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June  2020, 19(6): 3083-3091. doi: 10.3934/cpaa.2020133

A Lusin type result

Silvano Delladio

Department of Mathematics, University of Trento, via Sommarive 14, 38123 Trento, Italy

Received  May 2019 Revised  January 2020 Published  March 2020

By using the property known as Federer-Fleming conjecture (cf. [7, 3.1.17]), recently resolved by B. Bojarski, we prove the following Lusin type result:
Theorem. Let
$ A\subset {\mathbb{R}}^n $
 be a measurable set and let
$ k $
 be a nonnegative integer. Assume that to each
$ x\in A $
 corresponds a polynomial
$ P_x: {\mathbb{R}}^n\to {\mathbb{R}} $
 of degree less or equal to
$ k+1 $
 such that
$ \begin{equation*} {\rm{ap}}\lim\limits_{x\to a}\frac{(D^\alpha P_x)(x)-(D^\alpha P_a)(x)}{\vert x-a\vert} = 0 \end{equation*} $
holds for all
$ \alpha\in {\mathbb{N}}^n $
 such that
$ \vert\alpha\vert\leq k $
, at a.e.
$ a\in A $
. Then, for each
$ \varepsilon >0 $
, there exists
$ \varphi\in C^{k+1}( {\mathbb{R}}^n) $
 such that
$ {\mathcal L}^n \bigg(A\setminus \bigcap\limits_{\vert\alpha\vert\leq k+1}\{x\in A : D^\alpha\varphi (x) = (D^\alpha P_x)(x)\}\bigg)\leq\varepsilon. $
We will use such a theorem to provide a simple new proof of a well-known property of Sobolev functions.
Keywords: Lusin type property, approximate Taylor polynomial, approximately continuous functions, approximately differentiable functions, Sobolev functions, points of Lebesgue density.
Mathematics Subject Classification: Primary: 26B05, 54C08, 58B10; Secondary: 46E35.
Citation: Silvano Delladio. A Lusin type result. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3083-3091. doi: 10.3934/cpaa.2020133
References:
[1]

B. Bojarski, Differentiation of Measurable Functions and Whitney-Luzin Type Structure Theorems, Helsinki University of Technology Institute of Mathematics Research Reports, Vol. A572, 2009. Available from: http://math.tkk.fi/reports/a572.pdf. Google Scholar

[2]

B. Bojarski, Sobolev spaces and averaging I, Proc. A. Razmadze Math. Inst., 164 (2014), 19-44.   Google Scholar

[3]

B. Bojarski, P. Hajlasz and P. Strzelecki, Improved $C^{k, \lambda}$ approximation of higher order Sobolev functions in norm and capacity, Indiana Univ. Math. J., 51 (2002), 507-540.  doi: 10.1512/iumj.2002.51.2162.  Google Scholar

[4]

S. Delladio, A note on some topological properties of sets with finite perimeter, Glasg. Math. J., 58 (2016), 637-647.  doi: 10.1017/S0017089515000385.  Google Scholar

[5]

S. Delladio, The tangency of a $C^1$ smooth submanifold with respect to a non-involutive $C^1$ distribution has no superdensity points, Indiana Univ. Math. J., 68 (2019), 393-412.  doi: 10.1512/iumj.2019.68.7549.  Google Scholar

[6]

L. C. Evans and R. F. Gariepy, Lecture Notes on Measure Theory and Fine Properties of Functions, Studies in Advanced Math., CRC Press, 1992.  Google Scholar

[7]

H. Federer, Geometric Measure Theory, Springer-Verlag, 1969.  Google Scholar

[8]

R. F. Gariepy and W. P. Ziemer, Modern Real Analysis, PWS Publishing Company, 1995.  Google Scholar

[9]

R. Shakarchi and E. M. Stein, Real Analysis (measure theory, integration and Hilbert spaces), Princeton University Press, Princeton and Oxford, 2005.  Google Scholar

[10]

W. P. Ziemer, Weakly Differentiable Functions, GTM 120, Springer-Verlag, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

show all references

References:
[1]

B. Bojarski, Differentiation of Measurable Functions and Whitney-Luzin Type Structure Theorems, Helsinki University of Technology Institute of Mathematics Research Reports, Vol. A572, 2009. Available from: http://math.tkk.fi/reports/a572.pdf. Google Scholar

[2]

B. Bojarski, Sobolev spaces and averaging I, Proc. A. Razmadze Math. Inst., 164 (2014), 19-44.   Google Scholar

[3]

B. Bojarski, P. Hajlasz and P. Strzelecki, Improved $C^{k, \lambda}$ approximation of higher order Sobolev functions in norm and capacity, Indiana Univ. Math. J., 51 (2002), 507-540.  doi: 10.1512/iumj.2002.51.2162.  Google Scholar

[4]

S. Delladio, A note on some topological properties of sets with finite perimeter, Glasg. Math. J., 58 (2016), 637-647.  doi: 10.1017/S0017089515000385.  Google Scholar

[5]

S. Delladio, The tangency of a $C^1$ smooth submanifold with respect to a non-involutive $C^1$ distribution has no superdensity points, Indiana Univ. Math. J., 68 (2019), 393-412.  doi: 10.1512/iumj.2019.68.7549.  Google Scholar

[6]

L. C. Evans and R. F. Gariepy, Lecture Notes on Measure Theory and Fine Properties of Functions, Studies in Advanced Math., CRC Press, 1992.  Google Scholar

[7]

H. Federer, Geometric Measure Theory, Springer-Verlag, 1969.  Google Scholar

[8]

R. F. Gariepy and W. P. Ziemer, Modern Real Analysis, PWS Publishing Company, 1995.  Google Scholar

[9]

R. Shakarchi and E. M. Stein, Real Analysis (measure theory, integration and Hilbert spaces), Princeton University Press, Princeton and Oxford, 2005.  Google Scholar

[10]

W. P. Ziemer, Weakly Differentiable Functions, GTM 120, Springer-Verlag, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

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