\section{Introduction}

The notion of weak derivatives \cite{S63} was introduced as a generalization of classic solutions of PDEs. But the regularity of weak solutions is a long-standing complicated problem, see, for example \cite{LU68}, which is still open. In the present article we study fine regularity properties of Sobolev functions of the space $W^1_p(\Omega)$. Let $\Omega$ be an open subset of the Euclidean space $\mathbb R^n$, in the classical work \cite{C51} it was proved that functions in the Sobolev space $W^{1}_{p}(\Omega)$, $p>n$, are differentiable a.e. in $\Omega$ with respect to the uniform norm, see also works \cite{CZ62,Se61}. In the case $p=n$ the differentiability of monotone functions of the Sobolev space $W^1_n(\Omega)$ was obtained in \cite{V66}. This result was extended to the case of spaces $W^1_p(\Omega)$, $n-1<p<\infty$, in \cite{Man94}. 

In \cite{Re68,VGR79} was suggested the method which allows to consider differentiability of Sobolev functions in the topology associated with Sobolev norms. In the work \cite{Re68} it was proved that for a function $f\in W^1_p(\Omega)$, the formal differential $Df(x)$, $x\in\Omega$, defined by the vector gradient $\nabla f(x)$, is the differential with respect to convergence in $W^1_p(\Omega)$ for almost every $x\in\Omega$ with respect to Lebesgue measure. 
The main result of the present work is a refined version of the corresponding Lagrange formula with the remainder of the $W^1_p$ class. Corresponding result for $W^k_p$ classes are also proved.

The notion of refined Sobolev functions was introduced in \cite{HM72}, where it was proved that for a function $f\in W^1_p(\Omega)$ there exists a refined $p$-quasicontinuous function 
\begin{equation}
\label{eq:precise representation}
f^{*}(x)=\lim\limits_{r\to 0^+}\frac{1}{|B(x,r)|}\int\limits_{B(x,r)} f(z)~dz
\end{equation}
defined up to a set of $p$-capacity zero. We use the notation $|S|$ for the $n-$dimensional Lebesgue measure of a set $S\subset\mathbb{R}^n$. 
Recall that a function $f:\Omega\to \mathbb{R}$ is called p-quasicontinuous in an open set $\Omega\subset\mathbb R^n$ 
if and only if for every $\varepsilon >0$ there exists an open  set $U_{\varepsilon}\subset \Omega$ 
such that the $p$-capacity of $U_{\varepsilon}$ is less than $\varepsilon$ and on the set $\Omega\setminus U_{\varepsilon}$ 
the function  $f$ is continuous (see, for example \cite{HKM,M}). 

In the present work we prove that if a function $f\in W^1_p(\Omega)$ has a $p$-refined weak gradient, then its formal differential $Df(x)$, $x\in\Omega$, defined by the vector gradient $\nabla f(x)$, is the differential with respect to convergence in the topology of $W^1_{p,\loc}(\Omega)$ for every $x\in\Omega\setminus \mathcal{N}$, where $\mathcal{N}$ is a set of $p$-capacity zero. This result refines the the results of the work \cite{Re68} in the case of Sobolev functions with refined weak gradients. Note that it is known that the functions of the space $W^2_p(\Omega)$ have refined weak gradients, but we show that the class of functions $f\in W^1_p(\Omega)$ which have $p$-refined weak gradients is wider than the space $W^2_p(\Omega)$. This result is extended to $W^{k}_{p}$ spaces as well.

Note that the difference between almost everywhere and its capacitory version is significant because of the known connection between nonlinear capacity and Hausdorff dimension (measure). Hence, our refinement of the Reshetnyak result \cite{Re68} seems to be essential.
The $L_p$-approximate differentiability for functions of $W^{1}_{p,\loc}(\Omega)$ with $p$-refined weak gradients is considered also. 


\section{Sobolev spaces, capacity and refine functions; connections between capacity and Hausdorff measure}

\subsection{Sobolev spaces and capacity}

Let $\Omega$ be an open subset of $\mathbb R^n$. The Sobolev space 
$W^m_p(\Omega)$, $m\in \mathbb N$, $1\leq p<\infty$,
is defined as the normed
space of functions $f\in L_p(\Omega)$ such that the partial derivatives of order less than or equal to $m$
exist in the weak sense and belong to $L_p(\Omega)$. The space is equipped with the norm
\begin{equation}
\label{eq:definition of Sobolev norm W^m_p}
\|f\|_{W^m_p(\Omega)}=
\sum\limits_{|\alpha|\leq m}\left(\int\limits_{\Omega} |\nabla^{\alpha}f(x)|^p\,dx\right)^{\frac{1}{p}}<\infty,
\end{equation}  
$\nabla^{\alpha} f$ is the weak derivative of order $\alpha$ of the function $f$, 
where $\alpha=(\alpha_1,...,\alpha_n)$ multiindex, $\alpha_i\in \mathbb{N}\cup\{0\},1\leq i\leq n$. 

Sobolev spaces are Banach spaces of equivalence classes \cite{M}. To clarify the notion of equivalence classes of Sobolev functions we use the nonlinear $p$-capacity associated with Sobolev spaces \cite{GResh,HKM,M}. 
Suppose $\Omega$ is an open set in $\mathbb R^n$ and  $K\subset\Omega$ is a compact set. The $p$-capacity of $K$ with respect to $\Omega$ is defined by
\begin{equation*}
\cp_p(K;\Omega):=\inf\int_\Omega|\nabla u(x)|^p~dx,  
\end{equation*} 
where the infimum is taken over all functions $u\in C^\infty_c(\Omega)$, $u\geq 1$ on $K$, which are called {\it admissible functions} for the compact set $K\subset\Omega$. 
If 
$U \subset\Omega$ 
is an open set, we define
\begin{equation}
\cp_p(U;\Omega):=\sup\cp_p(K;\Omega), \,\,K\subset U,\,\,\text{$K$ is compact}.
\end{equation}
In the case of an arbitrary set 
$E\subset\Omega$
we define 
\begin{equation}
\cp_{p}(E;\Omega):=\inf\cp_{p}(U;\Omega),\,\,E\subset U\subset\Omega,\,\,\text{$U$ is open}.
\end{equation}
In case of $\Omega=\mathbb{R}^n$ we use the notation $\cp_{p}(E)=\cp_{p}(E;\mathbb{R}^n)$. It is well-know that if $\cp_p(E)=0$, then $|E|=0$ for every set $E\subset \mathbb{R}^n$ \cite{evans2015measure,K21}. The notion of the $p$-capacity permits us to refine the notion of Sobolev functions. Let  $f\in W^1_p(\Omega)$. Then the refined function $f^{*}$ (see $\eqref{eq:precise representation}$)
is defined quasieverywhere, i.~e. up to a set of $p$-capacity zero. This refined function $f^{*}\in W^1_p(\Omega)$ is called the unique quasicontinuous representation ({\it a canonical representation}) of the function 
$f\in W^1_p(\Omega)$. 

\subsection{Connections between weak derivation and the precise representative operation}

\begin{defn}
\label{def:precise representative}
Let $\Omega\subset\mathbb{R}^n$ be an open set and
$f\in L_{1,\loc}(\Omega).$
The {\it precise representative} of 
$f$ is defined by 
\begin{equation}
\label{precise representative}
f^*:\Omega\to \mathbb{R},\quad f^*(x):=
\begin{cases}
\lim_{r\to 0^+}\fint_{B(x,r)}f(y)dy,\quad & \text{if the limit exists};\\
0, \quad & \text{otherwise}.
\end{cases}
\end{equation}
\end{defn}
The symbol $\fint$ in the definition above stands for the average of the function 
$f$:
\begin{equation}
\fint_{B(x,r)}f(y)dy=\frac{1}{|B(x,r)|}\int_{B(x,r)}f(y)dy,
\end{equation}
where $B(x,r)$ stands for the open ball around $x$ with radius $r$.

Recall that since almost every point in $\Omega$ is a Lebesgue point with respect to Lebesgue measure for functions $f\in  L_{1,\loc}(\Omega)$, then $f(x)=f^*(x)$ for almost every point $x\in \Omega$ with respect to Lebesgue measure. Note also, that if $f,g\in L_{1,\loc}(\Omega)$ and $f=g$ almost everywhere in $\Omega$, then $f^*(x)=g^*(x)$ for every $x\in\Omega$. If $f$ is a continuous function, then $f(x)=f^*(x)$ for every point $x\in \Omega$. If $f\in W^{1}_{p}(\Omega)$, then 
$\nabla f=\nabla f^*$ almost everywhere in $\Omega$.   

Recall the following fine property of Sobolev functions $W^{1}_{p,\loc}(\Omega)$ \cite{evans2015measure,K21}:
\begin{thm}
\label{thm:fine property of Sobolev functions}
Let $\Omega\subset\mathbb R^n$ be an open set and $1\leq p<\infty$. If $f\in W^{1}_{p,\loc}(\Omega)$, then there exists a Borel set 
$\mathcal{N}\subset \Omega$ such that  
\begin{equation}
\cp_p(\mathcal{N})=0 \quad \wedge \quad  \lim_{r\to 0^+}\fint_{B(x,r)}|f(z)-f^*(x)|^pdz=0 \quad \forall x\in \Omega\setminus \mathcal{N}.
\end{equation} 
\end{thm}

\medskip

Let us remark that if $f\in W^{2}_{p,\loc}(\Omega)$, then  
$\left(\frac{\partial f}{\partial x_i}\right)^*=\left(\frac{\partial f^*}{\partial x_i}\right)^*$, $i=1,...,n$, for $\cp_p-$a.e. $x\in\Omega$: 
For any index $i\in \{1,...,n\}$, both functions 
$\frac{\partial f}{\partial x_i}$ and $\frac{\partial f^*}{\partial x_i}$ 
belong to the space $W^{1}_{p,\loc}(\Omega)$ and they are equal almost everywhere. Thus, by Theorem $\ref{thm:fine property of Sobolev functions}$, there exists a set $E$ such that $\cp_p(E)=0$ and 
\begin{equation*}
\lim_{r\to 0^+}\fint_{B(x,r)}\left|\frac{\partial f}{\partial x_i}(y)-\left(\frac{\partial f}{\partial x_i}\right)^*(x)\right|^pdy=0
\end{equation*}  
and
\begin{align*}
\lim_{r\to 0^+}&\fint_{B(x,r)}\left|\frac{\partial f^*}{\partial x_i}(y)-\left(\frac{\partial f^*}{\partial x_i}\right)^*(x)\right|^pdy
\\
&=\lim_{r\to 0^+}\fint_{B(x,r)}\left|\frac{\partial f}{\partial x_i}(y)-\left(\frac{\partial f^*}{\partial x_i}\right)^*(x)\right|^pdy=0
\end{align*}
for all $x\in \Omega\setminus E$. 
For every $x\in \Omega\setminus E$ and almost every $y\in \Omega$ we have 
\begin{multline*}
\left|\left(\frac{\partial f}{\partial x_i}\right)^*(x)-\left(\frac{\partial f^*}{\partial x_i}\right)^*(x)\right|^p\\
=\left|\left[\left(\frac{\partial f}{\partial x_i}\right)^*(x)-\frac{\partial f}{\partial x_i}(y)\right]+\left[\frac{\partial f}{\partial x_i}(y)-\left(\frac{\partial f^*}{\partial x_i}\right)^*(x)\right]\right|^p
\\
\leq 
2^{p-1}\left|\frac{\partial f}{\partial x_i}(y)-\left(\frac{\partial f}{\partial x_i}\right)^*(x)\right|^p
+2^{p-1}\left|\frac{\partial f}{\partial x_i}(y)-\left(\frac{\partial f^*}{\partial x_i}\right)^*(x)\right|^p.
\end{multline*}

