Previous |  Up |  Next


Title: Generalized Lebesgue points for Sobolev functions (English)
Author: Karak, Nijjwal
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 67
Issue: 1
Year: 2017
Pages: 143-150
Summary lang: English
Category: math
Summary: In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X,d,\mu )$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B(x,r)$ converge to $f(x)$ when $r$ converges to $0$. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function. We show that a function $f\in M^{s,p}(X)$, $0<s\leq 1$, $0<p<1$, where $X$ is a doubling metric measure space, has generalized Lebesgue points outside a set of $\mathcal {H}^h$-Hausdorff measure zero for a suitable gauge function $h$. (English)
Keyword: Sobolev space
Keyword: metric measure space
Keyword: median
Keyword: generalized Lebesgue point
MSC: 28A78
MSC: 46E35
idZBL: Zbl 06738509
idMR: MR3633003
DOI: 10.21136/CMJ.2017.0405-15
Date available: 2017-03-13T12:07:39Z
Last updated: 2020-01-05
Stable URL:
Reference: [1] Adams, D. R., Hedberg, L. I.: Function Spaces and Potential Theory.Grundlehren der Mathematischen Wissenschaften 314, Springer, Berlin (1996). Zbl 0834.46021, MR 1411441, 10.1007/978-3-662-03282-4
Reference: [2] Björn, J., Onninen, J.: Orlicz capacities and Hausdorff measures on metric spaces.Math. Z. 251 (2005), 131-146. Zbl 1084.31004, MR 2176468, 10.1007/s00209-005-0792-y
Reference: [3] Costea, Ş.: Besov capacity and Hausdorff measures in metric measure spaces.Publ. Mat. 53 (2009), 141-178. Zbl 1171.46025, MR 2474119, 10.5565/PUBLMAT_53109_07
Reference: [4] Evans, L. C., Gariepy, R. F.: Measure Theory and Fine Properties of Functions.Studies in Advanced Mathematics, CRC Press, Boca Raton (1992). Zbl 0804.28001, MR 1158660
Reference: [5] Federer, H.: Geometric Measure Theory.Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 153, Springer, New York (1969). Zbl 0874.49001, MR 0257325, 10.1007/978-3-642-62010-2
Reference: [6] Federer, H., Ziemer, W. P.: The Lebesgue set of a function whose distribution derivatives are $p$-th power summable.Math. J., Indiana Univ. 22 (1972), 139-158. Zbl 0238.28015, MR 0435361, 10.1512/iumj.1972.22.22013
Reference: [7] Fujii, N.: A condition for a two-weight norm inequality for singular integral operators.Stud. Math. 98 (1991), 175-190. Zbl 0732.42012, MR 1115188
Reference: [8] Hajłasz, P.: Sobolev spaces on an arbitrary metric space.Potential Anal. 5 (1996), 403-415. Zbl 0859.46022, MR 1401074, 10.1007/BF00275475
Reference: [9] Hajłasz, P., Kinnunen, J.: Hölder quasicontinuity of Sobolev functions on metric spaces.Rev. Mat. Iberoam. 14 (1998), 601-622. Zbl 1155.46306, MR 1681586, 10.4171/RMI/246
Reference: [10] Hajłasz, P., Koskela, P.: Sobolev met Poincaré.Mem. Am. Math. Soc. 145 (2000), 1-101. Zbl 0954.46022, MR 1683160, 10.1090/memo/0688
Reference: [11] Hedberg, L. I., Netrusov, Y.: An axiomatic approach to function spaces, spectral synthesis, and Luzin approximation.Mem. Am. Math. Soc. 188 (2007), 1-97. Zbl 1186.46028, MR 2326315, 2326315
Reference: [12] Heikkinen, T., Koskela, P., Tuominen, H.: Approximation and quasicontinuity of Besov and Triebel-Lizorkin functions.To appear in Trans Am. Math. Soc. MR 3605979
