A pointwise convergence for Sobolev space functions. (Q1428245)

The author proves an analogue of Lebesgue's differentiation theorem for smooth functions: Theorem. Let \(1<p<\infty\), let \(k\) and \(m\) be positive integers such that \(0\leq(k-2m)p\leq n\), let \(\Omega\) be an open set in \(\mathbb R^n\). There exist numbers \(c_j>0\) such that for every function \(f\) from the Sobolev space \(W^{k,p}(\Omega)\) \[ \lim_{r\to0}r^{-2m}| B(x,r)| ^{-1}\int_{B(x,r)}\biggl[f(y)-\sum_{j=0}^{m-1} c_jr^{2j}\Delta^jf(x)\biggr]dy=c_m\Delta^mf(x) \] \(B_{k-2m,p}\)-quasi everywhere in \(\Omega\), where \(B_{k-2m,p}\) denotes the Bessel capacity. An analogous result holds when the balls \(B(x,r)\) are replaced with the spheres \(S(x,r)\).