Thus, taking the average integral with respect to $dy$ on both sides of the last inequality on a ball $B(x,r)$, we obtain:
\begin{multline*}
\left|\left(\frac{\partial f}{\partial x_i}\right)^*(x)-\left(\frac{\partial f^*}{\partial x_i}\right)^*(x)\right|^p=\fint_{B(x,r)}\left|\left(\frac{\partial f}{\partial x_i}\right)^*(x)-\left(\frac{\partial f^*}{\partial x_i}\right)^*(x)\right|^pdy
\\
\leq 
2^{p-1}\fint_{B(x,r)}\left|\frac{\partial f}{\partial x_i}(y)-\left(\frac{\partial f}{\partial x_i}\right)^*(x)\right|^pdy
+2^{p-1}\fint_{B(x,r)}\left|\frac{\partial f}{\partial x_i}(y)-\left(\frac{\partial f^*}{\partial x_i}\right)^*(x)\right|^pdy,
\end{multline*}
and by taking the upper limit as $r\to 0^+$ on both sides of the last inequality we see that
\begin{equation*}
\left|\left(\frac{\partial f}{\partial x_i}\right)^*(x)-\left(\frac{\partial f^*}{\partial x_i}\right)^*(x)\right|=0.
\end{equation*}

In the above calculation we used the convexity of the function $s\longmapsto s^p$, $s\geq 0$, $1\leq p<\infty$, in the following way:
for $a,b\geq 0$
\begin{equation}
(a+b)^p=2^p\frac{1}{2^p}(a+b)^p=2^p\left(\frac
{1}{2}a+\frac{1}{2}b\right)^p\leq 2^{p-1}(a^p+b^p).
\end{equation}

\subsection{Connections between capacity and Hausdorff measure}
In this subsection we recall Frostman type theorems for capacity \cite{GResh,HKM}.
The $s-$dimensional Hausdorff measure is denoted by $\mathcal{H}^s$ and it is defined as follows: Let $A\subset\mathbb{R}^n$, $0\leq s<\infty$, $0<\delta\leq\infty$, define
\begin{equation*}
\mathcal{H}^s_\delta(A):=\inf\sum_{j=1}^\infty\alpha(s)\left(\frac{\diam C_j}{2}\right)^s, \,\,\text{where}\,\,A\subset \bigcup_{j=1}^\infty C_j,\diam C_j\leq \delta,
\end{equation*}
where $\diam C_j$ stands for the diameter of the set $C_j$ and 
\begin{equation*}
\alpha(s):=\frac{\pi^{s/2}}{\Gamma\left(\frac{s}{2}+1\right)},\quad \Gamma(s):=\int_0^\infty e^{-x}x^{s-1}dx,\quad s\in [0,\infty).
\end{equation*}
Then
\begin{equation*}
\mathcal{H}^s(A):=\lim_{\delta\to 0^+}\mathcal{H}^s_\delta(A)=\sup_{\delta>0}\mathcal{H}^s_\delta(A).
\end{equation*}

Recall that the Hausdorff measure is a Borel regular measure and for every set $E\subset\mathbb{R}^n$
\begin{equation}
\label{eq:controlling of capacity by Hausdorff measure}
\cp_p(E)\leq C(n,p)\mathcal{H}^{n-p}(E),
\end{equation}
where $1\leq p<n$ and $C(n,p)$ is a constant dependent on the parameters $p,n$ only.
For more information about Hausdorff measures and their connections to capacity see \cite{evans2015measure,Mattila}.

\section{Sobolev spaces with refined gradients}

In this section we define the class of Sobolev functions with refined gradients which is the main object of our study in the present paper.

\begin{defn}
\label{def:refined property of gradient}
Let $\Omega$ be an open subset of $\mathbb R^n$. Then for $1\leq p<\infty$ we define the refined Sobolev space $RW^{1}_{p}(\Omega)$ as a set of functions $f\in W^1_p(\Omega)$ which have refined gradients:
\begin{equation}
\label{eq:refined property of the gradient}
\lim_{r\to 0^+}\fint_{B(x,r)}|\nabla f(z)-(\nabla f)^*(x)|^pdz=0 \quad for\quad  \cp_p-a.e.\quad  x\in \Omega.
\end{equation}
\end{defn}

\begin{rem}The letter $R$ in the symbol of the set $RW^1_p$ was chosen to remind the notion of "Refinement".
\end{rem}

We will use \textbf{Jensen's inequality} in the following form: 
Let $A\subset\mathbb{R}^n$ be a set such that 
$0<|A|<\infty$. Let 
$H:[0,\infty)\to \mathbb{R}, H(s)=s^\alpha$, where $\alpha>0$.
If $\alpha\geq 1$, then $H$ is convex and
\begin{equation*}
H\left(\fint_{A}h(x)dx\right)\leq \fint_{A}H(h(x))dx,
\end{equation*}
and if $0<\alpha\leq 1$, then $H$ is concave and  
\begin{equation*}
H\left(\fint_{A}h(x)dx\right)\geq \fint_{A}H(h(x))dx,
\end{equation*}
for every measurable function $h:A\to [0,\infty),h\in L_1(A)$.

The following definition was introduced by Reshetnyak \cite{Re68}:
\begin{defn}
Let $\Omega\subset\mathbb{R}^n$ be an open set, $1\leq p<\infty$ and $h\in L_{p,\loc}(\Omega)$. Then a point $x\in \Omega$ is called an $L_p-$point of $h$ if 
\begin{equation}
\lim_{r\to 0^+}\fint_{B(x,r)}|h(z)-h^*(x)|^p~dz=0.
\end{equation} 
\end{defn}
Note that if $1\leq p<\infty$, $h\in L_{p,\loc}(\Omega)$ and $x\in \Omega$ is an $L_p-$point of 
$h$, then $h\in L_{q,\loc}(\Omega)$ and 
$x$ is an $L_q-$point of $h$ for every $1\leq q\leq p$ : 
since for $0<\alpha\leq 1$, the function $s\longmapsto s^\alpha$, $s\geq 0$, is concave, we get by Jensen's inequality 

\begin{multline*}
\fint_{B(x,r)}|h(z)-h^*(x)|^q~dz= \fint_{B(x,r)}\left(|h(z)-h^*(x)|^p\right)^{\frac{q}{p}}~dz
\\
\leq \left(\fint_{B(x,r)}|h(z)-h^*(x)|^p~dz\right)^{\frac{q}{p}}.
\end{multline*} 
 
\begin{rem}
In terms of $L_p$-points, the classical Theorem 
$\ref{thm:fine property of Sobolev functions}$ says that a Sobolev function has the property that $\cp_p-$ almost every point is an $L_p-$point. Definition 
$\ref{def:refined property of gradient}$ requires the same property for the weak derivative. In other words, the space $RW^1_p$ is the subset of functions from $W^1_p$ for which the weak derivative has an $L_p-$point almost everywhere with respect to the capacity.      
\end{rem}

Now we prove that the property of weak derivatives having an $L_p-$point almost everywhere with respect to the capacity gives refined Sobolev spaces.

\begin{thm}
\label{thm:RW^1_p is a vector subspace of W^1_p}
Let $\Omega\subset\mathbb{R}^n$ be an open set, $1\leq p<\infty$.
The set $RW^1_p(\Omega)$ is a vector subspace of $ W^1_p(\Omega)$.
\end{thm}

\begin{proof}
It is obvious that the zero function lies in $RW^1_p(\Omega)$. Now for any 
$\alpha\in\mathbb R$ and $f,g\in RW^1_p(\Omega)$ we have for $\cp_p-$a.e. $x\in \Omega$
\begin{equation*}
(\nabla(f+g))^*(x)=(\nabla f)^*(x)+(\nabla g)^*(x),\quad  (\nabla(\alpha f))^*(x)=\alpha(\nabla f)^*(x).
\end{equation*}
Indeed, since $f,g\in RW^1_p(\Omega)$, we obtain by the refined property $\eqref{eq:refined property of the gradient}$ the existence of a set 
$\mathcal{N}\subset \Omega$, $\cp_p(\mathcal{N})=0$, such that for every 
$x\in \Omega\setminus \mathcal{N}$
\begin{equation*}
(\nabla f)^*(x)=\lim_{r\to 0^+}\fint_{B(x,r)}\nabla f(z)dz,\quad (\nabla g)^*(x)=\lim_{r\to 0^+}\fint_{B(x,r)}\nabla g(z)dz.
\end{equation*}
Hence, by Definition~$\ref{def:precise representative}$ we get for  
$x\in \Omega\setminus \mathcal{N}$  
\begin{equation*}
(\nabla f)^*(x)+(\nabla g)^*(x)=\lim_{r\to 0^+}\fint_{B(x,r)}\nabla (f+g)(z)dz=(\nabla (f+g))^*(x),
\end{equation*}
and 
\begin{equation*}
\alpha(\nabla f)^*(x)=\lim_{r\to 0^+}\fint_{B(x,r)}\nabla (\alpha f)(z)dz=(\nabla(\alpha f))^*(x).
\end{equation*} 

Thus, we have for $x\in \Omega\setminus \mathcal{N}$
\begin{multline*}
\fint_{B(x,r)}|\nabla(f+g)(z)-(\nabla(f+g))^*(x)|^p~dz\\
\leq 2^{p-1}\fint_{B(x,r)}|\nabla f(z)-(\nabla f)^*(x)|^p~dz
+2^{p-1}\fint_{B(x,r)}|\nabla g(z)-(\nabla g)^*(x)|^pdz;\nonumber
\end{multline*}
and
\begin{equation*}
\fint_{B(x,r)}|\nabla(\alpha f)(z)-(\nabla(\alpha f))^*(x)|^pdz= |\alpha|^p
\fint_{B(x,r)}|\nabla f(z)-(\nabla f)^*(x)|^pdz.
\end{equation*}

Therefore, for any
$x\in \Omega\setminus \mathcal{N}$ 
\begin{multline*}
\limsup_{r\to 0^+}\fint_{B(x,r)}|\nabla(f+g)(z)-(\nabla(f+g))^*(x)|^pdz=0,\\
\lim_{r\to 0^+}\fint_{B(x,r)}|\nabla(\alpha f)(z)-(\nabla(\alpha f))^*(x)|^pdz=0,
\end{multline*}
which means that $f+g,\alpha f\in RW^1_p(\Omega)$.

Hence $RW^1_p(\Omega)$ is a vector subspace of $W^1_p(\Omega)$.
\end{proof}

\begin{rem}The set $RW^1_{p,\loc}(\Omega)$ is defined in a usual way: $f\in RW^1_{p,\loc}(\Omega)$ if and only if $f\in RW^1_p(\Omega_0)$ for every open and bounded set 
$\Omega_0\subset\Omega$ such that 
$\overline{\Omega}_0\subset\Omega$, where $\overline{\Omega}_0$ is the topological closure of $\Omega_0$. In the same way as above one can see that $RW^1_{p,\loc}(\Omega)$ is a vector subspace of $W^1_{p,\loc}(\Omega)$.
\end{rem}

Recall that for open set $\Omega\subset\mathbb{R}^n$ and $1\leq p\leq \infty$, if $f,g\in W^1_p(\Omega)\cap L_\infty(\Omega)$, then $fg\in W^1_p(\Omega)\cap L_\infty(\Omega)$, means that, the space $W^1_p(\Omega)\cap L_\infty(\Omega)$ is an algebra with respect to the pointwise product $(fg)(x)=f(x)g(x),x\in \Omega$ (for proof see for example \cite{evans2015measure}).

\begin{prop}
\label{prop: the refined Sobolev space is an algebra inside the bounded functions}
Let $\Omega\subset\mathbb{R}^n$ be an open set and 
$1\leq p<\infty$. The space $RW^1_p(\Omega)\cap L_\infty(\Omega)$ is an algebra.
\end{prop}

