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Subdifferentiable functions satisfy Lusin properties of class \(C^1\) or \(C^2\). (English) Zbl 1392.58005

Summary: Let \(f : \mathbb{R}^n \rightarrow \mathbb{R}\) be a function. Assume that for a measurable set \(\varOmega\) and almost every \(x \in \varOmega\) there exists a vector \(\xi_x \in \mathbb{R}^n\) such that \[ \mathop{\liminf}\limits_{h \rightarrow 0} \frac{f(x + h) - f(x) - \langle \xi_x, h \rangle}{| h |^2} > - \infty. \] Then we show that \(f\) satisfies a Lusin-type property of order 2 in \(\varOmega\), that is to say, for every \(\varepsilon > 0\) there exists a function \(g \in C^2(\mathbb{R}^n)\) such that \(\mathcal{L}^n(\{x \in \varOmega : f(x) \neq g(x) \}) \leq \varepsilon\). In particular every function which has a nonempty proximal subdifferential almost everywhere also has the Lusin property of class \(C^2\). We also obtain a similar result (replacing \(C^2\) with \(C^1\)) for the Fréchet subdifferential. Finally we provide some examples showing that these kinds of results are no longer true for Taylor subexpansions of higher order.

MSC:

58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
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[1] Alberti, G., A Lusin type theorem for gradients, J. Funct. Anal., 100, 1, 110-118 (1991) · Zbl 0752.46025
[2] Alberti, G., On the structure of singular sets of convex functions, Calc. Var. Partial Differential Equations, 2, 1, 17-27 (1994) · Zbl 0790.26010
[3] Alexandroff, A. D., Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, (Leningrad State Univ. Annals [Uchenye Zapiski] Math. Ser., Vol. 6 (1939)), 3-35, (in Russian)
[4] D. Azagra, P. Hajłasz, Lusin-type properties of convex functions, preprint, 2017.; D. Azagra, P. Hajłasz, Lusin-type properties of convex functions, preprint, 2017.
[5] Bagby, T.; Ziemer, W. P., Pointwise differentiability and absolute continuity, Trans. Amer. Math. Soc., 191, 129-148 (1974) · Zbl 0295.26013
[6] Bojarski, B.; Hajłasz, P., Pointwise inequalities for Sobolev functions, Studia Math., 106, 77-92 (1993) · Zbl 0810.46030
[7] Bojarski, B.; Hajłasz, P.; Strzelecki, P., Improved \(C^{k, \lambda}\) approximation of higher order Sobolev functions in norm and capacity, Indiana Univ. Math. J., 51, 507-540 (2002) · Zbl 1036.46023
[8] Bourgain, J.; Korobkov, M. V.; Kristensen, J., On the Morse-Sard property and level sets of \(W^{n, 1}\) Sobolev functions on \(R^n\), J. Reine Angew. Math., 700, 93-112 (2015) · Zbl 1322.46022
[9] Brown, J. B.; Kozlowski, G., Smooth interpolation, Hlder continuity, and the Takagivan der Waerden function, Amer. Math. Monthly, 110, 2, 142-147 (2003) · Zbl 1050.26004
[10] Busemann, H.; Feller, W., Krmmungseigenschaften Konvexer Flächen. (German), Acta Math., 66, 1, 1-47 (1936) · JFM 62.0832.02
[11] Calderón, A. P.; Zygmund, A., Local properties of solutions of elliptic partial differential equations, Studia Math., 20, 171-225 (1961) · Zbl 0099.30103
[12] Clarke, F. H.; Ledyaev, Yu. S.; Stern, R. J.; Wolenski, P. R., (Nonsmooth Analysis and Control Theory. Nonsmooth Analysis and Control Theory, Grad. Texts in Math, vol. 178 (1998), Springer) · Zbl 1047.49500
[13] Crandall, M. G.; Ishii, H.; Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27, 1-67 (1992) · Zbl 0755.35015
[14] Evans, L. C.; Gariepy, R. F., (Measure Theory and Fine Properties of Functions. Measure Theory and Fine Properties of Functions, Textbooks in Mathematics (2015), CRC Press: CRC Press Boca Raton, FL) · Zbl 1310.28001
[15] Federer, H., Surface area. II, Trans. Amer. Math. Soc., 55, 438-456 (1944) · Zbl 0060.14003
[16] Ferrera, J., An Introduction to Nonsmooth Analysis (2014), Elsevier/Academic Press: Elsevier/Academic Press Amsterdam · Zbl 1290.49001
[17] Ferrera, J.; Gomez-Gil, J., Generalized Takagi-Van der Waerden functions and their subdifferentials, J. Convex Anal., 25, 4 (2018), (in press) · Zbl 1406.26001
[18] Gora, P.; Stern, R. J., Subdifferential analysis of the Van der Waerden function, J. Convex Anal., 18, 3, 699-705 (2011) · Zbl 1225.26004
[19] Hardy, G. H., Weierstrass’s non-differentiable function, Trans. Amer. Math. Soc., 17, 3, 301-325 (1916) · JFM 46.0401.03
[20] Hirsch, M. W., (Differential Topology. Differential Topology, Graduate Texts in Mathematics, vol. 33 (1976), Springer-Verlag: Springer-Verlag New York) · Zbl 0356.57001
[21] Imomkulov, S. A., Twice differentiability of subharmonic functions, Izv. Ross. Akad. Nauk Ser. Mat., 56, 877-888 (1992), translation in Russian Acad. Sci. Izv. Math. 41 (1993) 157-167, (in Russian) · Zbl 0802.31003
[22] Isakov, N. M., A global property of approximately differentiable functions, Math. Notes Acad. Sci. USSR, 41, 280-285 (1987) · Zbl 0629.26007
[23] Kharazishvili, A. B., Strange functions in real analysis, (Pure and Applied Mathematics (Boca Raton), Vol. 272 (2006), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL) · Zbl 1097.26006
[24] Kocan, M.; Wang, X.-J., On the generalized Stepanov theorem, Proc. Amer. Math. Soc., 125, 2347-2352 (1997) · Zbl 0910.26006
[25] Liu, Fon-Che, A Luzin type property of Sobolev functions, Indiana Univ. Math. J., 26, 645-651 (1977) · Zbl 0368.46036
[26] Liu, F.-C.; Tai, W.-S., Approximate Taylor polynomials and differentiation of functions, Topol. Methods Nonlinear Anal., 3, 1, 189-196 (1994) · Zbl 0811.26005
[27] Lusin, N., Sur les propits des fonctions measurables, C. R. Acad. Sci., Paris, 154, 1688-1690 (1912) · JFM 43.0484.04
[28] Michael, J.; Ziemer, W. P., A Lusin type approximation of Sobolev functions by smooth functions, Contemp. Math., 42, 135-167 (1985) · Zbl 0592.41031
[29] Stein, E. M., (Singular Integrals and Differentiability Properties of Functions. Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30 (1970), Princeton University Press: Princeton University Press Princeton N.J.) · Zbl 0207.13501
[30] Takagi, T., A simple example of the continuous function without derivative, Proc. Phys. Math. Soc. Tokio Ser. II, 1, 176-177 (1903) · JFM 34.0410.05
[31] Whitney, H., Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36, 63-89 (1934) · JFM 60.0217.01
[32] Whitney, H., On totally differentiable and smooth functions, Pacific J. Math., 1, 143-159 (1951) · Zbl 0043.05803
[33] Ziemer, W. P., Weakly Differentiable Functions (1989), Springer-Verlag · Zbl 0692.46022
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