ such that $A\in {\mathcal{N}}(d')$ for
all $d'\in B$
\end{itemize}
\end{proposition}

After this proposition we know (see e.g. \cite[Theorem 3.2, p. 67]{D})
that there exists one and only one topology
${\tau}$ for ${\mathcal{I}}_{0}$ such that ${\mathcal{N}}(d)$ is,
for any $d\in {\mathcal{I}}_{0}$, the set of the nbds of $d$.

\begin{proof}
(i)
Let $A\in {\mathcal{N}}(d)$ and let ${\mathcal{U}}_{n,F}(d)\subseteq A$.
 Since $d\in {\mathcal{U}}_{n,F}(d)$, it is $d\in A$.

\noindent (ii)
Let ${\mathcal{U}}_{n,F}(d)\subseteq A$, ${\mathcal{U}}_{m,G}(d)\subseteq B$.
Then ${\mathcal{U}}_{\max (n,m),F\bigcup G}(d)\subseteq A\bigcap B$.

\noindent (iii)
Let $A\in {\mathcal{N}}(d)$ and let ${\mathcal{U}}_{n,F}(d)\subseteq A$. Since
$B\supseteq A$, then $B\supseteq {\mathcal{U}}_{n,F}(d)$;
 therefore $B\in  {\mathcal{N}}(d)$

\noindent(iv)
Let $A\in {\mathcal{N}}(d)$ and let ${\mathcal{U}}_{n,F}(d)\subseteq A$. We show that
(iv) is true with $B={\mathcal{U}}_{n,F}(d)$:
$$
d'\in {\mathcal{U}}_{n,F}(d)\quad  \Rightarrow \quad A\in {\mathcal{N}}(d')
$$
To this goal we need to find some ${\mathcal{U}}_{\nu,F}(d')$ such that
${\mathcal{U}}_{\nu,F}(d')\subset A$. Since $d'\in {\mathcal{U}}_{n,F}(d)$,
\begin{gather*}
|T'f-Tf|<\frac1n\quad\forall f\in F, \\
|S'f-Sf|<\frac1n\quad\forall f\in F\,.
\end{gather*}
Since $F$ is finite, we may consider $\nu\in\mathbb{N}$ such that
$$
\frac1\nu < \min\big\{ \min_F \{ \frac1n - |T'f-Tf| \},
\min_F \{ \frac1n - |S'f-Sf| \} \big\}.
$$
Let $d''\in {\mathcal{U}}_{\nu,F}(d')$. Then
\begin{gather*}
|T''f-T'f|<\frac1\nu\quad\forall f\in F,\\
|S''f-S'f|<\frac1\nu\quad\forall f\in F\,.
\end{gather*}
We have
$$
|T''f-Tf|\le |T''f-T'f|+|T'f-Tf|<\frac1\nu + |T'f-Tf|
<\frac1n \quad\forall f\in F
$$
and similarly $|S''f-Sf|<1/n$ for all $f\in F$. Therefore,
$d''\in {\mathcal{U}}_{n,F}(d)\subseteq A$.
\end{proof}

At this point we have a topology ${\tau}$ in ${\mathcal{I}}_{0}$.
The notion of convergence in this topology is, as usual,
$$
d_n\to d\text{ in } {\tau} \quad \Leftrightarrow\quad
\forall {\mathcal{U}}_{\nu,F}(d)\; \exists \, n_0\in\mathbb{N}:
d_n\in {\mathcal{U}}_{\nu,F}(d) \; \forall  n>n_0
$$
Set
$d_n=d_n(T_n,S_n)$, $d=d(T,S)$. It is readily seen that $d_n\to d$ if
and only if
\begin{equation}
\forall \, f\in \mathcal{C}_{0,+}^\infty(\Omega) \quad T_nf\to Tf
\quad\text{and}\quad S_nf\to Sf
\label{converg}
\end{equation}

 Now, we go back to the examples considered in Section 2, and we show how
 a suitable choice of the operators $T_n$, $S_n$, $T$, $S$ gives that
the considered inequalities
 converge to their respective limits in the sense of \eqref{converg}.

\subsection{Inequalities relating real numbers, part II}

Let $(a_{n})$, $(b_{n})$ be sequences of positive real numbers such that
$a_{n}\to a>0$, $b_{n}\to b>0$, and
\begin{equation}
a_{n}\le b_{n}\quad \forall n\in\mathbb{N}
\label{numerireali2}
\end{equation}
We cannot consider the most natural operators
$$
T_nf\equiv a_n\quad S_nf\equiv b_n\quad\forall
f\in \mathcal{C}_{0,+}^\infty(0,1)
$$
because they are not admissible (notice that property \eqref{incr}
does not hold).
Let us set
$$
T_nf\equiv a_n \sup_{(0,1)}f(x)\quad S_nf\equiv b_n\sup_{(0,1)}f(x)\quad
\forall f\in \mathcal{C}_{0,+}^\infty(0,1)
$$
Observe that now the operators $T_n$, $S_n$ are admissible.
The limit (in the sense of \eqref{converg}) of
$$
T_{n}f \le S_{n}f\quad\forall f\in \mathcal{C}_{0,+}^\infty(0,1)
$$
is
$$
Tf \le Sf\quad\forall f\in \mathcal{C}_{0,+}^\infty(0,1),
$$
where
$$
Tf\equiv a \sup_{(0,1)}f(x)\quad Sf\equiv b\sup_{(0,1)}f(x)
\quad\forall f\in \mathcal{C}_{0,+}^\infty(0,1).
$$

\subsection{Hardy's inequality, part II}\label{har}

We start with the homogenized version of inequality \eqref{hardy}:
\begin{equation}
\Big(\int_{0}^{1} \Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)^{p}dx\Big)^{1/p}
\le  \frac{p}{p-1} \Big(\int_{0}^{1} f^{p}(x)dx\Big)^{1/p}
\label{hardy2}
\end{equation}
As a first step, fix $n\in\mathbb{N}$ and
set $p=1+1/n$. Here, again, the most natural operators $T_n$, $S_n$,
respectively equal to the left hand side and the right hand side of
\eqref{hardy2}
do not work, due to the blowup of the right hand side. In this case
$T_n$ and $S_n$ would be admissible and it seems that there cannot be
a better choice.

The important consideration to be made at this point is to understand
$which$ inequality we wish to use before passing to the limit. In fact the standard
proof of Hardy's inequality leads to a better version of \eqref{hardy2}, containing
one more term, which is usually dropped. Taking into consideration such term,
we are able to prove that the limit of Hardy's inequality when $p\to 1+$ is
inequality \eqref{hardyp=1good} (actually, even a better one).

We stress that inequality \eqref{hardyp=1good} has an
independent, classical, simple proof in
\cite[vol.1, p.32, Theorem 13.15(iii)]{Z}.
However, our main intention here is to prove a limiting process and
to obtain, as a byproduct, a new tool for proving inequalities. We believe
that the following procedure has an independent interest.

For completeness, we start here with
the simple proof of \eqref{hardy2} (as usual, we are assuming here to deal
only with $\mathcal{C}_{0,+}^\infty$ functions, not identically zero).
We have
\begin{align*}
\int_{0}^{1}\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)^{p}dx
&=\int_{0}^{1} \Big( \int_{0}^{x}f(t)dt \Big)^{p}x^{-p}dx \\
&=\int_{0}^{1} \Big( \int_{0}^{x}f(t)dt \Big)^{p}d
\big(\frac{x^{1-p}}{1-p}\big) \\
&=\Big[-\frac{x^{1-p}\big( \int_{0}^{x}f(t)dt \big)^{p}}{p-1}
\Big]_{x=0}^{x=1}-
\int_0^1 \big(\frac{x^{1-p}}{1-p}\big) d\Big( \int_{0}^{x}f(t)dt \Big)^{p}\\
&=-\frac{\big( \int_{0}^{1}f(t)dt \big)^{p}}{p-1}
+\frac{p}{p-1}\int_{0}^{1}\Big( \frac{1}{x} \int_{0}^{x}f(t)dt
\Big)^{p-1}f(x)dx
\end{align*}
Applying Holder's inequality,
\begin{align*}
&\int_{0}^{1}\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)^{p}dx\\
&\le -\frac{\big( \int_{0}^{1}f(t)dt \big)^{p}}{p-1}+\frac{p}{p-1}
\Big[\int_{0}^{1}\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)^{p}dx\Big]^{1-1/p}
\Big( \int_{0}^{1}f(t)^pdt \Big)^{1/p}
\end{align*}
At this point Hardy's inequality \eqref{hardy2} is readily obtained
dropping the first added and multiplying each side by
$\Big[\int_{0}^{1}
\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)^{p}dx\Big]^{1/p-1}$.
We now $do$ $not$ drop the first added, which plays a key role when
passing to the limit for $p\to 1+$, and we raise both sides to the power
$1/p$, getting an inequality relating two homogeneous operators,
which we call respectively $T_nf$ and $S_nf$.

