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 & 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. 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.  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. 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.  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

[1]

Thomas Bartsch, Anna Maria Micheletti, Angela Pistoia. The Morse property for functions of Kirchhoff-Routh path type. Discrete & 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 & Continuous Dynamical Systems - A, 2000, 6 (3) : 683-690. doi: 10.3934/dcds.2000.6.683

[3]

Ian D. Morris. Ergodic optimization for generic continuous functions. Discrete & Continuous Dynamical Systems - A, 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 & 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 & 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 & Continuous Dynamical Systems - S, 2010, 3 (4) : 623-641. doi: 10.3934/dcdss.2010.3.623

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

Katsukuni Nakagawa. Compactness of transfer operators and spectral representation of Ruelle zeta functions for super-continuous functions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6331-6350. doi: 10.3934/dcds.2020282

[19]

Yunho Kim, Luminita A. Vese. Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability. Inverse Problems & Imaging, 2009, 3 (1) : 43-68. doi: 10.3934/ipi.2009.3.43

[20]

Davide Addona, Giorgio Menegatti, Michele Miranda jr.. $ BV $ functions on open domains: the Wiener case and a Fomin differentiable case. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2679-2711. doi: 10.3934/cpaa.2020117

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (76)
  • HTML views (91)
  • Cited by (0)

Other articles
by authors

[Back to Top]