Before proving Proposition~\ref{prop: the refined Sobolev space is an algebra inside the bounded functions} we need the following technical lemma:

\begin{lem}
\label{lem:Lp points for products}
Let $f,g\in L_{p,\loc}(\Omega)$ for an open set $\Omega\subset \mathbb{R}^n$ and $1\leq p\leq \infty$. Assume that   
$x\in\Omega$ is an $L_p-$point of $f$ and $g$, and assume the existence of an open set 
$\Omega_0$ such that $x\in \Omega_0\subset \Omega$ and at least one of the functions $f,g$ is bounded in $\Omega_0$. Then, $x$ is an $L_p-$point of the product $fg$ and
\begin{equation*}
f^*(x)g^*(x)=(fg)^*(x).
\end{equation*}
\end{lem} 

\begin{proof}
Assume that $f$ is bounded on $\Omega_0$. 
For almost every $y\in\Omega$ we have:
\begin{multline*}
|(fg)(y)-f^*(x)g^*(x)|^p=|\left((fg)(y)-f(y)g^*(x)\right)+\left(f(y)g^*(x)-f^*(x)g^*(x)\right)|^p\\
\leq
2^{p-1}|f(y)|^p|g(y)-g^*(x)|^p+2^{p-1}|g^*(x)|^p|f(y)-f^*(x)|^p. 
\end{multline*}
Taking the average integral with respect to $dy$ on both sides of the last inequality on a ball 
$B(x,r)\subset \Omega_0$, we get:
\begin{multline*}
\fint_{B(x,r)}|(fg)(y)-f^*(x)g^*(x)|^pdy\leq
\\
2^{p-1}||f||^p_{L_\infty(\Omega_0)}\fint_{B(x,r)}|g(y)-g^*(x)|^pdy+2^{p-1}|g^*(x)|^p\fint_{B(x,r)}|f(y)-f^*(x)|^pdy.
\end{multline*}

Thus, taking the limit as $r\to 0^+$ in the last inequality we obtain:
\begin{equation*}
\lim_{r\to 0^+}\fint_{B(x,r)}|(fg)(y)-f^*(x)g^*(x)|^pdy=0.
\end{equation*}
In particular, the limit 
$$
\lim_{r\to 0^+}\fint_{B(x,r)}(fg)(y)dy
$$ 
exists and equals to $f^*(x)g^*(x)$. By Definition $\ref{def:precise representative}$ 
\begin{equation*}
f^*(x)g^*(x)=\lim_{r\to 0^+}\fint_{B(x,r)}(fg)(y)dy=(fg)^*(x),
\end{equation*}
and so $x$ is an $L_p-$point of the product $fg$ 
\end{proof}

Notice that Lemma~\ref{lem:Lp points for products} is valid also if one of the functions $f,g$ is a vector valued function.  

We prove now Proposition $\ref{prop: the refined Sobolev space is an algebra inside the bounded functions}$:

\begin{proof}
Let $f,g\in RW^1_p(\Omega)\cap L_\infty(\Omega)$. Since $fg\in W^1_p(\Omega)\cap L_\infty(\Omega)$, it is sufficient to prove that 
\begin{equation}
\label{eq:L_p point for derivative of a product of functions}
\lim_{r\to 0^+}\fint_{B(x,r)}|\nabla(fg)(z)-(\nabla(fg))^*(x)|^pdz=0
\end{equation}
for $\cp_p-$a.e. $x\in \Omega$.
 
By the assumption that $f,g\in RW^1_p(\Omega)$ and by Theorem $\ref{thm:fine property of Sobolev functions}$, there exists a set $E\subset\Omega$ such that 
$\cp_p(E)=0$ and every $x\in \Omega\setminus E$ is an $L_p-$point of $f,g,\nabla f,\nabla g$. 

Since $f,g\in L_\infty(\Omega)$, we obtain from Lemma 
$\ref{lem:Lp points for products}$ 
\begin{equation*}
f^*(x)(\nabla g)^*(x)=(f\nabla g)^*(x),\quad g^*(x)(\nabla f)^*(x)=(g\nabla f)^*(x).
\end{equation*}
In the proof of Theorem 
$\ref{thm:RW^1_p is a vector subspace of W^1_p}$ was proved that at $L_p-$points the precise representative of a sum of functions equals to the sum of the precise representatives. Hence we get for every $x\in \Omega\setminus E$ 
\begin{multline*}
f^*(x)(\nabla g)^*(x)+g^*(x)(\nabla f)^*(x)=(f\nabla g)^*(x)+(g\nabla f)^*(x)
\\
=(f\nabla g+g\nabla f)^*(x)=(\nabla (f g))^*(x).
\end{multline*}

In the following calculation we omit the point $y$ in the functions, for example we write $f\nabla g$ in place of $f(y)\nabla g(y)$, $\nabla f$ in place of $\nabla f(y)$ et cetera.
For almost every $y$ in 
$\Omega$ and every $x\in\Omega\setminus E$, we get:
\begin{multline}
\label{eq:a bound on derivative of product}
\left|\nabla(fg)-(\nabla(fg))^*(x)\right|^p\\
=\left|f\nabla g+g\nabla f-\left[f^*(x)(\nabla g)^*(x)+g^*(x)(\nabla f)^*(x)\right]\right|^p
\\
\leq 2^{p-1}\left|f\nabla g-f^*(x)(\nabla g)^*(x)\right|^p+2^{p-1}\left|g\nabla f-g^*(x)(\nabla f)^*(x)\right|^p
\\
=2^{p-1}\left|\left[f\nabla g-f(\nabla g)^*(x)\right]+\left[f(\nabla g)^*(x)-f^*(x)(\nabla g)^*(x)\right]\right|^p
\\
+2^{p-1}\left|\left[g\nabla f-g(\nabla f)^*(x)\right]+\left[g(\nabla f)^*(x)-g^*(x)(\nabla f)^*(x)\right]\right|^p
\\
\leq 2^{2p-2}\left(|f|^p|\nabla g-(\nabla g)^*(x)|^p+|(\nabla g)^*(x)|^p|f-f^*(x)|^p\right)
\\
+2^{2p-2}\left(|g|^p|\nabla f-(\nabla f)^*(x)|^p+|(\nabla f)^*(x)|^p|g-g^*(x)|^p\right).
\end{multline}

Therefore, taking the average on a ball $B(x,r)$ on both sides on the last inequality and using that $f,g\in L_\infty(\Omega)$, we get
\begin{align*}
&\fint_{B(x,r)}\left|\nabla(fg)(y)-(\nabla(fg))^*(x)\right|^pdy
\\
&\leq 2^{2p-2}\|f\|_{L_\infty(\Omega)}^p\fint_{B(x,r)}|\nabla g(y)-(\nabla g)^*(x)|^pdy
\\
&+2^{2p-2}|(\nabla g)^*(x)|^p\fint_{B(x,r)}|f(y)-f^*(x)|^pdy
\\
&+2^{2p-2}\|g\|_{L_\infty(\Omega)}^p\fint_{B(x,r)}|\nabla f(y)-(\nabla f)^*(x)|^pdy
\\
&+2^{2p-2}|(\nabla f)^*(x)|^p\fint_{B(x,r)}|g(y)-g^*(x)|^pdy.
\end{align*}

Taking the limit in the last inequality as $r\to 0^+$, we conclude that $x$ is an $L_p-$point of $\nabla (fg)$.
\end{proof}

\begin{prop}
\label{prop:second order Sobolev space is continuously embedded in F^{1,p}}
Let $\Omega\subset\mathbb R^n$ be an open set and let $1\leq p<\infty$. Then the space $W^{2}_{p,\loc}(\Omega)$ is a vector subspace of the vector space $RW^1_{p,\loc}(\Omega)$.
\end{prop}

\begin{proof}
Since $W^{2}_{p,\loc}(\Omega)$ and $RW^1_{p,\loc}(\Omega)$ are vector spaces, it is enough to show that $W^{2}_{p,\loc}(\Omega)$ is a subset of $RW^1_{p,\loc}(\Omega)$. 

Indeed, let $f\in W^{2}_{p,\loc}(\Omega)$, then $\nabla f\in W^{1}_{p,\loc}(\Omega,\mathbb R^n)$, hence by Theorem $\ref{thm:fine property of Sobolev functions}$ it follows that $\cp_p-$a.e. $x\in \Omega$ is an $L_p-$point of $\nabla f$, thus  $f\in RW^1_{p,\loc}(\Omega)$.  
\end{proof}
         
Now we demonstrate by simple examples that the inclusions
\begin{equation*}
W^{2}_{p,\loc}(\Omega)\subset  RW^1_{p,\loc}(\Omega)\subset  W^{1}_{p,\loc}(\Omega)
\end{equation*} 
are strict.

On the first step we construct a function $f\in RW^1_{p,\loc}(\Omega)\setminus W^{2}_{p,\loc}(\Omega)$. We choose $\Omega=\mathbb R^n$, $n>1$, $p=1$ and let us look at the function $f:\mathbb R^n\to \mathbb R$ defined by the rule 
$$
f(x)=|x|, \,\, x\in\mathbb R^n.
$$
Since $f$ is a Lipschitz function, then $f\in W^{1}_{\infty}(\mathbb R^n)$ and, in particular, $f\in W^{1}_{1,\loc}(\mathbb R^n)$. The weak gradient of $f$  
$$
\nabla f(x)=\frac{x}{|x|}\notin W^{1}_{1,\loc}(\mathbb R^n,\mathbb R^n),
$$
therefore $f\notin W^{2}_{1,\loc}(\mathbb R^n)$.

Since every point $x\neq 0$ is a continuous point of $\nabla f$, then it is a Lebesgue point, so
\begin{equation*}
(\nabla f)^*(x)=\lim_{r\to 0^+}\fint_{B(x,r)}\nabla f(z)dz=\frac{x}{|x|},\quad \forall x\in \mathbb R^n\setminus\{0\}.
\end{equation*}
Therefore 
\begin{equation*}
\lim_{r\to 0^+}\fint_{B(x,r)}\left|\frac{z}{|z|}-\frac{x}{|x|}\right|dz=0,\quad \forall x\in \mathbb R^n\setminus\{0\},\quad \cp_1(\{0\})=0.
\end{equation*}
Thus, $f\in RW^1_{1,\loc}(\mathbb R^n)$.
We use the assumption $n>1$ to get 
$\cp_1(\{0\})=0$ from 
$\mathcal{H}^{n-1}(\{0\})=0$ using the inequality $\eqref{eq:controlling of capacity by Hausdorff measure}$.

To construct a function $f\in W^1_{p,\loc}(\Omega)\setminus RW^1_{p,\loc}(\Omega)$
we choose $\Omega=\mathbb R$, $p>1$ and the same function as above 
$f:\mathbb R\to\mathbb R, f(x)=|x|$. 
It follows as above
\begin{equation*}
\lim_{r\to 0^+}\fint_{B(x,r)}\left|\frac{z}{|z|}-\frac{x}{|x|}\right|dz=0,\quad \forall x\in \mathbb R\setminus\{0\},
\end{equation*}
and 
\begin{equation*}
(\nabla f)^*(0)=\lim_{r\to 0^+}\fint_{B(0,r)}\frac{z}{|z|}dz=0,\quad \lim_{r\to 0^+}\fint_{B(0,r)}\left|\frac{z}{|z|}-0\right|dz=1\neq 0.
\end{equation*}
Since $p>1$ we have $\cp_p(\{0\})>0$ as the (outer) measure $\cp_p$ is an atomic measure in the case where the parameter $p$ is strictly bigger than the dimension $n$ (for proof see for example \cite{K21}). Thus $f\notin RW^1_{p,\loc}(\mathbb R)$.

In fact, $f\in RW^1_{p,\loc}(\Omega)$ for $p>n$ if and only if 
$f\in W^{1}_{p,\loc}(\Omega)$ and \textbf{every} point $x\in \Omega$ is an $L_p-$point of $\nabla f$.

