Averaging operator in \(L_ p\) spaces for \(0<p<1\) (Q1110770)

The main aim of this paper is to construct some operator \(B_{\delta}\) such that \(B_{\delta}f\in C^{\infty}(R^ n)\), \(\| B_{\delta}f\|_{L_ p(R^ n)}\leq 2^{1/p-1}\| f\|_{L_ p(E)}\), \(\lim_{\delta \to 0+}\| B_{\delta}f-f\|_{L_ p(E)}=0\) for \(f\in L_ p(E)\), where \(0<p<1\) and E is an arbitrary measurable subset of \(R^ n\). This very important theorem has an interesting proof and moreover this result is connected with a similar theorem, but for the Sobolev operator.