We now compute the limit of $T_nf$ and $S_nf$. It is immediate to see that
the limit of $T_nf$ is exactly the left hand side of inequality
 \eqref{hardyp=1good}.
As for $S_nf$, we have
\begin{align*}
(S_nf)^p
&=-\frac{\big( \int_{0}^{1}f(t)dt \big)^{p}}{p-1}
 +\frac{p}{p-1}\Big[\int_{0}^{1} \Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)^{p}
dx\Big]^{1-1/p}\Big( \int_{0}^{1}f(t)^pdt \Big)^{1/p} \\
&= \frac{\big( \int_{0}^{1}f(t)^pdt \big)^{1/p}
 -\big( \int_{0}^{1}f(t)dt \big)^{p}}{p-1} \\
&\quad +\frac{p\big[\int_{0}^{1} \big( \frac{1}{x} \int_{0}^{x}f(t)dt \big)^{p}dx
\big]^{1-1/p}-1}{p-1}\Big( \int_{0}^{1}f(t)^pdt \Big)^{1/p}
\end{align*}
We can now compute the limit as $p\to 1+$, taking into account that
the quotients are in fact difference-quotients
(therefore it suffices to compute derivatives in $p$ and set $p=1$).
We have:
\begin{align*}
&\lim_{p\to 1+}\frac{\big( \int_{0}^{1}f(t)^pdt \big)^{1/p}
 -\big( \int_{0}^{1}f(t)dt \big)^{p}}{p-1}\\
&= \int_0^1 f(x)\log f(x)dx-2\Big( \int_{0}^{1}f(t)dt \Big)\log
\Big( \int_{0}^{1}f(t)dt \Big)
\end{align*}
and
$$
\lim_{p\to 1+}\frac{p\Big[\int_{0}^{1}
\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)^{p}dx\Big]^{1-1/p}-1}{p-1}
=1+\log\Big(\int_{0}^{1}\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)dx\Big).
$$
Hence the limit as $p\to 1+$ is the inequality
\begin{align*}
&\int_{0}^{1}\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)dx \\
&\le \int_0^1 f(x)\log f(x)dx-2\Big( \int_{0}^{1}f(t)dt \Big)
 \log\Big( \int_{0}^{1}f(t)dt \Big)+\int_{0}^{1}f(t)dt \\
&\quad +\Big( \int_{0}^{1}f(t)dt \Big)
 \log\Big(\int_{0}^{1}\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)dx\Big)
\end{align*}
We will prove that this inequality is finer than inequality \eqref{appl3},
 which leads to inequality \eqref{hardyp=1good}(see Application 3 in Section 4).
We consider two cases.

\noindent  First case:
$\int_{0}^{1}\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)dx\le1$
so that the last term of our inequality is nonpositive:
$$
\Big( \int_{0}^{1}f(t)dt \Big)\log\Big(\int_{0}^{1}
\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)dx\Big)\le 0
$$
We can drop it and observe that since
$$
\sup_{t>0} -2t\log t+t=M_1<\infty,
$$
we get
$$
\int_{0}^{1}\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)dx \le
\int_0^1 f(x)\log f(x)dx+M_1
$$
which is an inequality of the type \eqref{appl3}.

\noindent Second case:
$\int_{0}^{1} \Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)dx>1$.
By Young's inequality
$$
ab\le (a+\frac12)\log (1+2a)-a+\frac12(e^b-b-1)\quad\forall a,b\ge0
$$
and therefore, setting
$$
a=\int_{0}^{1}f(t)dt \quad b=\log\Big(\int_{0}^{1}
\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)dx\Big)
$$
the following inequality holds
\begin{align*}
&\Big( \int_{0}^{1}f(t)dt \Big)\log\Big(\int_{0}^{1}
\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)dx\Big) \\
&\le \Big(\int_{0}^{1}f(t)dt +\frac12\Big)
\log \Big(1+2\int_{0}^{1}f(t)dt \Big)
-\int_{0}^{1}f(t)dt +\frac12\int_{0}^{1}
\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)dx.
\end{align*}
Substituting this into our limit inequality we get
\begin{align*}
\int_{0}^{1}\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)dx
&\le \int_0^1 f(x)\log f(x)dx-2\Big( \int_{0}^{1}f(t)dt \Big)
\log\Big(\int_{0}^{1}f(t)dt \Big)\\
&\quad +\int_{0}^{1}f(t)dt+ \Big(\int_{0}^{1}f(t)dt +\frac12\Big)
\log \Big(1+2\int_{0}^{1}f(t)dt \Big)\\
&\quad -\int_{0}^{1}f(t)dt+\frac12\int_{0}^{1}
\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)dx
\end{align*}
i.e.,
\begin{align*}
\int_{0}^{1}\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)dx
&\le 2\int_0^1 f(x)\log f(x)dx-4\Big(\int_{0}^{1}f(t)dt \Big)
\log\Big(\int_{0}^{1}f(t)dt \Big) \\
&\quad +\Big(2\int_{0}^{1}f(t)dt +1\Big)
\log \Big(1+2\int_{0}^{1}f(t)dt \Big).
\end{align*}
Finally, since
$\sup_{t>0} -4t\log t+(2t+1)\log (1+2t)=M_2<\infty$,
we get
$$
\int_{0}^{1}\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)dx \le
2\int_0^1 f(x)\log f(x)dx+M_2
$$
which is exactly an inequality of the type \eqref{appl3}.

\begin{remark} \label{remk5.2} \rm
The passages to the limit in $p$, similar to that one made above,
are made in \cite[ 6.8, p. 139]{HLP}, where a logarithm appears after
the limiting process.
Much more recently,
a similar passage to the limit has been fruitful when studying maximal functions
and related weight classes, see \cite{SW}. In both cases the limit of the
Lebesgue quasinorm in $L^r$ has been studied when $r\to 0$. The appearance of
the logarithm in a limit for $r\to 1$, like in our case, has been noted and used in
\cite{MoS}. Finally, let us recall that the same procedure has been used to derive
the $LlogL$ integrability of the Jacobian (see \cite[(8.44) p. 186)]{IM}.
\end{remark}

\subsection{Sobolev inequalities for fractional Sobolev spaces, part II}
\label{sobo}

