Fine properties of functions from Hajłasz-Sobolev classes \(M_\alpha^p\), \(p > 0\). II: Lusin’s approximation. (English. Russian original) Zbl 1458.46026
J. Contemp. Math. Anal., Armen. Acad. Sci. 52, No. 1, 30-37 (2017); translation from Izv. Nats. Akad. Nauk Armen., Mat. 52, No. 1, 26-37 (2017).
Summary: The present paper is devoted to the Lusin’s approximation of functions from Hajłasz-Sobolev classes \(M_\alpha^p(X)\) for \(p > 0\). It is proved that for any \(f\in M_\alpha^p(X)\) and any \(\varepsilon > 0\) there exist an open set \(O_\varepsilon\subset X\) with measure less than \(\varepsilon\) (as a measure can be taken the corresponding capacity or Hausdorff content) and an approximating function \(f_\varepsilon\) such that \(f = f_\varepsilon\) on \(X\setminus O_\varepsilon\). Moreover, the correcting function \(f_\varepsilon\) is regular (that is, it belongs to the underlying space \(M_\alpha^p(X)\) and it is a locally Hölder function), and it approximates the original function in the metric of the space \(M_\alpha^p(X)\).
For Part I see [the authors, ibid. 51, No. 6, 282–295 (2016; Zbl 1458.46025)].
For Part I see [the authors, ibid. 51, No. 6, 282–295 (2016; Zbl 1458.46025)].
MSC:
| 46E36 | Sobolev (and similar kinds of) spaces of functions on metric spaces; analysis on metric spaces |
| 26B35 | Special properties of functions of several variables, Hölder conditions, etc. |
| 43A85 | Harmonic analysis on homogeneous spaces |
Keywords:
metric measure space; doubling condition; Lusin’s approximation; Sobolev space; capacity; outer measure; Hausdorff measure and dimensionCitations:
Zbl 1458.46025References:
| [1] | S. A. Bondarev, V. G. Krotov, “Fine properties of functions from Hajlasz-Sobolev classes αp, p > 0, I. Lebesgue points”, J. Contemp. Math. Anal., 51 (6), 282-295, 2016). · Zbl 1458.46025 · doi:10.3103/S1068362316060029 |
| [2] | P. Hajlasz, “Sobolev spaces on an arbitrary metric spaces”, Potential Anal., 5 (4), 403-415, 1996). · Zbl 0859.46022 |
| [3] | P. Hajlasz, J. Kinnunen, “Hölder qasicontinuity of Sobolev functions on metric spaces”, Rev.Mat. Iberoam., 14 (3), 601-622, 1998). · Zbl 1155.46306 · doi:10.4171/RMI/246 |
| [4] | J. Kinnunen, H. Tuominen, Pointwise behaviour of M1,1 Sobolev functions, Math. Zeit., 257 (3), 613-630, 2007). · Zbl 1128.46018 |
| [5] | V. G. Krotov, M. A. Prokhorovich, “The Rate of Convergence of Steklov Means on Metric Measure Spaces and Hausdorff Dimension”, Mat. Zametki, 89 (1), 145-148, 2011). · Zbl 1243.42032 · doi:10.4213/mzm8927 |
| [6] | T. Heikkinen, H. Tuominen, “Approximation by Holder functions in Besov and Triebel-Lizorkin spaces”, preprint 2015, http://arxiv.org/abs/1504.02585. · Zbl 1365.46028 |
| [7] | R. A. Macias, C. Segovia “A decomposition into atoms of distributions on spaces of homogeneous type”, Advances in Mathematics, 33, 271-309, 1979). · Zbl 0431.46019 · doi:10.1016/0001-8708(79)90013-6 |
| [8] | V. G. Krotov, S.A. Bondarev, “Fine properties of functions from Hajlasz-Sobolev classes Wp a,<Emphasis Type=”Italic“>p > 0”, Intern. conf. “Functional Spaces and Approximation Theory” (Moscow,May 25-29, 2015). |
| [9] | V. G. Krotov, A. I. Porabkovich, Estimates of Lp-Oscillations of Functions for <Emphasis Type=”Italic“>p > 0”, Mat. Zametki, 97 (3), 407-420, 2015). · Zbl 1319.42015 |
| [10] | V. G. Krotov, “Weighted Lp-inequalities for sharp-maximal functions on metric spaces with measure”, J. Contemp.Math. Anal., 41 (2), 25-42, 2006). |
| [11] | J. Kinnunen, V. Latvala, “Lebesgue points for Sobolev functions on metric spaces”, Rev. Mat. Iberoam., 18 (3), 685-700, 2002). · Zbl 1037.46031 · doi:10.4171/RMI/332 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
