Generalized Lebesgue points for Sobolev functions

Nijjwal Karak

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 1, page 143-150
  • ISSN: 0011-4642

Abstract

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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 , μ ) 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 M s , p ( X ) , 0 < s 1 , 0 < p < 1 , where X is a doubling metric measure space, has generalized Lebesgue points outside a set of h -Hausdorff measure zero for a suitable gauge function h .

How to cite

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Karak, Nijjwal. "Generalized Lebesgue points for Sobolev functions." Czechoslovak Mathematical Journal 67.1 (2017): 143-150. <http://eudml.org/doc/287898>.

@article{Karak2017,
abstract = {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 $f\in M^\{s,p\}(X)$, $0<s\le 1$, $0<p<1$, 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 $h$.},
author = {Karak, Nijjwal},
journal = {Czechoslovak Mathematical Journal},
keywords = {Sobolev space; metric measure space; median; generalized Lebesgue point},
language = {eng},
number = {1},
pages = {143-150},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Generalized Lebesgue points for Sobolev functions},
url = {http://eudml.org/doc/287898},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Karak, Nijjwal
TI - Generalized Lebesgue points for Sobolev functions
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 1
SP - 143
EP - 150
AB - 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 $f\in M^{s,p}(X)$, $0<s\le 1$, $0<p<1$, 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 $h$.
LA - eng
KW - Sobolev space; metric measure space; median; generalized Lebesgue point
UR - http://eudml.org/doc/287898
ER -

References

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  1. Adams, D. R., Hedberg, L. I., 10.1007/978-3-662-03282-4, Grundlehren der Mathematischen Wissenschaften 314, Springer, Berlin (1996). (1996) Zbl0834.46021MR1411441DOI10.1007/978-3-662-03282-4
  2. Björn, J., Onninen, J., 10.1007/s00209-005-0792-y, Math. Z. 251 (2005), 131-146. (2005) Zbl1084.31004MR2176468DOI10.1007/s00209-005-0792-y
  3. Costea, Ş., 10.5565/PUBLMAT_53109_07, Publ. Mat. 53 (2009), 141-178. (2009) Zbl1171.46025MR2474119DOI10.5565/PUBLMAT_53109_07
  4. Evans, L. C., Gariepy, R. F., Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton (1992). (1992) Zbl0804.28001MR1158660
  5. Federer, H., 10.1007/978-3-642-62010-2, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 153, Springer, New York (1969). (1969) Zbl0874.49001MR0257325DOI10.1007/978-3-642-62010-2
  6. Federer, H., Ziemer, W. P., 10.1512/iumj.1972.22.22013, Math. J., Indiana Univ. 22 (1972), 139-158. (1972) Zbl0238.28015MR0435361DOI10.1512/iumj.1972.22.22013
  7. Fujii, N., A condition for a two-weight norm inequality for singular integral operators, Stud. Math. 98 (1991), 175-190. (1991) Zbl0732.42012MR1115188
  8. Hajłasz, P., 10.1007/BF00275475, Potential Anal. 5 (1996), 403-415. (1996) Zbl0859.46022MR1401074DOI10.1007/BF00275475
  9. Hajłasz, P., Kinnunen, J., 10.4171/RMI/246, Rev. Mat. Iberoam. 14 (1998), 601-622. (1998) Zbl1155.46306MR1681586DOI10.4171/RMI/246
  10. Hajłasz, P., Koskela, P., 10.1090/memo/0688, Mem. Am. Math. Soc. 145 (2000), 1-101. (2000) Zbl0954.46022MR1683160DOI10.1090/memo/0688
  11. Hedberg, L. I., Netrusov, Y., 2326315, Mem. Am. Math. Soc. 188 (2007), 1-97. (2007) Zbl1186.46028MR2326315DOI2326315
  12. Heikkinen, T., Koskela, P., Tuominen, H., Approximation and quasicontinuity of Besov and Triebel-Lizorkin functions, To appear in Trans Am. Math. Soc. MR3605979
  13. Heinonen, J., Kilpeläinen, T., Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Mineola (2006). (2006) Zbl1115.31001MR2305115
  14. Karak, N., Koskela, P., 10.1007/s13163-015-0174-x, Rev. Mat. Complut. 28 (2015), 733-740. (2015) Zbl1325.31004MR3379045DOI10.1007/s13163-015-0174-x
  15. Karak, N., Koskela, P., 10.1007/s11425-015-5001-9, Sci. China Math. 58 (2015), 1697-1706. (2015) Zbl06485640MR3368175DOI10.1007/s11425-015-5001-9
  16. Kinnunen, J., Korte, R., Shanmugalingam, N., Tuominen, H., 10.1512/iumj.2008.57.3168, Indiana Univ. Math. J. 57 (2008), 401-430. (2008) Zbl1146.46018MR2400262DOI10.1512/iumj.2008.57.3168
  17. Kinnunen, J., Latvala, V., 10.4171/RMI/332, Rev. Mat. Iberoam. 18 (2002), 685-700. (2002) Zbl1037.46031MR1954868DOI10.4171/RMI/332
  18. Koskela, P., Saksman, E., 10.4310/MRL.2008.v15.n4.a11, Math. Res. Lett. 15 (2008), 727-744. (2008) Zbl1165.46013MR2424909DOI10.4310/MRL.2008.v15.n4.a11
  19. Koskela, P., Yang, D., Zhou, Y., 2764899, Adv. Math. 226 (2011), 3579-3621. (2011) Zbl1217.46019MR2764899DOI2764899
  20. Maz'ya, V. G., Khavin, V. P., 10.1070/rm1972v027n06ABEH001393, Russ. Math. Surv. 27 (1972), 71-148. (1972) Zbl0269.31004DOI10.1070/rm1972v027n06ABEH001393
  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. (1990) Zbl0719.46018MR1006450
  22. Orobitg, J., 10.1007/BFb0086804, Harmonic Analysis and Partial Differential Equations Proc. Int. Conf., El Escorial, 1987, Lect. Notes Math. 1384, Springer, Berlin (1989), 202-206. (1989) Zbl0699.46018MR1013826DOI10.1007/BFb0086804
  23. Poelhuis, J., Torchinsky, A., 10.4064/sm213-3-3, Stud. Math. 213 (2012), 227-242. (2012) Zbl1277.42024MR3024312DOI10.4064/sm213-3-3
  24. Shanmugalingam, N., 10.4171/RMI/275, Rev. Mat. Iberoam. 16 (2000), 243-279. (2000) Zbl0974.46038MR1809341DOI10.4171/RMI/275
  25. Strömberg, J.-O., 10.1512/iumj.1979.28.28037, Indiana Univ. Math. J. 28 (1979), 511-544. (1979) Zbl0429.46016MR529683DOI10.1512/iumj.1979.28.28037
  26. Yang, D., 10.1360/02ys0343, Sci. China, Ser. A 46 (2003), 675-689. (2003) Zbl1092.46026MR2025934DOI10.1360/02ys0343
  27. Ziemer, W. P., 10.1007/978-1-4612-1015-3, Graduate Texts in Mathematics 120, Springer, New York (1989). (1989) Zbl0692.46022MR1014685DOI10.1007/978-1-4612-1015-3

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