The problem of the ``not natural'' blowup
of the norm of $W_0^{s,p}$, $p<N$, when $s\uparrow 1$ has been studied
in \cite{BBM, BBM2, MS}. The following relation, established in \cite{BBM2}
for any $p\in[1,\infty[$,
shows that the factor $(1-s)$ permits to compute exactly the limit
$$
\lim_{s\uparrow 1} (1-s)
\int_\Omega
\int_\Omega \frac{ |f(x)-f(y)|^p}{|x-y|^{N+sp}}\,dx\,dy
=c(p)\|\nabla f\|^p_{L^p(\Omega)}.
$$
Here $\Omega$ stands for the cube
$\Omega=\{ x\in \mathbb{R}^N: |x_j|<1/2, 1\le i\le N\}$,
and, as usual, we restrict our attention to the functions $f$ in
$\mathcal{C}_{0,+}^\infty(\Omega)$.
Therefore, when $s\uparrow 1$, the limit of
the embedding inequality, established in \cite{BBM}
\begin{equation}
\|f\|_{L^q(\Omega)}\le c_1(N)^{1/p}\frac{(1-s)^{1/p}}{(N-sp)^{(p-1)/p}} \|f\|_{W_0^{s,p}(\Omega)}
\label{sobembed2}
\end{equation}
where $p<N$, $0<s<1$, $q=Np/(N-sp)$, is
\begin{equation}
\|f\|_{L^{Np/(N-p)}(\Omega)}\le c_2(p,N) \|f\|_{W_0^{1,p}(\Omega)}
\label{sobembed3}
\end{equation}
Let us set now
$s=1-1/n$  $(n>2)$ in \eqref{sobembed2} and consider
\begin{gather*}
T_nf= \|f\|_{L^{nNp/(nN-p)}(\Omega)}, \\
S_nf=c_1(N)^{1/p}\frac{(1/n)^{1/p}}{(N-p/n)^{(p-1)/p}}
\|f\|_{W_0^{1-1/n,p}(\Omega)}, \\
Tf= \|f\|_{L^{Np/(N-p)}(\Omega)}, \\
Sf=c_2(p,N) \|f\|_{W_0^{1,p}(\Omega)}
\end{gather*}
The limit (in the sense of \eqref{converg})
 of \eqref{sobembed2} is \eqref{sobembed3}.


\section{A topology on Inequalities}

The topology on ${\mathcal{I}}_{0}$ is not satisfactory for our purposes,
since, for instance, in ${\mathcal{I}}_{0}$ the inequalities
\begin{gather*}
Tf\le Sf\quad \forall f\in \mathcal{C}_{0,+}^\infty(\Omega), \\
2Tf\le 2Sf\quad \forall f\in \mathcal{C}_{0,+}^\infty(\Omega)
\end{gather*}
are \emph{different} objects. We are going to build up an abstract setting in which we identify
all equivalent inequalities, and we will consider a topology on these new objects.
Of course the convergence proved in the previous Section will be preserved
in this new setting.


\subsection{Equivalence of inequalities}


   \begin{definition} \label{equineq} \rm
   Let $T_1,S_1,T_2,S_2\in  \mathcal{O}$ and let the inequalities
   \begin{gather*}
d_1:\quad T_1f\le S_1f\quad\forall f\in \mathcal{C}_{0,+}^\infty(\Omega), \\
d_2:\quad T_2f\le S_2f\quad\forall f\in \mathcal{C}_{0,+}^\infty(\Omega)
\end{gather*}
   be given. We will say that $d_1$ is equivalent to $d_2$, and we will write
   $$
   d_1\sim d_2
   $$
   if it is possible to deduce $d_2$ from $d_1$ or $d_1$ from $d_2$ by combining a finite number of
   the following two operations:
\begin{itemize}
\item  There exists $W \in  \mathcal{O}$ such that $T_2f=T_1f+Wf$ and
$S_2f=S_1f+Wf$ for all $f\in \mathcal{C}_{0,+}^\infty(\Omega)$

\item There exists $\Phi$ nonnegative, strictly increasing function on
   $[0,\infty[$ such that $T_2f=\Phi(T_1f)$, $S_2f=\Phi(S_1f)$
for all $f\in \mathcal{C}_{0,+}^\infty(\Omega)$
\end{itemize}
     \label{equivalence}
   \end{definition}

   We immediately observe that such definition is well-posed, in fact it is
   trivial to prove the following

   \begin{proposition} \label{prop6.2}
The notion of equivalence introduced in Definition \ref{equivalence}
   is reflexive, symmetric, transitive.
   \label{wellposed}
\end{proposition}


\subsection{The quotient space}

Definition \ref{equivalence} leads naturally to consider classes of
equivalent inequalities.
Let us consider
 the set of inequalities
$$
{\mathcal{I}}_{0}=\{ d(T,S) : T,S\in {\mathcal{O}}\}
$$
and in such set we consider the classes of equivalence given by $\sim$:
$$
{\mathcal{I}}=\frac{{\mathcal{I}}_{0}}{\sim}
$$
An element of ${\mathcal{I}}$
 will be denoted by $[d]$, which is the class of all inequalities
$d_1\in {\mathcal{I}}_{0}$ equivalent to
 the inequality $d\in {\mathcal{I}}_{0}$:
$$
[d]=\{ d_1\in  {\mathcal{I}}_{0} : d\sim d_1\}
$$
In order to remember that $[d]$ is not an inequality, but a class of inequalities,
we will refer to it, in the sequel,  as ``Inequality''. The notion of convergence of inequalities and the introduction of
a topology for inequalities have much more sense when dealing with \sl Inequalities \rm rather than \sl inequalities. \rm

