About DML-CZ | FAQ | Conditions of Use | Math Archives | Contact Us
  Previous |  Up |  Next

Article

Karak, Nijjwal
Generalized Lebesgue points for Sobolev functions. (English). Czechoslovak Mathematical Journal, vol. 67 (2017), issue 1, pp. 143-150
MSC: 28A78, 46E35 | MR 3633003 | Zbl 06738509 | DOI: 10.21136/CMJ.2017.0405-15
Keywords:
Sobolev space; metric measure space; median; generalized Lebesgue point
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 $h$.
References:
[1] Adams, D. R., Hedberg, L. I.: Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften 314, Springer, Berlin (1996). DOI 10.1007/978-3-662-03282-4 | MR 1411441 | Zbl 0834.46021
[2] Björn, J., Onninen, J.: Orlicz capacities and Hausdorff measures on metric spaces. Math. Z. 251 (2005), 131-146. DOI 10.1007/s00209-005-0792-y | MR 2176468 | Zbl 1084.31004
[3] Costea, Ş.: Besov capacity and Hausdorff measures in metric measure spaces. Publ. Mat. 53 (2009), 141-178. DOI 10.5565/PUBLMAT_53109_07 | MR 2474119 | Zbl 1171.46025
[4] Evans, L. C., Gariepy, R. F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics, CRC Press, Boca Raton (1992). MR 1158660 | Zbl 0804.28001
[5] Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 153, Springer, New York (1969). DOI 10.1007/978-3-642-62010-2 | MR 0257325 | Zbl 0874.49001
[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. DOI 10.1512/iumj.1972.22.22013 | MR 0435361 | Zbl 0238.28015
[7] Fujii, N.: A condition for a two-weight norm inequality for singular integral operators. Stud. Math. 98 (1991), 175-190. DOI 10.4064/sm-98-3-175-190 | MR 1115188 | Zbl 0732.42012
[8] Hajłasz, P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5 (1996), 403-415. DOI 10.1007/BF00275475 | MR 1401074 | Zbl 0859.46022
[9] Hajłasz, P., Kinnunen, J.: Hölder quasicontinuity of Sobolev functions on metric spaces. Rev. Mat. Iberoam. 14 (1998), 601-622. DOI 10.4171/RMI/246 | MR 1681586 | Zbl 1155.46306
[10] Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145 (2000), 1-101. DOI 10.1090/memo/0688 | MR 1683160 | Zbl 0954.46022
[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. DOI 2326315 | MR 2326315 | Zbl 1186.46028
[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
[13] Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Dover Publications, Mineola (2006). MR 2305115 | Zbl 1115.31001
[14] Karak, N., Koskela, P.: Capacities and Hausdorff measures on metric spaces. Rev. Mat. Complut. 28 (2015), 733-740. DOI 10.1007/s13163-015-0174-x | MR 3379045 | Zbl 1325.31004
[15] Karak, N., Koskela, P.: Lebesgue points via the Poincaré inequality. Sci. China Math. 58 (2015), 1697-1706. DOI 10.1007/s11425-015-5001-9 | MR 3368175 | Zbl 06485640
[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. DOI 10.1512/iumj.2008.57.3168 | MR 2400262 | Zbl 1146.46018
[17] Kinnunen, J., Latvala, V.: Lebesgue points for Sobolev functions on metric spaces. Rev. Mat. Iberoam. 18 (2002), 685-700. DOI 10.4171/RMI/332 | MR 1954868 | Zbl 1037.46031
[18] Koskela, P., Saksman, E.: Pointwise characterizations of Hardy-Sobolev functions. Math. Res. Lett. 15 (2008), 727-744. DOI 10.4310/MRL.2008.v15.n4.a11 | MR 2424909 | Zbl 1165.46013
[19] Koskela, P., Yang, D., Zhou, Y.: Pointwise characterizations of Besov and Triebel-Lizorkin spaces and quasiconformal mappings. Adv. Math. 226 (2011), 3579-3621. DOI 2764899 | MR 2764899 | Zbl 1217.46019
[20] Maz'ya, V. G., Khavin, V. P.: Non-linear potential theory. Russ. Math. Surv. 27 (1972), 71-148. DOI 10.1070/rm1972v027n06ABEH001393 | Zbl 0269.31004
[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. MR 1006450 | Zbl 0719.46018
[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. DOI 10.1007/BFb0086804 | MR 1013826 | Zbl 0699.46018
[23] Poelhuis, J., Torchinsky, A.: Medians, continuity, and vanishing oscillation. Stud. Math. 213 (2012), 227-242. DOI 10.4064/sm213-3-3 | MR 3024312 | Zbl 1277.42024
[24] Shanmugalingam, N.: Newtonian spaces: An extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoam. 16 (2000), 243-279. DOI 10.4171/RMI/275 | MR 1809341 | Zbl 0974.46038
[25] Strömberg, J.-O.: Bounded mean oscillation with Orlicz norms and duality of Hardy spaces. Indiana Univ. Math. J. 28 (1979), 511-544. DOI 10.1512/iumj.1979.28.28037 | MR 529683 | Zbl 0429.46016
[26] Yang, D.: New characterizations of Hajłasz-Sobolev spaces on metric spaces. Sci. China, Ser. A 46 (2003), 675-689. DOI 10.1360/02ys0343 | MR 2025934 | Zbl 1092.46026
[27] Ziemer, W. P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics 120, Springer, New York (1989). DOI 10.1007/978-1-4612-1015-3 | MR 1014685 | Zbl 0692.46022
Partner of
EuDML logo