Leoni and Spector recently proved the following result, which is an interesting result:<\/p>\n
Let $\\Omega \\subset R^{N}$ be open, let $\\rho_{\\epsilon}$ satisfy
\n\\[
\n\\rho_{\\epsilon}\\geq 0,\\quad \\int_{R^{N}}\\rho_{\\epsilon}(x)dx=1
\n\\]
\nand
\n\\[
\n\\lim \\limits_{\\epsilon \\rightarrow 0}\\int_{|x|>\\delta}\\rho_{\\epsilon}(x)dx=0,\\quad \\text{for all } \\delta>0.
\n\\]
\nWhat’s more, let
\n\\[
\nC_{\\delta}(v)=\\bigg\\{ w \\in R^{N}\\backslash{0}:\\frac{v}{|v|}\\frac{w}{|w|}>1-\\delta\\bigg\\},
\n\\]
\nthen we require that $\\rho_\\eps$ satisfy
\n\\[
\n\\lim \\inf \\limits_{\\epsilon \\rightarrow 0}\\int_{C_{\\delta(v_{i})}}\\rho_{\\epsilon}(x)dx>0 \\text{ for all } i=1,…,N.
\n\\]
\nLet $1< p< \\infty$, 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\n\\[\n\\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\n\\]\nThen $f \\in W_{loc}^{1,p}(\\Omega)$ 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 $0<\\eta <\\frac{\\lambda}{3}$,\n\\begin{align*}\n&\\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\\\\\n&\\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.\n\\end{align*}\n<\/p>\n","protected":false},"excerpt":{"rendered":"
Leoni and Spector recently proved the fo …<\/p>\n