The authors prove a generalization of a theorem by J. H. Michael and W. P. Ziemer [Contemp. Math. 42, 135–167 (1985; Zbl 0592.41031)] which says that any function from the Sobolev space \(W^{k,p}(\mathbb R^n)\) is, up to a set of small Bessel \(B_{k-m,p}\)-capacity, of class \(C^m\), \(k<m\). Moreover, the function thus redefined is close to the original one in \(W^{m,p}\)-norm. The main result of the paper under review is as follows: Let \(\Omega\subset\mathbb R^n\) be an open set, \(1<p<\infty\) and \(m=0,1,2,\ldots k-1\). Then for any \(u\in W^{k,p}_{loc}(\Omega)\) and \(\varepsilon>0\) there is a closed \(F\subset\Omega\) and a function \(w\in C^{m}(\Omega)\cap W^{m+1,p}_{\text{loc}}(\Omega)\) such that: (i) the Bessel capacity \(B_{k-m-\lambda, p}(\Omega\setminus F)<\varepsilon\), (ii) \(D^\alpha u = D^\alpha w\) on the set \(F\) and all multiindices \(\alpha \leq m\), (iii) \(u-w\in W_0^{m+1,p}(\Omega)\), and (iv) \(u-w_{W^{m+1,p}(\Omega)} \leq Cu_{W^{m+1,p} (\Omega \setminus F)}\) with a constant \(C\) only depending on \(n,k, \lambda\) and \(p\).
A version of this theorem using \(C^{m, \lambda}\)-redefinitions is also proved.
This result is remarkable as one gets approximations in higher-order Sobolev spaces without changing the capacitary estimate. The proof of this result does not use the classical Whitney extension theorem (which had been used in all previous versions of theorems of this type) but a result which the authors call {Whitney smoothing} (the point here is that with the Whitney extension theorem applied to a Sobolev function \(uF\) restricted to \(F\), one loses all information outside \(F\), while the new construction approximates – “smoothes out” – \(uF^c\)). The authors conjecture that it is even impossible to obtain the above result with the classical Whitney theorem. Moreover, the proof draws heavily upon the pointwise estimates for (the canonical Borel representatives of) Sobolev functions which were pioneered by B. Bojarski and P. Hajłasz [Studia Math. 106, 137–152 (1993; Zbl 0810.46030)].