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 $0p_0$, 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

1$, and
let $u\in \mathcal{C}_{0,+}^\infty(\Omega)$. Let $q,r,p$ be such that
$$
1\le r0$.
Fix
$$
1\le r