We introduce in ${\mathcal{I}}$ the topology of the quotient space
${{\mathcal{I}}_{0}}/{\sim}$ (see e.g. \cite[p. 125]{D}): if we call
$P$ the projection
$$
P: {{\mathcal{I}}_{0}} \; \rightarrow\; {\mathcal{I}}
=\frac{{\mathcal{I}}_{0}}{\sim}
$$
then
$$
{\mathcal{A}}\subseteq {\mathcal{I}}  \text{ is open  in } {\mathcal{I}}
$$
if and only if
$$
P^{-1}[{\mathcal{A}}]=\cup \{ A :  A\in {\mathcal{A}} \}
\text{ is  open  in } {{\mathcal{I}}_{0}}.
$$
It is well-known that
$$
d_n\to d \quad \text{in } {{\mathcal{I}}_{0}} \; \Rightarrow \;
 [d_n]\to [d] \quad\text{in }{\mathcal{I}}
$$
therefore the convergence already shown in the previous Section still
hold in $ {\mathcal{I}} $.

We conclude writing explicitly, in terms of the admissible operators
in $\mathcal{O}$, what does it mean that $[d_n]\to [d]$ in ${{\mathcal{I}}}$.
The following notion of convergence represents the answer we
wanted to find to our original question, settled in the Introduction.

\begin{quote}
Let $[d_n]$, $[d]$ in ${{\mathcal{I}}}$.
Write $d=d(T,S)$, $d_n=d_n(T_n,S_n)$ $\forall\, n\in\mathbb{N}$. It is
$[d_n]\to [d]$ in ${{\mathcal{I}}}$ if for any $d'=d'(T',S')$,
$d'\sim d$, for any $n'=n'(T',S')\in\mathbb{N}$, for any
$F'=F'(T',S')\subset \mathcal{C}_{0,+}^\infty(\Omega)$ finite,
there exists $\nu\in\mathbb{N}$ such that for $n>\nu$ the following holds:
$$
\forall\, d'_n(T'_n,S'_n)\sim d_n(T_n,S_n)\; \exists d'(T',S')\sim d(T,S) :
 d'_n(T'_n,S'_n)\in {\mathcal{U}}_{n',F'}(d'(T',S'))
$$
\end{quote}

\subsection{The three main examples}

We now make a few comments on the examples discussed in
Sections 5.1, 5.2, and 5.3. In the first case, the sequence of
inequalities was a sequence of relations between norms, therefore,
the step made in this last Section has no a relevant meaning.
In the other two cases, we changed, for convenience, the sequences
of inequalities: in the case of Hardy's inequality, we preferred
to deal with \eqref{hardy2} rather than \eqref{hardy}; in the case of
the Sobolev inequalities for fractional Sobolev spaces, we dealt with
inequality \eqref{sobembed2}, and we used a relation of limit
involving the right norms, up to the factor $(1-s)$. It is trivial
that such transformations (to raise the inequalities to a certain power,
and to multiply the inequalities by a constant) can be done,
giving of course \emph{equivalent}
inequalities. But without this last step (the construction of a topology on
Inequalities, made in this Section 6), the trivial transformations
would lead to $different$ inequalities, and this would be not natural
for the problem we wished to study.

\section{Computing limits}

In this last Section we wish to provide some tools to compute explicitly
the limits of some Inequalities. Some of them have been implicitly
proved or used in the previous Sections, others come as a byproduct
from Function Space Theory. We stress here that
the novelty of the limits we are going to show is not in the difficulty
of the computations, but in the new light given by our construction:
more or less common ``passages to the limit'' are in fact concrete
limits in a suitable topology. We will conclude the Section giving
two applications,
which show how the construction of the topology leads to the proof
of new results.

\subsection{Some basic tools}

Given a sequence of true inequalities, it is evident that the explicit
computation of the limit must be carried out by passing through
equivalent inequalities (namely, the operations described in the
Definition \ref{equineq}), and by computing the limits of the left hand
side and the right hand side. Therefore the basic tools rely upon
the study of sequences of admissible operators, rather than the
inequalities themselves. Moreover, we observe that since the
admissible operators have real values, the standard theorems on
operation of limits (for instance, the limit of a sum, of a product, the composition
with a continuous nonnegative real function) can be applied.
We are going to show some first ``bricks'' that can be used in applications.
For our purposes it will be sufficient to confine ourselves
to functions defined in domains having measure $1$.

The first tool, which has been already used in Sections \ref{har} and \ref{sobo}, is completely
standard, and it can be found in \cite{HLP}, n. 194, p. 143.

\begin{proposition}
Let $0<p$, $p_0<\infty$. Then
$$\Big(\int_{0}^{1}f(t)^pdt\Big)^{1/p}\to
\Big(\int_{0}^{1}f(t)^{p_0}dt\Big)^{1/{p_0}}
\quad \text{as } p\to p_0\quad\forall f\in \mathcal{C}_{0,+}^\infty(0,1)
$$
\label{primo}
\end{proposition}

For completeness, we state here the case when $p_0=0$ (we refer to
\cite[n. 187, p. 139]{HLP},  and  \cite[Section 6.7]{HLP} for the
exact meaning of the limit expression).

\begin{proposition} \label{prop7.2}
Let $0<p<\infty$. Then
$$
\Big(\int_{0}^{1}f(t)^pdt\Big)^{1/p}\to \exp \Big(\int_{0}^{1}\log f(t) dt\Big)
\quad \text{as } p\to 0\quad\forall f\in \mathcal{C}_{0,+}^\infty(0,1)
$$
\end{proposition}

Now let $\Omega\subset \mathbb{R}^N$ be a bounded regular domain and
let $1\le p<\infty$. Since by
the Poincar\'e's inequality the expression $\| |\nabla f|\|_{L^{p}(\Omega)}$
is equivalent to
the norm $\| f \|_{W_0^{1,p}(\Omega)}$ of the Sobolev space
$W_0^{1,p}(\Omega)$, from the Proposition
\ref{primo} we get immediately that

\begin{proposition}
Let $1\le p$, $p_0<\infty$. Then
$$
\| f \|_{W_0^{1,p}(\Omega)}\to \| f \|_{W_0^{1,{p_0}}(\Omega)}
\quad \text{as } p\to p_0\quad\forall f\in \mathcal{C}_{0,+}^\infty(0,1)
$$
\end{proposition}

We consider now other two convergences, for which the deduction
of the limit is less trivial. Since our goal is just to provide some tools for
explicit computations, we omit the corresponding statements for Sobolev functions.
Both of them can be deduced by a standard computation
(it suffices to take into account that the limit to be computed is of a difference-quotient).
The first one is used in \cite[Section 8.6]{IM}, and
\cite[Section 6.8, p.139]{HLP}.

\begin{proposition}
Let $0<p<\infty$. Then
$$
\frac{1}{p}\Big(\int_{0}^{1}f(t)^pdt-1\Big)\to
\int_{0}^{1}\log f(t)dt \quad \text{as } p\to 0\quad\forall
f\in \mathcal{C}_{0,+}^\infty(0,1)
$$
\end{proposition}

The second convergence result will be stated for the parameter $p$
approaching $p_0$ from the right, because the expression involved,
if $p>p_0$, is an admissible operator in the sense
of Section \ref{homog}, by virtue of the classical H\"older's inequality.

\begin{proposition}
Let $1\le p_0<p<\infty$. Then
$$
\frac{\|  f\|_{L^{p}(\Omega)}-\|  f\|_{L^{p_0}(\Omega)}
}{p-p_0}\to \frac{1}{p_0}
\Big( \frac{\|f^{p_0}\log f\|_{L^{1}(\Omega)}
}{\|f\|_{L^{p_0}(\Omega)}^{p_0-1}}
-\|f\|_{L^{p_0}(\Omega)}\log \|f\|_{L^{p_0}(\Omega)}\Big)
$$
as $p\to p_0+$, for all $f\in \mathcal{C}_{0,+}^\infty(\Omega)$.
\end{proposition}

We conclude this subsection stating explicitly the following result, proved
and used in Section \ref{har}.

\begin{proposition}
Let $1< p<\infty$. Then
$$
\frac{\|  f\|_{L^{p}(\Omega)}-\|  f\|^p_{L^{1}(\Omega)}
}{p-1}\to \|f\log f\|_{L^{1}(\Omega)}
-2\|f\|_{L^{1}(\Omega)}\log \|f\|_{L^{1}(\Omega)}
$$
as $p\to 1$ for all $f\in \mathcal{C}_{0,+}^\infty(\Omega)$.
\end{proposition}

\subsection{Application 1: A refinement of the endpoint Hardy's inequality}

In this section we highlight the following result, obtained in Section \ref{har}. It is an inequality,
involved in the limit of the Hardy's inequalities when the exponent goes to $1$, which has been shown to be sharper
than the classical one. In order to give a meaning to the right hand side, we recall that
the expression $f(x)\log f(x)$ (which is an integrand in the right hand side) has to be understood equal to zero
whenever $f(x)=0$.

\begin{proposition} \label{prop7.7}
The following inequality holds for every nonnegative, measurable function
$f$ on $(0,1)$:
\begin{align*}
\int_{0}^{1}\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)dx
&\le \int_0^1 f(x)\log f(x)dx-2\Big(\int_{0}^{1}f(t)dt \Big)
\log\Big(\int_{0}^{1}f(t)dt \Big)\\
&\quad +\int_{0}^{1}f(t)dt
+\Big(\int_{0}^{1}f(t)dt \Big)\log\Big(\int_{0}^{1}
\Big( \frac{1}{x} \int_{0}^{x}f(t)dt \Big)dx\Big).
\end{align*}
\end{proposition}

