Lebesgue points for Sobolev functions on metric spaces. (English) Zbl 1037.46031

Let \(M^{1,p}(X)\) be the Sobolev space of functions \(u\) defined on a metric measure space \((X,d)\) with a doubling measure \(\mu\). The space was introduced by P. Hajlasz [Potential Anal. 5, 403–415 (1999; Zbl 0859.46022)] and it is based on the Lipschitz type representation \(|(u(x)- u(y)|\leq d(x,y)(g(x)+ g(y))\), where \(g\) can be regarded as a derivative of \(u\). It is shown that the quasicontinuous representative of a function \(u\) has Lebesgue points except on a set of \(p\)-capacity zero. The proof makes use of a discrete Hardy-Littlewood maximal function and capacitary weak type estimates; the Besicovitch covering theorem and representations formulas for classical Sobolev functions are not available in metric spaces.


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems


Zbl 0859.46022
Full Text: DOI EuDML


[1] Buckley, S.: Is the maximal function of a Lipschitz function continuous? Ann. Acad. Sci. Fenn. Math. 24 (1999), 519-528. · Zbl 0935.42010
[2] Björn, J., MacManus, P. and Shanmugalingam, N.: Fat sets and pointwise boundary estimates for p-harmonic functions in metric spaces. J. Anal. Math. 85 (2001), 339-369. · Zbl 1003.31004
[3] Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9 (1999) 428-517. · Zbl 0942.58018
[4] Coifman, R. R. and Weiss, G.: Analyse Harmonique Non-Commutative sur Certain Espaces Homogenés. Lecture Notes in Mathematics 242, Springer-Verlag, 1971. · Zbl 0224.43006
[5] Evans, L.C. and Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, 1992. · Zbl 0804.28001
[6] Fabes, E.B., Kenig, C.E. and Serapioni, R.P.: The local regular- ity of solutions of degenerate elliptic equations Comm. Partial Differential Equations 7 (1982), 77-116. · Zbl 0498.35042
[7] Federer, H. and Ziemer, W. P.: The Lebesgue set of a function whose distribution derivatives are p-th power integrable. Indiana Univ. Math. J. 22 (1972), 139-158. · Zbl 0238.28015
[8] Hajlasz, P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5 (1995), 403-415. · Zbl 0859.46022
[9] Hajlasz, P. and Kinnunen, J.: Hölder quasicontinuity of Sobolev func- tions on metric spaces. Rev. Mat. Iberoamericana 14 (1998), 601-622. · Zbl 1155.46306
[10] Hajlasz, P. and Koskela, P.: Sobolev meets Poincaré. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 1211-1215. · Zbl 0837.46024
[11] Hajlasz, P. and Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000), 1-101.
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