\section{$L_p$-approximate differentiability for functions $RW^1_{p,\loc}(\Omega)$}

The notion of the (strong) $L_1$-approximate differentiability was introduced  in \cite{AFP} in connections to $BV$-functions. 
In this section we introduce a notion of $L_p$-approximate differentiability which is a generalization of the notion of the $L_1-$approximate differentiability and is useful for study of functions of 
$RW^1_{p,\loc}(\Omega)$.

\begin{defn}
Let $\Omega\subset\mathbb R^n$ be an open set, $1\leq p<\infty$ and $f\in L_{p,\loc}(\Omega)$. 
We say that $f$ is {\it $L_p$-approximately differentiable} at $x\in \Omega$ if there exists a linear mapping 
$L:\mathbb R^n\to \mathbb R$ such that
\begin{equation*}
\lim_{r\to 0^+}\fint_{B(x,r)}\frac{|f(y)-f^*(x)-L(y-x)|^p}{r^p}dy=0.
\end{equation*}
This linear mapping $L$ is called the {\it $L_p$-approximate differential} of the function $f$ at the point $x$.
We denote by $\mathcal{D}_f\subset \Omega$ the set of points $x\in\Omega$ where the function $f$ in the class $L_{p,\loc}(\Omega)$
is $L_p-$ approximately differentiable and by $L=D_af(x)$ its $L_p-$approximate differential at the point $x\in\Omega$.
\end{defn}

Note that if $x\in \Omega$ is a point of $L_p-$approximate differentiability of $f\in L_{p,\loc}(\Omega)$, then it is also a point of $L_q-$approximate differentiability for every 
$1\leq q\leq p$ by Jensen's inequality for concave functions:

\begin{multline*}
\fint_{B(x,r)}\frac{|f(y)-f^*(x)-L(y-x)|^q}{r^q}dy\\
=\fint_{B(x,r)}\left(\frac{|f(y)-f^*(x)-L(y-x)|^p}{r^p}\right)^{\frac{q}{p}}dy
\\
\leq \left(\fint_{B(x,r)}\frac{|f(y)-f^*(x)-L(y-x)|^p}{r^p}dy\right)^{\frac{q}{p}}.
\end{multline*}


\medskip

\noindent 
\textbf{Uniqueness of the $L_p-$approximate differential.} 
Assume $L$ and $L'$ are two $L_p-$approximate differentials of $f$ at $x$.  Then for any 
$\varepsilon>0$ there exists $r>0$ such that
\begin{equation*}
\fint_{B(x,r)}\frac{|f(y)-f^*(x)-L(y-x)|^p}{r^p}dy\leq \varepsilon
\end{equation*}
and 
\begin{equation*}
\fint_{B(x,r)}\frac{|f(y)-f^*(x)-L'(y-x)|^p}{r^p}dy\leq \varepsilon.
\end{equation*}
Therefore, 
\begin{multline*}
\fint_{B(x,r)}|L(y-x)-L'(y-x)|^pdy\leq 2^{p-1}\fint_{B(x,r)}|f(y)-f^*(x)-L(y-x)|^pdy
\\
+2^{p-1}\fint_{B(x,r)}|f(y)-f^*(x)-L'(y-x)|^pdy\leq 2^p\varepsilon r^p.\nonumber
\end{multline*}
Hence, by changing variable $y=x+rz$ on the left hand side of the last inequality we obtain
\begin{equation*}
\fint_{B(0,1)}|L(z)-L'(z)|^pdz\leq 2^p\varepsilon.
\end{equation*}
Therefore, $L(z)=L'(z)$ for all $z\in B(0,1)$ and since $L,L'$ are linear mappings they are equal everywhere: 
if $w\notin B(0,1)$, let us choose any $0<\delta<1$; then 
$L\left(\frac{\delta w}{|w|}\right)=L'\left(\frac{\delta w}{|w|}\right)$, so by the linearity of $L,L'$ we get $L(w)=L'(w)$.

The notion of the $L_p-$ approximate differentiability is stronger than the notion of approximate differentiability. Let us recall the classical notion of approximate differentiability \cite{F69}:

\begin{defn}
Let $f:\Omega\to\mathbb{R}$ be a measurable function, $x\in \Omega$.  
We say that $f$ is approximately differentiable at the point $x$ if there exists a linear mapping $L:\mathbb{R}^n\to \mathbb{R}$ such that for every 
$\varepsilon>0$ the set 
\begin{equation}
\label{apdif}
A_\varepsilon=\{y\in \Omega\setminus \{x\} : D_x(y)>\varepsilon\},\quad \text{where}\,\,D_x(y):=\frac{|f(y)-f^*(x)-L(y-x)|}{|y-x|},
\end{equation}
has density zero at the point $x$, means that:
\begin{equation*}
\lim_{r\to 0^+}\frac{\left|A_\varepsilon\cap B(x,r)\right|}{\left|B(x,r)\right|}=0.
\end{equation*}
\end{defn} 

Recall also the {\it Chebyshev inequality}:
Let $A\subset \mathbb{R}^n$ be a measurable set, $1\leq p<\infty$ and $h\in L_p(A)$. Then, for every 
$\varepsilon>0$ 
\begin{equation}
\left|\left\{x\in A:|h(x)|>\varepsilon\right\}\right|\leq \frac{1}{\varepsilon^p}\int_{A}|h(x)|^pdx.
\end{equation}

Let $x\in \Omega$ be a point of $L_p-$ approximate differentiability of $f\in L_{p,\loc}(\Omega)$. We prove that $A_\varepsilon$ has density zero at $x$ for every $\varepsilon>0$.  
Define $A_\varepsilon,D_x$ by the formula~(\ref{apdif}). By contradiction we assume that there exists $\varepsilon>0$ such that the upper density of the set $A_\varepsilon$ at the point $x$ is positive, means that 
\begin{equation*}
\limsup_{r\to 0^+}\frac{\left|A_\varepsilon\cap B(x,r)\right|}{\left|B(x,r)\right|}>0.
\end{equation*}
So there exists a positive number 
$\alpha>0$ and a sequence $r_i \to 0^+$ as $i\to \infty$  
such that the density quotient of $A_\varepsilon$ at $x$ for $r_i$ is bigger than $\alpha$ for each $i$, means that 
\begin{equation*}
\frac{\left|A_\varepsilon\cap B(x,r_i)\right|}{\left|B(x,r_i)\right|}>\alpha,\quad \forall i\in \mathbb{N}.
\end{equation*}
For any $0<\sigma<1$ we have by measurability of the ball 
$B(x,\sigma r_i)$  
$$
\left|A_\varepsilon\cap \left(B(x,r_i)\setminus B(x,\sigma r_i)\right)\right|=\left|A_\varepsilon\cap B(x,r_i)\right|-\left|A_\varepsilon\cap  B(x,\sigma r_i)\right|.
$$
Thus,
\begin{equation*}
\frac{\left|A_\varepsilon\cap \left(B(x,r_i)\setminus B(x,\sigma r_i)\right)\right|}{\left|B(x,r_i)\right|}>\alpha-\frac{\left|A_\varepsilon\cap  B(x,\sigma r_i)\right|}{\left|B(x,r_i)\right|}
,\quad \forall i\in \mathbb{N}.
\end{equation*}

Hence, since 
$$
\frac{\left|A_\varepsilon\cap  B(x,\sigma r_i)\right|}{\left|B(x,r_i)\right|}\leq \sigma^n,\quad \forall i\in \mathbb{N}
$$
we obtain for $\sigma^n<\frac{\alpha}{2}$
\begin{equation*}
\frac{\left|A_\varepsilon\cap \left[B(x,r_i)\setminus B(x,\sigma r_i)\right]\right|}{\left|B(x,r_i)\right|}>\frac{\alpha}{2},\quad \forall i\in \mathbb{N}.
\end{equation*}
Therefore, by Chebyshev inequality we get for every  $i\in \mathbb{N}$
\begin{multline*}
\frac{\alpha}{2}<\frac{\left|A_\varepsilon\cap \left[B(x,r_i)\setminus B(x,\sigma r_i)\right]\right|}{\left|B(x,r_i)\right|}
\\
\leq \frac{\left|\{y\in B(x,r_i):|f(y)-f^*(x)-L(y-x)|>\varepsilon \sigma r_i\}\right|}{\left|B(x,r_i)\right|}
\\
\leq \frac{1}{(\varepsilon \sigma)^p}\fint_{B(x,r_i)}\frac{|f(y)-f^*(x)-L(y-x)|^p}{r_i^p}dy.
\end{multline*}
The last inequality contradicts the assumption that $x\in \mathcal{D}_f$. It proves that the set $A_\varepsilon$ has density zero at $x$. 

\begin{prop}
Assume that for every $x\in \Omega$ the function $y\mapsto D_x(y)$ as defined in $\eqref{apdif}$ is locally bounded. Then the definitions of approximate differentiability and $L_p-$approximate differentiability are equivalent.
\end{prop}

\begin{proof}
We have already proved above that $L_p-$approximate differentiability implies approximate differentiability. Let $f\in L_{p,\loc}(\Omega)$ and $x\in \Omega$. Let $\Omega_0$ be an open set such that $x\in \Omega_0\subset \Omega$ and $D_x\in L_\infty(\Omega_0)$.  

Assume $x$ is a point of approximate differentiability of $f$ and denote by $L$ the approximate differential of $f$ at $x$. It follows for every $r>0$ such that $B(x,r)\subset \Omega_0$ that
\begin{multline*}
\fint_{B(x,r)}\frac{|f(y)-f^*(x)-L(y-x)|^p}{r^p}dy\leq \fint_{B(x,r)}\frac{|f(y)-f^*(x)-L(y-x)|^p}{|y-x|^p}dy
\\
=\frac{1}{\left|B(x,r)\right|}\int_{B(x,r)\cap A_\varepsilon}\left(D_x(y)\right)^pdy
+\frac{1}{\left|B(x,r)\right|}\int_{B(x,r)\setminus A_\varepsilon}\left(D_x(y)\right)^pdy\nonumber
\\
\leq M^p\frac{\left|A_\varepsilon\cap B(x,r)\right|}{\left|B(x,r)\right|}+\varepsilon^p,\nonumber
\end{multline*}
where $M$ is a bound on $D_x$ in $\Omega_0$ and $A_\varepsilon$ is defined as in $\eqref{apdif}$. Since $x$ is a point of approximate differentiability and 
$\varepsilon>0$ is arbitrary, we obtain $x\in \mathcal{D}_f$.

Therefore, $x$ is a point of approximate differentiability of $f$ if and only if $x\in \mathcal{D}_f$.
\end{proof}

Recall the notion of standard mollifier (see, for example, \cite{M,evans2015measure}): Consider 
\begin{equation*}
\eta:\mathbb{R}^n\to \mathbb{R},\quad \eta(x):=
\begin{cases}
C\exp\left(\frac{1}{|x|^2-1}\right)\quad & |x|<1
\\
0\quad &|x|\geq 1
\end{cases}.
\end{equation*}
The function $\eta$ is smooth and the constant $C$ is chosen for having
$\int_{\mathbb{R}^n}\eta(x)dx=1$. For each 
$\varepsilon>0$ we define the function
$$
\eta_\varepsilon:\mathbb{R}^n\to \mathbb{R},\quad \eta_\varepsilon(x):=\frac{1}{\varepsilon^n}\eta\left(\frac{x}{\varepsilon}\right).
$$
The family of functions $\eta_\varepsilon$ is called the {\it standard mollifier}.