\subsection{Application 2: Misproving inequalities without counterexamples}

We present here a new application of our limits of inequalities, in which the so-called
small Lebesgue spaces are involved. Such spaces, introduced by the second author in
\cite{F} and then studied in \cite{CF}, turned out to be useful in Sobolev-type estimates in borderline cases and
in questions of regularity for quasilinear equations (see \cite{FR1}). Our main goal now is
to discuss an inequality, close to the classical multiplicative embedding inequality for Sobolev functions
(see \cite{DIB}, chap. IX, n. 1, p. 423), which is in a class of inequalities
useful for variational problems with critical exponent (see the recent paper \cite{FR2}). A complete
picture of this class of inequalities is in the paper \cite{FR3}, announced in \cite{FR2}.

Let $\Omega$  be a bounded, open, connected, smooth set in $\mathbb{R}^n$, $n>1$, and
let $u\in  \mathcal{C}_{0,+}^\infty(\Omega)$. Let $q,r,p$ be such that
$$
1\le r<n \quad 1<p\le q< \frac{nr}{n-r}
$$
and set
$$
a=\frac{\frac1p-\frac1q}{\frac1n-\frac1r+\frac1p}
$$
Let us consider the inequality
$$
\|u\|_{(q}\le \beta \| |\nabla u| \|_{(r}^a  \|u\|_{p}^{1-a}\leqno (I_a)
$$
where the symbol $\|u\|_{p}$ denotes the usual norm in the Lebesgue space
$L^{p}(\Omega)$ and
$\|\cdot\|_{(q}$ stands for the norm of the small Lebesgue space
$L^{(q}(\Omega)$ (see e.g. \cite{CF} and references therein).

The problem is to know whether such inequality, when $0\le a\le 1$, is true
or not, in the sense that there exists a constant $\beta$, independent of
$u\in \mathcal{C}_{0,+}^\infty(\Omega)$, possibly depending
on $q,r,p,n, \Omega$, such that $(I_a)$ holds. We begin our discussion
observing that the inequality $(I_a)$ is
false when $a=0$, true when $a=1$: in fact in the first case it is $p=q$,
and the inequality reduces to
\begin{equation}
\|u\|_{(p}\le \beta  \|u\|_{p}
\label{falsee}
\end{equation}
and this is false, because the embedding $L^p(\Omega)\subset L^{(p}(\Omega)$
 does not hold
(see e.g. \cite{F}); on the other hand, when $a=1$, the inequality reduces to
$$
\|u\|_{(q}\le \beta \| |\nabla u| \|_{(r}
$$
and this is true in view of the classical Sobolev embedding theorem:
$$
\|u\|_{(q}\le \beta_1 \|u\|_{nr/n-r}\le \beta_2 \| |\nabla u |\|_{r}\le
\beta_3\| |\nabla u| \|_{(r}
$$
(here $\beta_i$, $i=1,2,3$ are independent of $u$).

The question in the cases $0< a< 1$ will be analyzed, also in a much
more general framework, in  \cite{FR3}, where other applications
involving PDEs will be given.
Here we will show - by using a simple argument based on our limit processes
 - that for $a$ sufficiently small the inequality $(I_a)$ cannot
hold with a uniform bound $\beta$.
The importance of this fact is immediately understood after noticing that in the classical case
$$
\|u\|_{q}\le \beta \| |\nabla u| \|_{r}^a  \|u\|_{p}^{1-a}
$$
a uniform bound for $\beta$ exists (with respect to $a$; see e.g. \cite{LU}).

We are now ready to prove the
following result.

\begin{proposition}
There exists $a_0$, $0<a_0<1$, such that $(I_a)$
cannot hold with a bound $\beta$  uniform with respect to $0\le a<a_0$.
\end{proposition}

\begin{proof}
 We make an argument by contradiction. Suppose, on the contrary, that
there exists a sequence $(a_n)_{n\in \mathbb{N}}$, $0<a_n<1$, $a_n\to 0$, such that $(I_{a_n})$ is true
for every $n$, for some $\beta>0$.
Fix
$$
1\le r<n \quad 1<q< \frac{nr}{n-r}
$$
and define $p_n$ by
$$
a_n=\frac{\frac1{p_n}-\frac1q}{\frac1n-\frac1r+\frac1{p_n}}
$$
Set $T_nu=\|u\|_{(q}$, $S_nu=\| |\nabla u| \|_{(r}^{a_n}  \|u\|_{p}^{1-{a_n}}$.
We observe that both $T_n$, $S_n$ are admissible operators in the sense
of Section \ref{homog}. The limit of $d_n=d(T_n,S_n)$ is, in the sense of  \eqref{converg}, the inequality
\eqref{falsee}, which is false. The proposition is therefore proven.
\end{proof}


\begin{thebibliography}{00}

 \bibitem{A} Adams, {\sl Sobolev spaces},  Pure and Applied Mathematics vol. 65, Academic Press (1975)
\bibitem{BS} C. Bennett, R. Sharpley,  {\sl Interpolation of operators},
Pure and Applied Mathematics vol. 129,
Academic Press (1988)

