| Title:
|
Coarea integration in metric spaces (English) |
| Author:
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Malý, Jan |
| Language:
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English |
| Journal:
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Nonlinear Analysis, Function Spaces and Applications |
| Volume:
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Vol. 7 |
| Issue:
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2002 |
| Year:
|
|
| Pages:
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149-192 |
| . |
| Category:
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math |
| . |
| Summary:
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Let $X$ be a metric space with a doubling measure, $Y$ be a boundedly compact metric space and $u:X\to Y$ be a Lebesgue precise mapping whose upper gradient $g$ belongs to the Lorentz space $L_{m,1}$, $m\ge 1$. Let $E\subset X$ be a set of measure zero. Then $\widehat{\Cal H}_m(E\cap u^{-1}(y))=0$ for $\Cal H_m$-a.e.\ $y\in Y$, where $\Cal H_m$ is the $m$-dimensional Hausdorff measure and $\widehat{\Cal H}_m$ is the $m$-codimensional Hausdorff measure. This property is closely related to the coarea formula and implies a version of the Eilenberg inequality. The result relies on estimates of Hausdorff content of level sets of mappings between metric spaces and analysis of their Lebesgue points. Adapted versions of the Frostman lemma and of the Muckenhoupt-Wheeden inequality appear as essential tools. (English) |
| Keyword:
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coarea formula |
| Keyword:
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Eilenberg inequality |
| Keyword:
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Hausdorff content |
| Keyword:
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Hausdorff measure |
| Keyword:
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Lebesgue points |
| Keyword:
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Riesz potentials |
| Keyword:
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Lorentz space |
| Keyword:
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upper gradient |
| Keyword:
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Poincaré inequality |
| Keyword:
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space of homogenous type |
| Keyword:
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metric space |
| Keyword:
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doubling measure |
| MSC:
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28A75 |
| MSC:
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28A78 |
| MSC:
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31C15 |
| MSC:
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42B25 |
| MSC:
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43A85 |
| MSC:
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46E35 |
| MSC:
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54E99 |
| . |
| Date available:
|
2009-10-08T09:50:19Z |
| Last updated:
|
2012-08-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/702476 |
| . |
| Reference:
|
[AH] Adams D. R., Hedberg L. I.: Function spaces and potential theory.Grundlehren der Mathematischen Wissenschaften 314, Springer-Verlag, Berlin, 1995.Zbl 0834.46021. Zbl 0834.46021, MR 1411441 |
| Reference:
|
[A] Aïssaoui N.: Maximal operators, Lebesgue points and quasicontinuity in strongly nonlinear potential theory.Acta Math. Univ. Comenian. 71 (2002), 35–50. MR 1 943 014. Zbl 1052.46019, MR 1943014 |
| Reference:
|
[AT] Ambrosio L., Tilli P.: Selected topics on analysis on metric spaces.Scuola Normale Superiore Pisa, 2000. MR 2012736 |
| Reference:
|
[BZ] Bagby T., Ziemer W. P.: Pointwise differentiability and absolute continuity.Trans. Amer. Math. Soc. 191 (1974), 129–148. Zbl 0295.26013, MR 49 #9129. MR 0344390 |
| Reference:
|
[BS] Bennett C., Sharpley R.: Interpolation of operators.Pure and Applied Mathematics 129, Academic Press, Inc., Boston, MA, 1988. Zbl 0647.46057, MR 89e:46001. Zbl 0647.46057, MR 0928802 |