Let $\Omega\subset\mathbb{R}^n$ be an open set. We denote 
$\Omega_\varepsilon:=\left\{x\in \Omega: \dist(x,\partial \Omega)>\varepsilon\right\}$. It is known that for a function 
$f\in L_{1,\loc}(\Omega)$ the convolution 
$$
f_\varepsilon(x):=f*\eta_\varepsilon(x)=\int_{\Omega}f(y)\eta_\varepsilon(x-y)dy
$$ 
is a smooth function in $\Omega_\varepsilon$ and $f_\varepsilon$ converges to $f$ almost everywhere in $\Omega$ as $\varepsilon\to 0^+$; if $f\in W^1_{p,\loc}(\Omega)$, $1\leq p<\infty$, then $f_\varepsilon$ converges to $f$ as $\varepsilon\to 0^+$ in the topology of $W^1_{p,\loc}(\Omega)$ and $\nabla f_\varepsilon=\nabla f*\eta_\varepsilon$; for more details see, for example, \cite{evans2015measure,M}. Recall also that if $f\in L_p(\Omega)$, $1\leq p\leq \infty$, then 
$$
\|f*\eta_\varepsilon\|_{L_p(\Omega)}\leq \|\eta_\varepsilon\|_{L_1(\Omega)}\|f\|_{L_p(\Omega)}=\|f\|_{L_p(\Omega)}, 
\,\,\|\eta_\varepsilon\|_{L_1(\Omega)}=1.
$$

\begin{rem}
\label{rem:every L_p point is a converging point of the mollified family}
Notice that for every $L_p-$point $w\in \Omega$ of a function $f\in L_{p,\loc}(\Omega)$ we have 
$\lim_{\varepsilon\to 0^+}f_\varepsilon(w)=f^*(w)$ by the following calculation:
\begin{multline*}
|f_\varepsilon(w)-f^*(w)|^p=\left|\int_{B(w,\varepsilon)}\left(f(z)-f^*(w)\right)\eta_\varepsilon(w-z)dz\right|^p
\\
\leq \left(\frac{1}{\varepsilon^n}\int_{B(w,\varepsilon)}\left|f(z)-f^*(w)\right|\eta\left(\frac{w-z}{\varepsilon}\right)dz\right)^p
\\
\leq \|\eta\|^p_{L_\infty(\mathbb{R}^n)}\omega_n^p\left(\fint_{B(w,\varepsilon)}\left|f(z)-f^*(w)\right|dz\right)^p
\\
\leq \|\eta\|^p_{L_\infty(\mathbb{R}^n)}\omega_n^p\fint_{B(w,\varepsilon)}\left|f(z)-f^*(w)\right|^pdz,
\end{multline*}
where $\omega_n=|B(0,1)|$ is the volume of the unit ball $B(0,1)\subset\mathbb R^n$.
Thus, by theorem $\ref{thm:fine property of Sobolev functions}$, for a function 
$f\in W^1_{p,\loc}(\Omega)$ we have 
$\lim_{\varepsilon\to 0^+}f_\varepsilon(w)=f^*(w)$ for $\cp_p-$a.e. in $\Omega$. In particular, for every Sobolev function $f\in W^1_{p,\loc}(\Omega)$ there exists a sequence of smooth functions which converges to $f^*$ $\cp_p-$a.e. in $\Omega$.
\end{rem}

In the proof of the next theorem we use the following lemma which concerns Lebesgue spaces $L_p$ (for proof see for example \cite{F69}).

\begin{lem}
\label{lem:characterization of the L^p norm by the dual space}
Assume $E\subset \mathbb R^n$ is a measurable set. Suppose either (i) $1<p<\infty$ and $q=\frac{p}{p-1}$; or (ii) $p=1$ and $q=\infty$; or (iii) $p=\infty$ and $q=1$. 
If $f\in L_p(E)$, then
\begin{equation*}
\|f\|_{L_p(E)}=\sup\int_Ef(x)g(x)dx,
\end{equation*}
where the supremum is taken over all $g\in L_q(E)$ with $\|g\|_{L_q(E)}=1$.
\end{lem} 

The following theorem is a capacitary version of a well-known theorem about the $L_p-$approximate differentiability almost everywhere(with respect to Lebesgue measure) for Sobolev functions $W^1_p$ (for proof see, for example, \cite{evans2015measure}).  
   
\begin{thm}
\label{thm:L^p differentiability for functions in F^{1,p}}
Let $\Omega\subset\mathbb R^n$ be an open set. If $f\in RW^1_{p,\loc}(\Omega)$, $1\leq p<\infty$,
then $f$ is $L_p-$approximately differentiable at $\cp_p-$a.e. $x\in\Omega$ and  $D_af(x)=(\nabla f)^*(x)$.
\end{thm}

\begin{proof}
By the assumption that $f\in RW^1_{p,\loc}(\Omega)$ and by Theorem $\ref{thm:fine property of Sobolev functions}$, there exists a set $E$ such that 
$\cp_p(E)=0$ and  
\begin{multline}
\label{eq: refined property of the gradient}
\lim_{r\to 0^+}\fint_{B(x,r)}\left|f(z)-f^*(x)\right|^pdz\\
=\lim_{r\to 0^+}\fint_{B(x,r)}\left|\nabla f(z)-(\nabla f)^*(x)\right|^pdz=0,\quad \forall x\in \Omega\setminus E.
\end{multline}
Fix $x\in \Omega\setminus E$. Let $\phi\in L_q(B(x,r))$, where $p,q$ as in Lemma 
$\ref{lem:characterization of the L^p norm by the dual space}$ and $r>0$ is such that $\overline{B}(x,r)\subset\Omega$. Consider the function
$$
f_\varepsilon(x):=f*\eta_\varepsilon(x)=\int_{\Omega}f(y)\eta_\varepsilon(x-y)dy.
$$ 
By the fundamental theorem of calculus, the chain rule and Fubini's theorem we have for small enough $\varepsilon>0$ such that $B(x,r)\subset \Omega_\varepsilon$ 

\begin{multline}
\label{eq:eqution1}
\fint_{B(x,r)}\phi(y)\left(f_\varepsilon(y)-f_\varepsilon(x)- (\nabla f)^*(x)\cdot(y-x)\right)dy
\\
=\fint_{B(x,r)}\phi(y)\left(\int_0^1\frac{d}{ds}f_\varepsilon(x+s(y-x))ds-(\nabla f)^*(x)\cdot(y-x)\right)dy
\\
=\fint_{B(x,r)}\phi(y)\left(\int_0^1 \nabla f_\varepsilon(x+s(y-x))\cdot(y-x)ds-(\nabla f)^*(x)\cdot(y-x)\right)dy
\\
=\int_0^1\fint_{B(x,r)}\phi(y)\left(\nabla f_\varepsilon(x+s(y-x))-(\nabla f)^*(x)\right)\cdot(y-x)dyds.
\end{multline}
Let us assume first that $p>1$. By changing variables $z=x+s(y-x)$, H{\"o}lder inequality and $\eqref{eq:eqution1}$ we get
\begin{align}
\label{eq:estimate for case p=1}
&\fint_{B(x,r)}\phi(y)\left(f_\varepsilon(y)-f_\varepsilon(x)- (\nabla f)^*(x)\cdot(y-x)\right)dy 
\\
&=\int_0^1\frac{1}{s^{n+1}}\frac{1}{\omega_nr^n}\int_{B(x,sr)}\phi\left(\frac{z-x}{s}+x\right)\left(\nabla f_\varepsilon(z)-(\nabla f)^*(x)\right)\cdot(z-x)dzds \nonumber
\\
&=\int_0^1\frac{1}{s}\fint_{B(x,sr)}\phi\left(\frac{z-x}{s}+x\right)\left(\nabla f_\varepsilon(z)-(\nabla f)^*(x)\right)\cdot(z-x)dzds \nonumber
\\
&\leq r\int_0^1\fint_{B(x,sr)}\left|\phi\left(\frac{z-x}{s}+x\right)\right|\left|\nabla f_\varepsilon(z)-(\nabla f)^*(x)\right|dzds \nonumber
\\
&\leq r\int_0^1\left(\fint_{B(x,sr)}\left|\phi\left(\frac{z-x}{s}+x\right)\right|^qdz\right)^{\frac{1}{q}}\left(\fint_{B(x,sr)}\left|\nabla f_\varepsilon(z)-(\nabla f)^*(x)\right|^pdz\right)^{\frac{1}{p}} \nonumber
ds. 
\end{align}
Assume that
$\|\phi\|_{L_q\left(B(x,r)\right)}=1$. 
Then we obtain
\begin{multline}
\label{eq:equation2}
\fint_{B(x,sr)}\left|\phi\left(\frac{z-x}{s}+x\right)\right|^qdz=\frac{1}{\omega_n(rs)^n}\int_{B(x,sr)}\left|\phi\left(\frac{z-x}{s}+x\right)\right|^qdz
\\
=\frac{1}{\omega_n(rs)^n}s^n\int_{B(x,r)}\left|\phi\left(y\right)\right|^qdy
=\frac{1}{\omega_nr^n}. 
\end{multline}
Hence, by $\eqref{eq:estimate for case p=1}$ and $\eqref{eq:equation2}$ we get
\begin{multline}
\label{eq:inequality2}
\fint_{B(x,r)}\phi(y)\left(f_\varepsilon(y)-f_\varepsilon(x)-(\nabla f)^*(x)(y-x)\right)dy
\\
\leq r\left(\frac{1}{\omega_nr^n}\right)^{1/q}\int_0^1\left(\fint_{B(x,sr)}\left|\nabla f_\varepsilon(z)-(\nabla f)^*(x)\right|^pdz\right)^{1/p}ds.
\end{multline}
Taking the supremum in 
$\eqref{eq:inequality2}$ over all such functions $\phi$ and using Lemma 
$\ref{lem:characterization of the L^p norm by the dual space}$ we obtain
\begin{multline}
\label{eq:inequality5}
\frac{1}{\omega_nr^n}\left(\int_{B(x,r)}\left|f_\varepsilon(y)-f_\varepsilon(x)-(\nabla f)^*(x)(y-x)\right|^pdy\right)^{1/p}
\\
\leq r\left(\frac{1}{\omega_nr^n}\right)^{1/q}\int_0^1\left(\fint_{B(x,sr)}\left|\nabla f_\varepsilon(z)-(\nabla f)^*(x)\right|^pdz\right)^{1/p}ds.
\end{multline}
Using the identity 
$$
\frac{1}{\omega_nr^n}=\left(\frac{1}{\omega_nr^n}\right)^{1/p}\left(\frac{1}{\omega_nr^n}\right)^{1/q},
$$
we get from $\eqref{eq:inequality5}$ the inequality
\begin{multline}
\label{eq:inequality1}
\left(\fint_{B(x,r)}\left|f_\varepsilon(y)-f_\varepsilon(x)-(\nabla f)^*(x)(y-x)\right|^pdy\right)^{1/p}
\\
\leq r\int_0^1\left(\fint_{B(x,sr)}\left|\nabla f_\varepsilon(z)-(\nabla f)^*(x)\right|^pdz\right)^{1/p}ds.
\end{multline}

A similar inequality as 
$\eqref{eq:inequality1}$ can be obtained also for $p=1$: using 
$\eqref{eq:estimate for case p=1}$ we get

\begin{multline}
\label{eq:inequality6}
\fint_{B(x,r)}\phi(y)\left(f_\varepsilon(y)-f_\varepsilon(x)- (\nabla f)^*(x)\cdot(y-x)\right)dy 
\\
\leq r\int_0^1\fint_{B(x,sr)}\left|\phi\left(\frac{z-x}{s}+x\right)\right|\left|\nabla f_\varepsilon(z)-(\nabla f)^*(x)\right|dzds
\\
\leq r\int_0^1\fint_{B(x,sr)}\left|\nabla f_\varepsilon(z)-(\nabla f)^*(x)\right|dzds,
\end{multline}
where $\|\phi\|_{L_{\infty}(B(x,r))}=1$.
Taking the supremum in the inequality 
$\eqref{eq:inequality6}$ over all $\phi \in L_{\infty}(B(x,r))$ with $\|\phi\|_{L_{\infty}(B(x,r))}=1$ and using Lemma 
$\ref{lem:characterization of the L^p norm by the dual space}$, we obtain
inequality $\eqref{eq:inequality1}$ with $p=1$.

We intend to take the limit as $\varepsilon\to 0^+$ in inequality $\eqref{eq:inequality1}$. Since $x$ is an $L_p-$point of $f$, we get by Remark $\ref{rem:every L_p point is a converging point of the mollified family}$ that $\lim_{\varepsilon\to 0^+}f_\varepsilon(x)=f^*(x)$.  
Thus, since $f_\varepsilon$ converges to $f$ almost everywhere, we have by Fatou's lemma 

\begin{multline*}
\fint_{B(x,r)}\left|f(y)-f^*(x)-(\nabla f)^*(x)(y-x)\right|^pdy
\\
\leq \liminf_{\varepsilon\to 0^+}\fint_{B(x,r)}\left|f_\varepsilon(y)-f_\varepsilon(x)-(\nabla f)^*(x)(y-x)\right|^pdy.
\end{multline*}
Note that
\begin{multline*}
\fint_{B(x,sr)}\left|\nabla f_\varepsilon(z)-(\nabla f)^*(x)\right|^pdz
\leq 2^{p-1}\fint_{B(x,sr)}\left|\nabla f_\varepsilon(z)\right|^pdz
+2^{p-1}\left|(\nabla f)^*(x)\right|^p
\\
=2^{p-1}\frac{1}{\omega_n(sr)^n}\left\|\nabla f*\eta_\varepsilon\right\|^p_{L_p\left(B(x,sr)\right)}
+2^{p-1}\left|(\nabla f)^*(x)\right|^p
\\
\leq 2^{p-1}\frac{1}{\omega_n(sr)^n}\left\|\nabla f\right\|^p_{L_p\left(B(x,sr)\right)}
+2^{p-1}\left|(\nabla f)^*(x)\right|^p
\\
=2^{p-1}\fint_{B(x,sr)}\left|\nabla f(z)\right|^pdz
+2^{p-1}\left|(\nabla f)^*(x)\right|^p
\\
\leq 2^{2p-2}\fint_{B(x,sr)}\left|\nabla f(z)-(\nabla f)^*(x)\right|^pdz
+(2^{2p-2}+2^{p-1})\left|(\nabla f)^*(x)\right|^p.
\end{multline*}

The function $s\longmapsto \fint_{B(x,sr)}\left|\nabla f(z)-(\nabla f)^*(x)\right|^pdz$ is bounded on $(0,1)$ for every small enough $r>0$ becuase, by $\eqref{eq: refined property of the gradient}$, there exists $\delta>0$ such that 
\begin{equation}
\label{eq:controlling average for using D.C.T (1)}
\fint_{B(x,\rho)}\left|\nabla f(z)-(\nabla f)^*(x)\right|^pdz\leq 1,\quad \forall \rho\in (0,\delta).
\end{equation}
Hence, for every $0<r<\delta$ and $s\in (0,1)$ we have 
\begin{equation}
\label{eq:controlling average for using D.C.T (2)}
\fint_{B(x,sr)}\left|\nabla f(z)-(\nabla f)^*(x)\right|^pdz\leq 1. 
\end{equation}
Thus, we can use the dominated convergence theorem together with the convergence of 
$f_\varepsilon$ to $f$ in the topology of $W^1_{p,\loc}$ to obtain
\begin{multline*}
\lim_{\varepsilon\to 0^+}\int_0^1\left(\fint_{B(x,sr)}\left|\nabla f_\varepsilon(z)-(\nabla f)^*(x)\right|^pdz\right)^{1/p}ds
\\
=\int_0^1\lim_{\varepsilon\to 0^+}\left(\fint_{B(x,sr)}\left|\nabla f_\varepsilon(z)-(\nabla f)^*(x)\right|^pdz\right)^{1/p}ds
\\
=\int_0^1\left(\fint_{B(x,sr)}\left|\nabla f(z)-(\nabla f)^*(x)\right|^pdz\right)^{1/p}ds.
\end{multline*}
Therefore, taking the lower limit as 
$\varepsilon\to 0^+$ in 
$\eqref{eq:inequality1}$, we obtain
\begin{multline}
\label{eq:inequality17}
\left(\fint_{B(x,r)}\left|f(y)-f(x)-(\nabla f)^*(x)(y-x)\right|^pdy\right)^{1/p}
\\
\leq r\int_0^1\left(\fint_{B(x,sr)}\left|\nabla f(z)-(\nabla f)^*(x)\right|^pdz\right)^{1/p}ds.
\end{multline}

By Jensen's inequality and $\eqref{eq:inequality17}$ we obtain
\begin{multline*}
\fint_{B(x,r)}\frac{\left|f(y)-f(x)-(\nabla f)^*(x)(y-x)\right|^p}{r^p}dy
\\
\leq\left(\int_0^1\left(\fint_{B(x,sr)}\left|\nabla f(z)-(\nabla f)^*(x)\right|^pdz\right)^{1/p}ds\right)^p
\\
\leq \int_0^1\fint_{B(x,sr)}\left|\nabla f(z)-(\nabla f)^*(x)\right|^pdzds.
\end{multline*}
Therefore, since $x$ is an $L_p-$ point of $\nabla f$, we get by the dominated convergence theorem
\begin{multline*}
\limsup_{r\to 0^+}\fint_{B(x,r)}\frac{\left|f(y)-f(x)-(\nabla f)^*(x)(y-x)\right|^p}{r^p}dy
\\
\leq \limsup_{r\to 0^+} \int_0^1\fint_{B(x,sr)}\left|\nabla f(z)-(\nabla f)^*(x)\right|^pdzds=0.
\end{multline*}
Therefore, $f$ is $L_p-$ approximately differentiable at $x$ and by the uniqueness of the $L_p-$approximate differential we have $D_af(x)=(\nabla f)^*(x)$.
\end{proof}
\begin{cor}
Let 
$1\leq p<\infty$, $\Omega\subset\mathbb R^n$ be an open set and $f\in W^{2}_{p,\loc}(\Omega).$
Then $f$ is $L_p-$ approximately differentiable at
$\cp_p-$a.e $x\in \Omega$.
\end{cor}

\begin{proof}
The proof follows immediately by Proposition 
$\ref{prop:second order Sobolev space is continuously embedded in F^{1,p}}$ and Theorem $\ref{thm:L^p differentiability for functions in F^{1,p}}$.
\end{proof}


\section{Convergence of the difference quotient to the weak derivative in the topology of $W_{p,\loc}^{1}$}

Let $f\in W^1_p(\Omega)$ be a function  with a $p$-refined weak gradient. Then its refined gradient $(\nabla f)^*(x)$ defines the formal differential $Df(x)$ by the rule:
$$
Df(x):\mathbb R^n\to \mathbb R,\,\, Df(x)(z)=(\nabla f)^*(x)\cdot z
$$
in $\cp_p$-almost all pints $x\in\Omega$. We prove that this formal differential $Df(x)$ is the differential with respect to convergence in the topology of $W^1_{p,\loc}(\Omega)$ for $x\in\Omega\setminus \mathcal{N}$, where $\mathcal{N}$ is a set of $p$-capacity zero. 

Let $\Omega\subset \mathbb R^n$ be an open set and let $f\in W^{1}_{p,\loc}(\Omega)$ for some $1\leq p<\infty$. Let us define for $x\in \Omega$ and $t>0$ a function $f_{x,t}(z)$ by:
\begin{equation}
\label{function}
f_{x,t}(z):=\frac{f^*(x+t z)-f^*(x)}{t}.
\end{equation}
For each $x\in \Omega$ the family of functions $\{f_{x,t}\}_{t>0}$ is well defined on any nonempty bounded set of $\mathbb R^n$ for every sufficiently small $t>0$: 
Since $\Omega$ is open and $x\in \Omega$, there exists $r>0$ such that $B(x,r)\subset\Omega$. If $B\subset\mathbb{R}^n$ is an arbitrary nonempty bounded set, then for every $t$ such that $|t|<r/R, R:=\sup_{z\in B}|z|$, we have $x+tB\subset B(x,r)$. Thus, the function $f_{x,t}$ is defined on $B$ for every $0<t<r/R$.

\begin{lem}
\label{lem:fine property ia a point}
Let $\Omega\subset \mathbb R^n$ be an open set and let $f\in W^{1}_{p,\loc}(\Omega)$ for some $1\leq p<\infty$. 
Assume existence of a point $x\in \Omega$ such that
\begin{equation}
\label{eq:v is p-approximate limit of Df at x}
\lim_{r\to 0^+}\fint_{B(x,r)}|\nabla f(y)-(\nabla f)^*(x)|^pdy=0
\end{equation}
and
\begin{equation}
\label{eq:convergence at a point of the mollification}
\lim_{\varepsilon\to 0^+}f_\varepsilon(x)=f^*(x),
\end{equation}
where $f_\varepsilon=f*\eta_\varepsilon$ and $\eta_\varepsilon$ is the standard mollifier.
Then, the family of functions 
$\{f_{x,t}\}_{t>0}$ converges to the function $(\nabla f)^*(x):\mathbb R^n\to \mathbb R$ as $t\to 0^+$ in the topology of 
$W^{1}_{p,\loc}(\mathbb R^n)$; means that, for any open and bounded set 
$U\subset\mathbb R^n$ it follows that
\begin{equation*}
\lim_{t\to 0^+}\|f_{x,t}-(\nabla f)^*(x)\|_{W^{1}_p(U)}=0.
\end{equation*}
\end{lem}

\begin{proof}
During this proof we denote $v:=(\nabla f)^*(x)$. Notice that the assumption $\eqref{eq:v is p-approximate limit of Df at x}$ is equivalent to say that for every  open and bounded set $U\subset\mathbb R^n$ it follows that
\begin{equation}
\label{eq:equivalence for L_p point}
\lim_{s\to 0^+}\frac{1}{s^n}\int_{x+sU}\left|\nabla f(y)-v\right|^pdy=0. 
\end{equation}
Let $U\subset\mathbb R^n$ be any nonempty open and bounded set.
By the fundamental theorem of Calculus, Jensen's inequality, Fubini's theorem and the change of variable formula we have for small enough $t>0$
\begin{multline}
\label{eq:inequality4}
\int_{U}|(f_\varepsilon)_{x,t}(z)-v(z)|^pdz
=\int_{U}\left|\frac{\int_0^1\nabla f_\varepsilon(x+st z)\cdot t zds}{t}-v(z)\right|^pdz
\\
\leq \int_{U}\int_0^1\left|\nabla f_\varepsilon(x+st z)\cdot z-v(z)\right|^pdsdz
\leq \sup_{z\in U}|z|^p\int_0^1\int_{U}\left|\nabla f_\varepsilon(x+st z)-v\right|^pdzds
\\
=\sup_{z\in U}|z|^p\int_0^1\frac{1}{(st)^n}\int_{x+st U}\left|\nabla f_\varepsilon(y)-v\right|^pdyds.
\end{multline}

Since $f_\varepsilon$ converges to $f$ almost everywhere, $f=f^*$ almost everywhere and assumption 
$\eqref{eq:convergence at a point of the mollification}$, then we obtain for almost every $z\in U$
\begin{equation*}
\lim_{\varepsilon\to 0^+}(f_\varepsilon)_{x,t}(z)=f_{x,t}(z).
\end{equation*} 
By Fatou's lemma 
\begin{equation}
\label{eq:inequality7}
\int_{U}|f_{x,t}(z)-v(z)|^pdz=\int_{U}\lim_{\varepsilon\to 0^+}|(f_\varepsilon)_{x,t}(z)-v(z)|^pdz
\leq \liminf_{\varepsilon\to 0^+}\int_{U}|(f_\varepsilon)_{x,t}(z)-v(z)|^pdz.
\end{equation}

By dominated convergence theorem and the convergence of $f_\varepsilon$ to $f$ in the topology of 
$W^1_{p,\loc}$ we obtain 
\begin{align}
\label{eq:equality8}
\lim_{\varepsilon\to 0^+}\int_0^1\frac{1}{(st)^n}\int_{x+st U}\left|\nabla f_\varepsilon(y)-v\right|^pdyds=\int_0^1\frac{1}{(st)^n}\lim_{\varepsilon\to 0^+}\int_{x+st U}\left|\nabla f_\varepsilon(y)-v\right|^pdyds
\\
=\int_0^1\frac{1}{(st)^n}\int_{x+st U}\left|\nabla f(y)-v\right|^pdyds.\nonumber
\end{align}
Thus, taking the lower limit as 
$\varepsilon\to 0^+$ in 
$\eqref{eq:inequality4}$ and using $\eqref{eq:inequality7}$ and 
$\eqref{eq:equality8}$, we get
\begin{equation}
\label{eq:ineqality8}
\int_{U}|f_{x,t}(z)-v(z)|^pdz\leq \sup_{z\in U}|z|^p\int_0^1\frac{1}{(st)^n}\int_{x+st U}\left|\nabla f(y)-v\right|^pdyds.
\end{equation}
Therefore, by dominated convergence theorem, 
$\eqref{eq:equivalence for L_p point}$ and $\eqref{eq:ineqality8}$ we obtain
\begin{multline*}
\limsup_{t\to 0^+}\int_{U}|f_{x,t}(z)-v(z)|^pdz
\\
\leq \left(\sup_{z\in U}|z|^p\right)\limsup_{t\to 0^+}\int_0^1\frac{1}{(st)^n}\int_{x+st U}\left|\nabla f(y)-v\right|^pdyds=0.
\end{multline*}
Next, notice that for small enough 
$t>0$ and almost all  
$z\in U$
\begin{align*}
\nabla \left[f_{x,t}-v\right](z)=\nabla f(x+t z)-v.
\end{align*}
Hence, by the last result and the change of variable formula we obtain
\begin{multline*}
\int_{U}\left|\nabla\left[f_{x,t}-v\right](z)\right|^pdz=\int_{U}\left|\nabla f(x+t z)-v\right|^pdz
\\
=\frac{1}{t^n}\int_{x+t U}\left|\nabla f(y)-v\right|^pdy.
\end{multline*}
Therefore, we get by 
$\eqref{eq:equivalence for L_p point}$
\begin{align*}
\lim_{t\to 0^+}\int_{U}\left|\nabla\left[f_{x,t}-v\right](z)\right|^pdz=\lim_{t\to 0^+}\frac{1}{t^n}\int_{x+t U}\left|\nabla f(y)-v\right|^pdy=0.
\end{align*}
It completes the proof.
\end{proof}