\bibitem{BBM} J. Bourgain, H. Brezis, P. Mironescu,  {\sl Limiting
embedding theorems for $W\sp {s,p}$ when $s\uparrow1$ and
applications. Dedicated to the memory of Thomas H. Wolff},  J. Anal. Math. vol. 87, 77--101 (2002)

\bibitem{BBM2}
J. Bourgain, H. Brezis, P. Mironescu,
{\sl Another look at Sobolev spaces}, in
{\sl Optimal Control and Partial Differential Equations},
J.L. Menaldi et al. (Eds.), IOS Press (2001)

\bibitem{B} H. Brezis, {\sl Analyse fonctionnelle. Th\'eorie et applications.}
Collection Math\'ematiques Appliqu\'ees pour la Matr\^ise. Masson, Paris, (1983)

\bibitem{CF} C. Capone, A. Fiorenza,   {\sl
 On small Lebesgue spaces},
J. Funct. Spaces Appl., \bf 3 \rm (1)
(2005), 73-89


\bibitem{CAM} C. Capone, A. Fiorenza, M. Krbec,  {\sl On extrapolation blowups in the $L_p$ scale}, 
Journal of Inequalities and Applications, vol. 2006, Article ID 74960, 15 pages

\bibitem{DIB} Di Benedetto,   {\sl Real Analysis}, Birkh\"auser Advanced Texts, Birkh\"auser (2002)

\bibitem{D} J. Dugundji,   {\sl Topology}, Allyn and Bacon Series in Advanced Mathematics, Allyn and Bacon (1966)

\bibitem{F} A. Fiorenza,   {\sl Duality and Reflexivity in Grand Lebesgue Spaces},
Collect. Math., vol. \bf 51, \rm no. 2, (2000),
131-148



\bibitem{FKa} A. Fiorenza, G. E.  Karadzhov,
 {\sl Grand and small Lebesgue spaces and their analogs}
  Z. Anal. Anwendungen. vol. 23, (4), 657--681 (2004)

\bibitem{FK} A. Fiorenza, M. Krbec,  {\sl On optimal decompositions in Zygmund spaces},
Georgian Math. J. vol. 9, 271--286 (2002)

\bibitem{FR1}  A. Fiorenza, J.M. Rakotoson   {\sl New properties of small
Lebesgue spaces and their applications},  Mathematische Annalen, vol. {\bf 326,}
 (2003), 543-561


\bibitem{FR2}  A. Fiorenza, J.M. Rakotoson   {\sl Compactness, interpolation inequalities for small
Lebesgue-Sobolev spaces and applications}, Calculus of Variations and Partial Differential Equations,
vol. \bf 25 \rm (2), (2005), 187--203

\bibitem{FR3}  A. Fiorenza, J.M. Rakotoson, work in preparation

\bibitem{GT}  D. Gilbarg, N.S. Trudinger   {\sl Elliptic Partial Differential Equations of Second Order},
Springer, (2001)
\bibitem{HLP} G.H. Hardy, J.E. Littlewood, G. P\'olya, {\sl
Inequalities}, Cambridge University Press (1952)

\bibitem{IM} T. Iwaniec, G. Martin, {\sl
Geometric Function Theory and Non-linear Analysis}, Oxford Science Publications (2001)

\bibitem{K} A. Kufner, O. John, S. Fu\v cik,  {\sl Function Spaces},
Noordhoff International Publishing, Leyden (1977)

\bibitem{KR} O.~Kov\'a\v{c}ik, J.~R\'akosn\'\i k,  {\sl On spaces $L^{p(x)}$ and $W^{k,p(x)}$},
Czech. Math. Jour. vol. 41 (116), 592--618 (1991)

\bibitem{LU}  O.A. Ladyzhenskaya, N.N. Ural Tseva  {\sl Linear  and
quasilinear Elliptic Equations}, Mathematics in Sciences and
Ingeneering {\bf 46} Academic Press (1968).

\bibitem{MS} V. Maz'ya and T. Shaposhnikova, {\sl On the Bourgain, Brezis, and
Mironescu Theorem Concerning Limiting Embeddings of
Fractional Sobolev Spaces}, J. Funct. Anal. vol. 195, 230--238 (2002) and
vol. 210, 298--300 (2003)

\bibitem{MoS} G. Moscariello, C. Sbordone,  {\sl $A\sb \infty$
as a limit case of reverse-H\"older inequalities when the exponent tends to $1$.},
Ricerche Mat. vol. 44, (1), 131--144 (1995)

\bibitem{SW} C. Sbordone, I. Wik, {\sl Maximal functions and related weight classes},
Publ. Math. vol. 38, 127--155 (1994)

\bibitem{S} E.M. Stein, {\sl Singular Integrals and Differentiability
properties of functions}, Princeton University Press (1970)

\bibitem{T} H. Triebel, {\sl Theory of function spaces}, Birkh\"auser, Basel, Boston (1983)

\bibitem{Z} Zygmund, {\sl Trigonometric Series}, Cambridge Univ. Press (1959)

\end{thebibliography}
































\end{document}