A Lusin type result

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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 and 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.
[2]	B. Bojarski, Sobolev spaces and averaging I, Proc. A. Razmadze Math. Inst., 164 (2014), 19-44.
[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.
[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.
[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.
[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.
[7]	H. Federer, Geometric Measure Theory, Springer-Verlag, 1969.
[8]	R. F. Gariepy and W. P. Ziemer, Modern Real Analysis, PWS Publishing Company, 1995.
[9]	R. Shakarchi and E. M. Stein, Real Analysis (measure theory, integration and Hilbert spaces), Princeton University Press, Princeton and Oxford, 2005.
[10]	W. P. Ziemer, Weakly Differentiable Functions, GTM 120, Springer-Verlag, 1989. doi: 10.1007/978-1-4612-1015-3.

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.
[2]	B. Bojarski, Sobolev spaces and averaging I, Proc. A. Razmadze Math. Inst., 164 (2014), 19-44.
[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.
[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.
[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.
[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.
[7]	H. Federer, Geometric Measure Theory, Springer-Verlag, 1969.
[8]	R. F. Gariepy and W. P. Ziemer, Modern Real Analysis, PWS Publishing Company, 1995.
[9]	R. Shakarchi and E. M. Stein, Real Analysis (measure theory, integration and Hilbert spaces), Princeton University Press, Princeton and Oxford, 2005.
[10]	W. P. Ziemer, Weakly Differentiable Functions, GTM 120, Springer-Verlag, 1989. doi: 10.1007/978-1-4612-1015-3.

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