Using Lemma~$\ref{lem:fine property ia a point}$ we obtain the fine differentiability of Sobolev functions with refined gradients:

\begin{thm}
\label{cor:convergence of the difference quotient to its formal differential}
Let $1\leq p<\infty$, $\Omega\subset \mathbb R^n$ be an open set and let 
$f\in RW^1_{p,\loc}(\Omega)$.
Then for every open and bounded set $U\subset\mathbb R^n$
\begin{equation}
\lim_{t\to 0^+}\|f_{x,t}-(\nabla f)^*(x)\|_{W^1_p(U)}=0,\quad \text{for}\,\,\cp_p-a.e.\quad x\in \Omega,
\end{equation}
where $f_{x,t}$ is defined by (\ref{function}).
\end{thm}

\begin{proof}
Since $f\in RW^1_{p,\loc}(\Omega)$, then there exists a set $E\subset \Omega$ such that $\cp_p(E)=0$ and for every $x\in \Omega\setminus E$ 
\begin{equation*}
\lim_{r\to 0^+}\fint_{B(x,r)}|f(y)- f^*(x)|^pdy=0\quad \wedge\quad \lim_{r\to 0^+}\fint_{B(x,r)}|\nabla f(y)-(\nabla f)^*(x)|^pdy=0.
\end{equation*}
By remark $\ref{rem:every L_p point is a converging point of the mollified family}$ we also know that for every $x\in \Omega\setminus E$ 
\begin{equation*}
\lim_{\varepsilon\to 0^+}f_\varepsilon(x)=f^*(x).
\end{equation*}
The result now follows from the previous Lemma for each $x\in \Omega\setminus E$. 
\end{proof}

\subsection*{Generalization}
We say that $\alpha\in \mathbb{R}^n$ is a multi-index if $\alpha=(\alpha_1,...,\alpha_n)$, where for every $1\leq i\leq n$, $\alpha_i\in \mathbb{N}\cup \{0\}$. Recall the operations 
$|\alpha|=\alpha_1+...+\alpha_n$, $\alpha!=\alpha_1!\cdot...\cdot\alpha_n!$ and for $z=(z_1,...,z_n)\in \mathbb{R}^n$, $z^\alpha=z_1^{\alpha_1}\cdot...\cdot z_n^{\alpha_n}$. 

\begin{defn}
Let $\Omega$ be an open subset of 
$\mathbb R^n$. Then for $1\leq p<\infty$ we define the refined Sobolev space $RW^{k}_{p}(\Omega)$ as a set of functions $f\in W^k_p(\Omega)$ which have refined gradients of order $k$: for every multi-index $\alpha$ such that $|\alpha|=k$
\begin{equation*}
\label{eq:refined property of high order gradient}
\lim_{r\to 0^+}\fint_{B(x,r)}|\nabla^{\alpha} f(z)-(\nabla^{\alpha} f)^*(x)|^pdz=0 \quad for\quad  \cp_{p}-a.e.\quad  x\in \Omega.
\end{equation*}
The local space $RW^{k}_{p,\loc}(\Omega)$ is defined as follows: $f\in RW^{k}_{p,\loc}(\Omega)$ if and only if $f\in RW^{k}_{p}(\Omega_0)$ for every open and bounded set $\Omega_0$ with 
$\overline{\Omega}_0\subset \Omega$.
\end{defn}

\begin{rem}
The spaces $RW^{k}_{p}(\Omega)$, $RW^{k}_{p,\loc}(\Omega)$ are vector subspaces of $W^{k}_{p}(\Omega)$, $W^{k}_{p,\loc}(\Omega)$, respectively. 
\end{rem}

\begin{rem}
\label{rem:L_p points for all derivaties for function RW}
Note that for a function $f\in RW^{k}_{p,\loc}(\Omega)$, we get by Theorem $\ref{thm:fine property of Sobolev functions}$ that almost every point with respect to $\cp_p$ is an $L_p-$point of $\nabla^\alpha f$ for every multi-index $|\alpha|\leq k$.
\end{rem}

Recall Taylor formula with remainder of integral form for functions $f$ of the class $C^k$: if $\Omega\subset\mathbb{R}^n$ is an open set and 
$f\in C^k(\Omega)$ , then for every $x\in \Omega$ there exists $r>0$ such that $B(x,r)\subset\Omega$ and for every $y\in B(x,r)$ the following formula holds:
\begin{equation}
\label{eq:Taylor series for high derivatives of Sobolev functions}
f(y)=\sum_{|\alpha|\leq k-1}\frac{\nabla^\alpha f(x)}{\alpha!}(y-x)^{\alpha}
+\sum_{|\alpha|=k}\frac{k}{\alpha!}(y-x)^\alpha\int_{0}^1(1-t)^{k-1}\nabla^\alpha f(x+t(y-x))dt.
\end{equation}
Writing $y=x+hz$ for $|h|<r,z\in B(0,1)$, we get
\begin{equation*}
f(x+hz)=\sum_{|\alpha|\leq k-1}\frac{\nabla^\alpha f(x)}{\alpha!}(hz)^{\alpha}
+h^k\sum_{|\alpha|=k}\frac{k}{\alpha!}z^\alpha\int_{0}^1(1-t)^{k-1}\nabla^\alpha f(x+thz)dt.
\end{equation*}
The Taylor polynomial of order $k$ of $f$ around the point $x$ is given by
\begin{equation*}
\mathcal{P}^k_{f,x}:\mathbb{R}^n\to \mathbb{R},\quad \mathcal{P}^k_{f,x}(y):=\sum_{|\alpha|\leq k}\frac{\nabla^\alpha f(x)}{\alpha!}(y-x)^{\alpha},
\end{equation*}
and substituting $y=x+hz$ we get
\begin{equation*}
\mathcal{P}^k_{f,x}(x+hz)=\sum_{|\alpha|\leq k}\frac{\nabla^\alpha f(x)}{\alpha!}(hz)^{\alpha}.
\end{equation*}
The remainder of order $k$ of $f$ around $x$ is given by 
\begin{equation*}
\mathcal{R}^k_{f,x}:\Omega\to \mathbb{R},\quad \mathcal{R}^k_{f,x}(y):=f(y)-\mathcal{P}^k_{f,x}(y).
\end{equation*}
Substituting $y=x+hz$ we obtain
\begin{multline}
\label{eq:description for the remainder at x+hz}
\mathcal{R}^k_{f,x}(x+hz)=
h^k\sum_{|\alpha|=k}\frac{k}{\alpha!}z^\alpha\int_{0}^1(1-t)^{k-1}\nabla^\alpha f(x+thz)dt-\sum_{|\alpha|=k}\frac{\nabla^\alpha f(x)}{\alpha!}(hz)^{\alpha}
\\
=h^k\sum_{|\alpha|=k}\frac{k}{\alpha!}z^\alpha\int_{0}^1(1-t)^{k-1}\nabla^\alpha f(x+thz)dt-\sum_{|\alpha|=k}\frac{\nabla^\alpha f(x)}{\alpha!}(hz)^{\alpha}\left(k\int_0^1(1-t)^{k-1}dt\right)
\\
=kh^k\sum_{|\alpha|=k}\frac{z^{\alpha}}{\alpha!}\int_0^1(1-t)^{k-1}\left(\nabla^\alpha f(x+thz)-\nabla^\alpha f(x)\right)dt.
\end{multline}

Now we give definitions of the Taylor polynomial and the remainder for Sobolev functions $f\in W^k_p(\Omega)$ in terms of the precise representative:
\begin{defn}
Let $\Omega\subset\mathbb{R}^n$ be an open set and $k\in \mathbb{N}$. Let 
$f\in W^k_{1,\loc}(\Omega)$ and $x\in \Omega$. We define {\it Taylor polynomial of order $k$ of the function $f$ at the point $x$} to be the following function:
\begin{equation*}
\mathcal{P}^k_{f,x}:\mathbb{R}^n\to \mathbb{R},\quad \mathcal{P}^k_{f,x}(z):=\sum_{|\alpha|\leq k}\frac{(\nabla^\alpha f)^*(x)}{\alpha!}(z-x)^{\alpha}.
\end{equation*}
We define the    
{\it remainder of order $k$ of the function $f$ at the point $x$} to be the following function:
\begin{equation*}
\mathcal{R}^k_{f,x}:\Omega\to \mathbb{R},\quad \mathcal{R}^k_{f,x}(z):=f^*(z)-\mathcal{P}^k_{f,x}(z).
\end{equation*}
\end{defn}

\begin{rem}
\label{re:equivalence for a high order Sobolev norm}
Recall that the Sobolev norm $\|f\|_{W^k_p(U)}$ as defined in 
$\eqref{eq:definition of Sobolev norm W^m_p}$ is equivalent to the norm $\|f\|_{L_p(U)}+\sum_{|\alpha|=k}\|\nabla^\alpha f\|_{L_p(U)}$ for every open and bounded set $U\subset\mathbb{R}^n$ with Lipschitz boundary; means that, there exist constants $c,C$ such that for every 
$f\in W^k_p(U)$
\begin{equation*}
c\|f\|_{W^k_p(U)}\leq \|f\|_{L_p(U)}+\sum_{|\alpha|=k}\|\nabla^\alpha f\|_{L_p(U)}\leq C\|f\|_{W^k_p(U)}.
\end{equation*}
In particular, this equivalence is valid for open balls. 
A proof of this equivalence can be found in \cite{GResh}.
\end{rem}

Let us define for $f\in W^k_p(\Omega)$, $h>0$ and $x\in \Omega$ a function
\begin{equation*}
\label{eq:definition of Rx,h}
R^k_{f,x,h}(z):=\mathcal{R}^k_{f,x}(x+hz).
\end{equation*}
The function $z\longmapsto R^k_{f,x,h}(z)$ is defined on $\frac{\Omega-x}{h}$ and, in particular, the family of functions 
$\{R^k_{f,x,h}\}_{h>0}$ is defined on any bounded set $B\subset\mathbb{R}^n$ for every small enough $h>0$. 

