Hausdorff Dimension of Lebesgue Sets for $W^p_\alpha$ Classes on Metric Spaces

M. A. Prokhorovich

Matematičeskie zametki, Tome 82 (2007) no. 1, pp. 99-107.

Résumé

Let $(X,\mu,d)$ be a space of homogeneous type, where $d$ and $\mu$ are a metric and a measure, respectively, related to each other by the doubling condition with $\gamma>0$. Let $W^p_\alpha(X)$ be generalized Sobolev classes, let $\operatorname{Cap}_{\alpha,p}$ (where $p>1$ and $0\alpha\le 1$) be the corresponding capacity, and let $\dim_H$ be the Hausdorff dimension. We show that the capacity $\operatorname{Cap}_{\alpha,p}$ is related to the Hausdorff dimension and also prove that, for each function $u\in W^p_\alpha(X)$, $p>1$, $0\alpha\gamma/p$, there exists a set $E\subset X$ such that $\dim_H(E)\le\gamma-\alpha p$, the limit $$ \lim_{r\to +0}\frac{1}{\mu(B(x,r))}\int_{B(x,r)}u\,d\mu=u^*(x) $$ exists for each $x\in X\setminus E$, and moreover $$ \lim_{r\to+0}\frac{1}{\mu(B(x,r))}\int_{B(x,r)}|u-u^*(x)|^q\,d\mu=0,\qquad \frac{1}{q}=\frac{1}{p}-\frac{\alpha}{\gamma}. $$

Détail

Keywords: Sobolev class, capacity, metric space, Borel measure.
Mots-clés : Lebesgue set, Hausdorff dimension

Bibliographie

[1] R. R. Coifman, G. Weiss, “Analyse harmonique non-commutative sur certain espaces homogenés”, Lecture Notes in Math., v. 242, Springer-Verlag, Berlin–Heidelberg–New York, 1971 | MR | Zbl

[2] R. R. Coifman, G. Weiss, “Extensions of Hardy spaces and their use in analysis”, Bull. Amer. Math. Soc., 83:4 (1977), 569–645 | DOI | MR | Zbl

[3] P. Hajłasz, “Sobolev spaces on an arbitrary metric spaces”, Potential Anal., 5:4 (1996), 403–415 | MR | Zbl

[4] P. Hajłasz, P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc., 145{, 688}, American Mathematical Society, Providence, RI, 2000 | MR | Zbl

[5] J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York, 2001 | MR | Zbl

[6] J. Hu, “A note on Hajłasz–Sobolev spaces on fractals”, J. Math. Anal. Appl., 280:1 (2003), 91–101 | DOI | MR | Zbl

[7] D. Yang, “New haracterization of Hajłasz–Sobolev spaces on metric spaces”, Sci. China Ser. A, 46:5 (2003), 675–689 | DOI | MR | Zbl

[8] I. A. Ivanishko, “Obobschennye klassy Soboleva na metricheskikh prostranstvakh s meroi”, Matem. zametki, 77:6 (2005), 937–941 | MR | Zbl

[9] J. Kinnunen, O. Martio, “The Sobolev capacity on metric spaces”, Ann. Acad. Sci. Fenn. Math., 21:2 (1996), 367–382 | MR | Zbl

[10] J. Kinnunen, V. Latvala, “Lebesgue points for Sobolev functions on metric spaces”, Rev. Mat. Iberoamericana, 18:3 (2002), 685–700 | MR | Zbl

[11] P. Hajłasz, J. Kinnunen, “Hölder qasicontinuity of Sobolev functions on metric spaces”, Rev. Mat. Iberoamericana, 14:3 (1998), 601–622 | MR | Zbl

[12] M. A. Prokhorovich, “Emkosti i tochki Lebega dlya drobnykh klassov Khailasha–Soboleva na metricheskikh prostranstvakh s meroi”, Vestsi NAN Belarusi. Ser. fiz.-mat. navuk, 2006, no. 1, 19–23 | MR

[13] M. A. Prokhorovich, “Emkosti i razmernost Khausdorfa mnozhestva tochek Lebega dlya funktsii iz klassov Soboleva na prostranstvakh odnorodnogo tipa”, Sovremennye problemy teorii funktsii i ikh prilozheniya, Tezisy dokladov 13-i Saratovskoi zimnei shkoly, Saratov, 2006, 146–147

[14] H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, 153, Springer-Verlag, Berlin–Heidelberg–New York, 1969 | MR | Zbl

[15] D. R. Adams, L. I. Hedberg, “Function spaces and potential theory”, Grundlehren der Mathematischen Wissenschaften, 314, Springer-Verlag, Berlin–Heidelberg–New York, 1996 | MR | Zbl

[16] I. A. Ivanishko, V. G. Krotov, “Obobschennoe neravenstvo Puankare–Soboleva na metricheskikh prostranstvakh s meroi”, Tr. In-ta matem. NAN Belarusi, 14:1 (2006)

[17] E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Mathematical Series, 43, Princeton Univ. Press, Princeton, NJ, 1993 | MR | Zbl

[18] T. Bagby, W. P. Ziemer, “Pointwise differentiablity and absolute continuity”, Trans. Amer. Math. Soc., 191 (1974), 129–148 | DOI | MR | Zbl