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

A Lusin type result

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.
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. BojarskiP. 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. BojarskiP. 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.

[1]

Thomas Bartsch, Anna Maria Micheletti, Angela Pistoia. The Morse property for functions of Kirchhoff-Routh path type. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 1867-1877. doi: 10.3934/dcdss.2019123

[2]

Bernhard Kawohl. Symmetry results for functions yielding best constants in Sobolev-type inequalities. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 683-690. doi: 10.3934/dcds.2000.6.683

[3]

Ian D. Morris. Ergodic optimization for generic continuous functions. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 383-388. doi: 10.3934/dcds.2010.27.383

[4]

Shingo Takeuchi. The basis property of generalized Jacobian elliptic functions. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2675-2692. doi: 10.3934/cpaa.2014.13.2675

[5]

Noah H. Rhee, PaweŁ Góra, Majid Bani-Yaghoub. Predicting and estimating probability density functions of chaotic systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 297-319. doi: 10.3934/dcdsb.2017144

[6]

Xuemei Li, Rafael de la Llave. Convergence of differentiable functions on closed sets and remarks on the proofs of the "Converse Approximation Lemmas''. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 623-641. doi: 10.3934/dcdss.2010.3.623

[7]

Najeeb Abdulaleem. Optimality and duality for $ E $-differentiable multiobjective programming problems involving $ E $-type Ⅰ functions. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022004

[8]

P. Candito, S. A. Marano, D. Motreanu. Critical points for a class of nondifferentiable functions and applications. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 175-194. doi: 10.3934/dcds.2005.13.175

[9]

Piotr Fijałkowski. A global inversion theorem for functions with singular points. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 173-180. doi: 10.3934/dcdsb.2018011

[10]

Qian Liu, Xinmin Yang, Heung Wing Joseph Lee. On saddle points of a class of augmented lagrangian functions. Journal of Industrial and Management Optimization, 2007, 3 (4) : 693-700. doi: 10.3934/jimo.2007.3.693

[11]

Marko Kostić. Almost periodic type functions and densities. Evolution Equations and Control Theory, 2022, 11 (2) : 457-486. doi: 10.3934/eect.2021008

[12]

Moritz Egert, Patrick Tolksdorf. Characterizations of Sobolev functions that vanish on a part of the boundary. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 729-743. doi: 10.3934/dcdss.2017037

[13]

Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43

[14]

Krzysztof Frączek, M. Lemańczyk, E. Lesigne. Mild mixing property for special flows under piecewise constant functions. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 691-710. doi: 10.3934/dcds.2007.19.691

[15]

Leon Ehrenpreis. Special functions. Inverse Problems and Imaging, 2010, 4 (4) : 639-647. doi: 10.3934/ipi.2010.4.639

[16]

Yulin Song. Density functions of distribution dependent SDEs driven by Lévy noises. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2399-2419. doi: 10.3934/cpaa.2021087

[17]

Regina Sandra Burachik, Alex Rubinov. On the absence of duality gap for Lagrange-type functions. Journal of Industrial and Management Optimization, 2005, 1 (1) : 33-38. doi: 10.3934/jimo.2005.1.33

[18]

Xiwang Cao, Hao Chen, Sihem Mesnager. Further results on semi-bent functions in polynomial form. Advances in Mathematics of Communications, 2016, 10 (4) : 725-741. doi: 10.3934/amc.2016037

[19]

Kohei Ueno. Weighted Green functions of nondegenerate polynomial skew products on $\mathbb{C}^2$. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 985-996. doi: 10.3934/dcds.2011.31.985

[20]

Kohei Ueno. Weighted Green functions of polynomial skew products on $\mathbb{C}^2$. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2283-2305. doi: 10.3934/dcds.2014.34.2283

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (206)
  • HTML views (101)
  • Cited by (0)

Other articles
by authors

[Back to Top]