Local maximal operators on fractional Sobolev spaces (Q329680)

The Hardy-Littlewood maximal function NEWLINE\[NEWLINEM_\Omega f(x)= \sup_{n>0}\;\not\mkern-7mu\int_{B(x,n)} |f(y)|\,dy,\quad 0<n< \text{dist}(x,\partial\Omega),NEWLINE\]NEWLINE is studied in the fractional Sobolev space \(W^{2,p}(\Omega)\), where \(0<s<1\), \(1>p>\infty\). The main result is that for an arbitrary domain \(\Omega\) in \(\mathbb{R}^n\) we have NEWLINE\[NEWLINE\int_\Omega \int_\Omega{|M_\Omega f(x)-M_\Omega f(y)|^p\over |x-y|^{n+sp}}\,dx\,dy\leq C \int_\Omega \int_\Omega {|f(x)-f(y)|^p\over |x-y|^{n+sp}}\,dx\,dy,NEWLINE\]NEWLINE where \(C= (C(n,s,p)\). A simple, but ingenious, pointwise inequality is the main ingradient of the proof.NEWLINENEWLINE As an application, the fractional Hardy inequality is studied. A necessary and sufficient condition for its validity is given in terms of the so-called \((s,p)\)-capacity.