\begin{lem}
\label{lem:convergence of the remainder to zero}
Let $\Omega\subset \mathbb R^n$ be an open set, $1\leq p<\infty$, $k\in \mathbb{N}$ and $f\in W^k_{p,\loc}(\Omega)$. Suppose $x\in \Omega$ is a point such that for every multi-index 
$|\alpha|=k$ 
\begin{equation}
\label{eq:assumption1}
\lim_{r\to 0^+}\fint_{B(x,r)}|\nabla^\alpha f(y)-(\nabla^\alpha f)^*(x)|^pdy=0,
\end{equation}
and for every multi-index 
$|\alpha|\leq k-1$
\begin{equation}
\label{eq:assumption2}
\lim_{\varepsilon\to 0^+}\nabla ^\alpha f_\varepsilon(x)=(\nabla^\alpha f)^*(x),\,\,f_\varepsilon=f*\eta_\varepsilon.
\end{equation}
Then, for every open and bounded set 
$V\subset\mathbb R^n$ 
\begin{equation}
\lim_{h\to 0^+}\Big\|\frac{1}{h^k}R^k_{f,x,h}\Big\|_{W^k_p(V)}=0.
\end{equation}
\end{lem}

\begin{proof}
Using $\eqref{eq:description for the remainder at x+hz}$ for the smooth function $f_\varepsilon$ we get: 
\begin{align*}
\frac{1}{h^k}R^k_{f_\varepsilon,x,h}(z)&=k\sum_{|\alpha|=k}\frac{z^{\alpha}}{\alpha!}\int_0^1(1-t)^{k-1}\left(\nabla^\alpha f_\varepsilon(x+thz)-\nabla^\alpha f_\varepsilon(x)\right)dt.
\end{align*}
Therefore,
\begin{multline}
\label{eq:inequality9}
\left|\frac{1}{h^k}R^k_{f_\varepsilon,x,h}(z)\right|^p
=\left|k\sum_{|\alpha|=k}\frac{z^{\alpha}}{\alpha!}\int_0^1(1-t)^{k-1}\left(\nabla^\alpha f_\varepsilon(x+thz)-\nabla^\alpha f_\varepsilon(x)\right)dt\right|^p
\\
\leq k^p|z|^{pk}\left(\sum_{|\alpha|=k}\frac{1}{\alpha!}\int_0^1(1-t)^{k-1}\left|\nabla^\alpha f_\varepsilon(x+thz)-\nabla^\alpha f_\varepsilon(x)\right|dt\right)^p
\\
\leq k^p|z|^{pk}C(k,p)\sum_{|\alpha|=k}\left(\frac{1}{\alpha!}\right)^p\int_0^1(1-t)^{(k-1)p}\left|\nabla^\alpha f_\varepsilon(x+thz)-\nabla^\alpha f_\varepsilon(x)\right|^pdt
\\
\leq k^p|z|^{pk}C(k,p)\sum_{|\alpha|=k}\left(\frac{1}{\alpha!}\right)^p\int_0^1\left|\nabla^\alpha f_\varepsilon(x+thz)-\nabla^\alpha f_\varepsilon(x)\right|^pdt,
\end{multline}
where $C(k,p)$ is a constant dependent on $k,p$ only.

Let $U\subset\mathbb{R}^n$ be an open ball. Then, by Fubini's theorem, the change of variables formula and 
$\eqref{eq:inequality9}$
\begin{multline}
\label{eq:inequality3}
\int_{U}\left|\frac{1}{h^k}R^k_{f_\varepsilon,x,h}(z)\right|^pdz\leq
\\
k^pC(k,p)\sup_{w\in U}|w|^{pk}\sum_{|\alpha|=k}\left(\frac{1}{\alpha!}\right)^p\int_{0}^1\left(\int_{U}\left|\nabla^\alpha f_\varepsilon(x+thz)-\nabla^\alpha f_\varepsilon(x)\right|^pdz\right)dt=
\\
k^pC(k,p)\sup_{w\in U}|w|^{pk}\sum_{|\alpha|=k}\left(\frac{1}{\alpha!}\right)^p\int_{0}^1\left(\frac{1}{(th)^n}\int_{x+thU}\left|\nabla^\alpha f_\varepsilon(y)-\nabla^\alpha f_\varepsilon(x)\right|^pdy\right)dt.
\end{multline}
Note that for almost every $z\in U$ we have
\begin{align}
\label{eq:equality9}
\lim_{\varepsilon\to 0^+}R^k_{f_\varepsilon,x,h}(z)&=\lim_{\varepsilon\to 0^+}\left(f_\varepsilon(x+hz)-\mathcal{P}^k_{f_\varepsilon,x}(x+hz)\right)
\\
&=f^*(x+hz)-\mathcal{P}^k_{f,x}(x+hz)=R^k_{f,x,h}(z).\nonumber
\end{align}
Indeed, since $f_\varepsilon$ converges to $f$ almost everywhere in $\Omega$ and $f=f^*$ almost everywhere, then 
$\lim_{\varepsilon\to 0^+}f_\varepsilon(x+hz)=f^*(x+hz)$ for almost every $z\in U$; by the assumption $\eqref{eq:assumption1}$, Remark $\ref{rem:every L_p point is a converging point of the mollified family}$ and the identity $\nabla^\alpha f_\varepsilon=(\nabla^\alpha f)*\eta_\varepsilon=(\nabla^\alpha f)_\varepsilon$, we have for every multi-index $|\alpha|=k$
\begin{equation*}
\lim_{\varepsilon\to 0^+}\nabla^\alpha f_\varepsilon(x)=(\nabla^\alpha f)^*(x).
\end{equation*}
Thus, taking into account the assumption 
$\eqref{eq:assumption2}$, we obtain for every $z\in \mathbb{R}^n$
\begin{align*}
\lim_{\varepsilon\to 0^+}\mathcal{P}^k_{f_\varepsilon,x}(x+hz)=
\lim_{\varepsilon\to 0^+}\sum_{|\alpha|\leq k}\frac{\nabla^\alpha f_\varepsilon(x)}{\alpha!}(hz)^{\alpha}
\\
=\sum_{|\alpha|\leq k}\frac{(\nabla^\alpha f)^*(x)}{\alpha!}(hz)^{\alpha}=\mathcal{P}^k_{f,x}(x+hz).
\end{align*} 
 
Thus, by $\eqref{eq:equality9}$ and Fatou's lemma 
\begin{equation}
\label{eq:inequality10}
\int_{U}\left|\frac{1}{h^k}R^k_{f,x,h}(z)\right|^pdz
\leq \liminf_{\varepsilon\to 0^+}\int_{U}\left|\frac{1}{h^k}R^k_{f_\varepsilon,x,h}(z)\right|^pdz.
\end{equation}

For every multi-index $\alpha$ such that $|\alpha|=k$ we get by the dominated convergence theorem, the convergence of $f_\varepsilon$ to $f$ in the topology of $W^k_{p,\loc}$ and the fact that $\nabla^\alpha f_\varepsilon(x)$ converges to 
$(\nabla^\alpha f)^*(x)$ as 
$\varepsilon\to 0^+$
\begin{align}
\label{eq:equality10}
\lim_{\varepsilon\to 0^+}\int_{0}^1\left(\frac{1}{(th)^n}\int_{x+thU}\left|\nabla^\alpha f_\varepsilon(y)-\nabla^\alpha f_\varepsilon(x)\right|^pdy\right)dt
\\
=\int_{0}^1\left(\frac{1}{(th)^n}\int_{x+thU}\left|\nabla^\alpha f(y)-(\nabla^\alpha f)^*(x)\right|^pdy\right)dt.\nonumber
\end{align}

Therefore, by taking the lower limit as 
$\varepsilon\to 0^+$ in the inequality 
$\eqref{eq:inequality3}$ and using 
$\eqref{eq:inequality10},\eqref{eq:equality10}$, we obtain 

\begin{multline}
\label{eq:inequality11}
\int_{U}\left|\frac{1}{h^k}R^k_{f,x,h}(z)\right|^pdz\leq
\\
k^pC(k,p)\sup_{w\in U}|w|^{pk}\sum_{|\alpha|=k}\left(\frac{1}{\alpha!}\right)^p\int_{0}^1\left(\frac{1}{(th)^n}\int_{x+thU}\left|\nabla^\alpha f(y)-(\nabla^\alpha f)^*(x)\right|^pdy\right)dt.
\end{multline}

By dominated convergence theorem, assumption $\eqref{eq:assumption1}$ and $\eqref{eq:inequality11}$, we obtain
\begin{equation}
\label{eq:limit of Rx,h in L_p to zero}
\lim_{h\to 0^+}\int_{U}\left|\frac{1}{h^k}R^k_{f,x,h}(z)\right|^pdz=0.
\end{equation}

Let $\alpha$ be a multi-index such that $|\alpha|=k$. Then, for almost every 
$z\in U$
\begin{equation*}
\nabla^\alpha \left(\frac{1}{h^k}R^k_{f,x,h}\right)(z)=\nabla^\alpha f(x+hz)-(\nabla^\alpha f)^*(x).
\end{equation*}
Thus, by the last equation and the change of variable formula we get
\begin{multline*}
\int_{U}\left|\nabla^\alpha \left(\frac{1}{h^k}R^k_{f,x,h}\right)(z)\right|^pdz=\int_{U}\left|\nabla^\alpha f(x+hz)-(\nabla^\alpha f)^*(x)\right|^pdz
\\
=\frac{1}{h^n}\int_{x+hU}\left|\nabla^\alpha f(y)-(\nabla^\alpha f)^*(x)\right|^pdy.
\end{multline*}
Taking the limit as $h\to 0^+$ on both sides of the last equation and using assumption $\eqref{eq:assumption1}$ we get
\begin{equation}
\label{eq:limit of the alpha order derivative of Rx,h in L_p to zero}
\lim_{h\to 0^+}\int_{U}\left|\nabla^\alpha \left(\frac{1}{h^k}R^k_{f,x,h}\right)(z)\right|^pdz=0.
\end{equation}
Now, let $V\subset\mathbb{R}^n$ be any open and bounded set. Let $U$ be an open ball such that $V\subset U$. Using Remark $\ref{re:equivalence for a high order Sobolev norm}$, there exists a constant $C$ such that
\begin{align*}
\Big\|\frac{1}{h^k}R^k_{f,x,h}\Big\|_{W^k_p(V)}\leq \Big\|\frac{1}{h^k}R^k_{f,x,h}\Big\|_{W^k_p(U)}
\\
\leq C\left(\left\|\frac{1}{h^k}R^k_{f,x,h}\right\|_{L_p(U)}+\sum_{|\alpha|=k}\left\|\nabla^\alpha \left(\frac{1}{h^k}R^k_{f,x,h}\right)\right\|_{L_p(U)}\right).
\end{align*}
Taking the limit as $h\to 0^+$ in the last inequality and using $\eqref{eq:limit of Rx,h in L_p to zero},\eqref{eq:limit of the alpha order derivative of Rx,h in L_p to zero}$, we obtain
\begin{equation*}
\lim_{h\to 0^+}\Big\|\frac{1}{h^k}R^k_{f,x,h}\Big\|_{W^k_p(V)}=0.
\end{equation*}
\end{proof}

\begin{thm}
\label{thm:convergence of the remainder to zero}
Let $\Omega\subset \mathbb R^n$ be an open set, $1\leq p<\infty$, $k\in \mathbb{N}$ and $f\in RW^k_{p,\loc}(\Omega)$. 
Then, for every open and bounded set 
$V\subset\mathbb R^n$ 
\begin{equation*}
\lim_{h\to 0^+}\Big\|\frac{1}{h^k}R^k_{f,x,h}\Big\|_{W^k_p(V)}=0\quad \text{for}\quad \cp_p-a.e. \quad x\in \Omega.
\end{equation*}
\end{thm}

\begin{proof}
Since $f\in RW^k_{p,\loc}(\Omega)$ and Remark $\ref{rem:L_p points for all derivaties for function RW}$, there exists $E\subset \Omega$ such that $\cp_p(E)=0$ and for every $x\in \Omega\setminus E$ and multi-index 
$|\alpha|\leq k$ we get
\begin{equation*}
\lim_{r\to 0^+}\fint_{B(x,r)}|\nabla^\alpha f(y)-(\nabla^\alpha f)^*(x)|^pdy=0.
\end{equation*}
By remark $\ref{rem:every L_p point is a converging point of the mollified family}$ and the fact that 
$\nabla ^\alpha f_\varepsilon=(\nabla ^\alpha f)*\eta_\varepsilon=(\nabla ^\alpha f)_\varepsilon$
we also know that for every $x\in \Omega\setminus E$ and every multi-index 
$|\alpha|\leq k$ 
\begin{equation*}
\lim_{\varepsilon\to 0^+}\nabla ^\alpha f_\varepsilon(x)=(\nabla^\alpha f)^*(x).
\end{equation*}
The result now follows from the previous lemma for each $x\in \Omega\setminus E$. 
\end{proof}

\vskip 0.3cm