| Reference:
|
[BG] Burkholder D. L., Gundy R. F.: Extrapolation and interpolation of quasilinear operators on martingales.Acta Math. 124 (1970), 249–304. Zbl 0223.60021, MR 55 #13567. MR 0440695 |
| Reference:
|
[Ce] Cesari L.: Sulle funzioni assolutamente continue in due variabili.Ann. Scuola Norm. Sup. Pisa, II. Ser. 10 (1941), 91–101. Zbl 0025.31301, MR 3,230e. Zbl 0025.31301, MR 0005911 |
| Reference:
|
[Ch] Cheeger J.: Differentiability of Lipschitz functions on metric measure spaces.Geom. Funct. Anal. 9 (1999), 428–517. Zbl 0942.58018, MR 2000g:53043. Zbl 0942.58018, MR 1708448 |
| Reference:
|
[CW] Coifman R. R., Weiss G.: Analyse harmonique non-commutative sur certain espaces homogènes.Lecture Notes in Math. 242. Springer-Verlag, Berlin, 1971. Zbl 0224.43006, MR 58 #17690. MR 0499948 |
| Reference:
|
[E] Eilenberg S.: On $\varphi $ measures.Ann. Soc. Pol. Math. 17 (1938), 251–252. |
| Reference:
|
[F] Federer H.: Geometric measure theory.Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 153. Springer-Verlag, New York, 1969. Zbl 0176.00801, MR 41 #1976. Zbl 0176.00801, MR 0257325 |
| Reference:
|
[FZ] Federer H., Ziemer W. P.: The Lebesgue set of a function whose partial derivatives are $p$-th power summable.Indiana Univ. Math. J. 22 (1972), 139–158. Zbl 0238.28015, MR 55 #8321. MR 0435361 |
| Reference:
|
[FP] Fiorenza A., Prignet A.: Orlicz capacities and applications to some existence questions for elliptic PDEs having measure data.ESAIM: Control, Optim. and Calc. Var. 9 (2003), 317–341. Zbl 1075.35012, MR 1966536 |
| Reference:
|
[Fl] Fleming W. H.: Functions whose partial derivatives are measures.Illinois J. Math. 4 (1960), 452–478. Zbl 0151.05402, MR 24 #A202. Zbl 0151.05402, MR 0130338 |
| Reference:
|
[FR] Fleming W. H., Rishel R.: An integral formula for the total variation.Arch. Math. 111 (1960), 218–222. Zbl 0094.26301, MR 22 #5710. MR 0114892 |
| Reference:
|
[Fr] Frostman O.: Potentiel d’équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions.Medd. Lunds Univ. Mat. Sem. 3 (1935), 1–118. Zbl 0013.06302. Zbl 0013.06302 |
| Reference:
|
[GGKK] Genebashvili I., Gogatishvili A., Kokilashvili V., Krbec M.: Weight theory for integral transforms on spaces of homogeneous type.Pitman Monographs and Surveys in Pure and Applied Mathematics 92. Longman, Harlow, 1998. Zbl 0955.42001, MR 2003b:42002. Zbl 0955.42001, MR 1791462 |
| Reference:
|
[Ha] Hajłasz P.: Sobolev mappings, co-area formula and related topics.In: Proceedings on analysis and geometry. International conference in honor of the 70th birthday of Professor Yu. G. Reshetnyak, Novosibirsk, Russia, August 30–September 3, 1999 (S. K. Vodop’yanov, ed.). Izdatel’stvo Instituta Matematiki Im. S. L. Soboleva SO RAN, Novosibirsk, 2000, pp. 227–254. Zbl 0988.28002, MR 2002h:28005. MR 1847519 |
| Reference:
|
[HaKi] Hajłasz P., Kinnunen J.: Hölder quasicontinuity of Sobolev functions on metric spaces.Rev. Mat. Iberoamericana 14 (1998), 601–622. Zbl pre01275454, MR 2000e:46046. Zbl 1155.46306, MR 1681586 |
| Reference:
|
[HaK] Hajłasz P., Koskela P.: Sobolev met Poincaré.Memoirs Amer. Math. Soc. 688. Zbl 0954.46022, MR 2000j:46063. |
| Reference:
|
[He] Heinonen J.: Lectures on analysis on metric spaces.Universitext. Springer-Verlag, New York, 2001. Zbl 0985.46008, MR 2002c:30028. Zbl 0985.46008, MR 1800917 |
| Reference:
|
[HeK] Heinonen J., Koskela P.: Quasiconformal maps in metric spaces with controlled geometry.Acta Math. 181 (1998), 1–61. Zbl 0915.30018, MR 99j:30025. Zbl 0915.30018, MR 1654771 |
| Reference:
|
[HKST] Heinonen J., Koskela P., Shanmugalingam N., Tyson J.: Sobolev classes of Banach space-valued functions and quasiconformal mappings.J. Anal. Math. 85 (2001), 87–139. Zbl pre01765855, MR 2002k:46090. Zbl 1013.46023, MR 1869604 |
| Reference:
|
[HM] Hencl S., Malý J.: Mapping of bounded distortion: Hausdorff measure of zero sets.Math. Ann. 324 (2002), 451–464. MR 1 938 454. MR 1938454 |
| Reference:
|
[Ho] Honzík P.: Estimates of norms of operators in weighted spaces.(Czech). