A Lusin type approximation of Sobolev functions by smooth functions.

*(English)*Zbl 0592.41031
Classical real analysis, Proc. Spec. Sess., 794th Meet. AMS, Madison/Wis. 1982, Contemp. Math. 42, 135-167 (1985).

[For the entire collection see Zbl 0565.00007.]

This paper is concerned with the approximation of a Sobolev function f by a smooth function g in such a way that \(| f-g|\) is small and the set where f and g disagree has small capacity. The following is the main theorem. Let \(1\leq p<\infty\), let \(\ell,m\) be positive integers, with \(1\leq m\leq \ell\) and \((\ell -m)p<n\) and let \(\Omega\) be a non-empty open set of \(R^ n\). Let \(f\in W^{\ell,p}(\Omega)\) and be approximately continuous at each point of \(\Omega\) except for a set E with Riesz capacity \(R_{\ell -m,p}(E)=0\). Let \(\epsilon >0\). Then there exists a \(C^ m\) function g on \(\Omega\), such that (a) the set \(F=\{x;x\in \Omega\) and f(x)\(\neq g(x)\}\) has \(R_{\ell -m,p}(F)<\epsilon\) and (b) \(| f-g|_{m,h}<\epsilon.\)

In another theorem it is shown that each \(f\in W^{\ell,p}(\Omega)\) can be represented by a function which is approximately continuous except for a set E, with \(R_{\ell -m,p}(E)=0\). Since \(R_{0,p}\) is equivalent to Lebesgue measure, the above theorem generalises a result of Fon-Che Liu [Indiana Univ. Math. J. 26, 645-651 (1977; Zbl 0368.46036)].

This paper is concerned with the approximation of a Sobolev function f by a smooth function g in such a way that \(| f-g|\) is small and the set where f and g disagree has small capacity. The following is the main theorem. Let \(1\leq p<\infty\), let \(\ell,m\) be positive integers, with \(1\leq m\leq \ell\) and \((\ell -m)p<n\) and let \(\Omega\) be a non-empty open set of \(R^ n\). Let \(f\in W^{\ell,p}(\Omega)\) and be approximately continuous at each point of \(\Omega\) except for a set E with Riesz capacity \(R_{\ell -m,p}(E)=0\). Let \(\epsilon >0\). Then there exists a \(C^ m\) function g on \(\Omega\), such that (a) the set \(F=\{x;x\in \Omega\) and f(x)\(\neq g(x)\}\) has \(R_{\ell -m,p}(F)<\epsilon\) and (b) \(| f-g|_{m,h}<\epsilon.\)

In another theorem it is shown that each \(f\in W^{\ell,p}(\Omega)\) can be represented by a function which is approximately continuous except for a set E, with \(R_{\ell -m,p}(E)=0\). Since \(R_{0,p}\) is equivalent to Lebesgue measure, the above theorem generalises a result of Fon-Che Liu [Indiana Univ. Math. J. 26, 645-651 (1977; Zbl 0368.46036)].

##### MSC:

41A30 | Approximation by other special function classes |