Reference: [13] Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations.Dover Publications, Mineola (2006). Zbl 1115.31001, MR 2305115
Reference: [14] Karak, N., Koskela, P.: Capacities and Hausdorff measures on metric spaces.Rev. Mat. Complut. 28 (2015), 733-740. Zbl 1325.31004, MR 3379045, 10.1007/s13163-015-0174-x
Reference: [15] Karak, N., Koskela, P.: Lebesgue points via the Poincaré inequality.Sci. China Math. 58 (2015), 1697-1706. Zbl 06485640, MR 3368175, 10.1007/s11425-015-5001-9
Reference: [16] Kinnunen, J., Korte, R., Shanmugalingam, N., Tuominen, H.: Lebesgue points and capacities via the boxing inequality in metric spaces.Indiana Univ. Math. J. 57 (2008), 401-430. Zbl 1146.46018, MR 2400262, 10.1512/iumj.2008.57.3168
Reference: [17] Kinnunen, J., Latvala, V.: Lebesgue points for Sobolev functions on metric spaces.Rev. Mat. Iberoam. 18 (2002), 685-700. Zbl 1037.46031, MR 1954868, 10.4171/RMI/332
Reference: [18] Koskela, P., Saksman, E.: Pointwise characterizations of Hardy-Sobolev functions.Math. Res. Lett. 15 (2008), 727-744. Zbl 1165.46013, MR 2424909, 10.4310/MRL.2008.v15.n4.a11
Reference: [19] Koskela, P., Yang, D., Zhou, Y.: Pointwise characterizations of Besov and Triebel-Lizorkin spaces and quasiconformal mappings.Adv. Math. 226 (2011), 3579-3621. Zbl 1217.46019, MR 2764899, 2764899
Reference: [20] Maz'ya, V. G., Khavin, V. P.: Non-linear potential theory.Russ. Math. Surv. 27 (1972), 71-148. Zbl 0269.31004, 10.1070/rm1972v027n06ABEH001393
Reference: [21] Netrusov, Yu. V.: Sets of singularities of functions in spaces of Besov and Lizorkin-Triebel type.Proc. Steklov Inst. Math. 187 (1990), 185-203 187 1989 162-177 Translation from Tr. Mat. Inst. Steklova. Zbl 0719.46018, MR 1006450
Reference: [22] Orobitg, J.: Spectral synthesis in spaces of functions with derivatives in $H^1$.Harmonic Analysis and Partial Differential Equations Proc. Int. Conf., El Escorial, 1987, Lect. Notes Math. 1384, Springer, Berlin (1989), 202-206. Zbl 0699.46018, MR 1013826, 10.1007/BFb0086804
Reference: [23] Poelhuis, J., Torchinsky, A.: Medians, continuity, and vanishing oscillation.Stud. Math. 213 (2012), 227-242. Zbl 1277.42024, MR 3024312, 10.4064/sm213-3-3
Reference: [24] Shanmugalingam, N.: Newtonian spaces: An extension of Sobolev spaces to metric measure spaces.Rev. Mat. Iberoam. 16 (2000), 243-279. Zbl 0974.46038, MR 1809341, 10.4171/RMI/275
Reference: [25] Strömberg, J.-O.: Bounded mean oscillation with Orlicz norms and duality of Hardy spaces.Indiana Univ. Math. J. 28 (1979), 511-544. Zbl 0429.46016, MR 529683, 10.1512/iumj.1979.28.28037
Reference: [26] Yang, D.: New characterizations of Hajłasz-Sobolev spaces on metric spaces.Sci. China, Ser. A 46 (2003), 675-689. Zbl 1092.46026, MR 2025934, 10.1360/02ys0343
Reference: [27] Ziemer, W. P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation.Graduate Texts in Mathematics 120, Springer, New York (1989). Zbl 0692.46022, MR 1014685, 10.1007/978-1-4612-1015-3


Files Size Format View
CzechMathJ_67-2017-1_10.pdf 288.7Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo