Leoni and Spector’s Characterization of Sobolev and BV spaces

Leoni and Spector recently proved the following result, which is an interesting result:

Let $\Omega \subset R^{N}$ be open, let $\rho_{\epsilon}$ satisfy
\[
\rho_{\epsilon}\geq 0,\quad \int_{R^{N}}\rho_{\epsilon}(x)dx=1
\]
and
\[
\lim \limits_{\epsilon \rightarrow 0}\int_{|x|>\delta}\rho_{\epsilon}(x)dx=0,\quad \text{for all } \delta>0.
\]
What’s more, let
\[
C_{\delta}(v)=\bigg\{ w \in R^{N}\backslash{0}:\frac{v}{|v|}\frac{w}{|w|}>1-\delta\bigg\},
\]
then we require that $\rho_\eps$ satisfy
\[
\lim \inf \limits_{\epsilon \rightarrow 0}\int_{C_{\delta(v_{i})}}\rho_{\epsilon}(x)dx>0 \text{ for all } i=1,…,N.
\]
Let $1< p< \infty$, and $1\leq q<\infty$, with $1\leq q\leq \frac{N}{N-p}$ if $p< N$, and let f $\in L_{loc}^{1}(\Omega)$. Assume \[ \lim \limits_{\lambda \rightarrow 0}\lim \sup \limits_{\epsilon \rightarrow 0}\int_{\Omega_{\lambda}}\biggl(\int_{\Omega_{\lambda}}\biggl(\frac{|f(x)-f(y)|^{p}}{|x-y|^{p}}\biggr)^{q}\rho_{\epsilon}(x-y)dy \biggr)^{\frac{1}{q}}dx<\infty \] Then $f \in W_{loc}^{1,p}(\Omega)$ and $\nabla f \in L^{p}(\Omega ;R^{N})$. Moreover, there exist $\epsilon_{j} \rightarrow 0^{+}$ and a probability measure $\mu \in$ M($S^{N-1}$) such that for all $0<\eta <\frac{\lambda}{3}$, \begin{align*} &\lim \limits_{\lambda \rightarrow 0}\lim \sup \limits_{\epsilon \rightarrow 0}\int_{\Omega_{\lambda}^{\eta}}\biggl(\int_{\Omega_{\lambda}^{2\eta}}\biggl(\frac{|f(x)-f(y)|^{p}}{|x-y|^{p}}\biggr)^{q}\rho_{\epsilon}(x-y)dy \biggr)^{\frac{1}{q}}dx\\ &\qquad\qquad\geq \int_{\Omega}\biggl(\int_{S_{N-1}}(|\nabla f(x)\cdot \sigma|^{p})^{q}d\mu(\sigma) \biggr)^{\frac{1}{q}}dx. \end{align*}

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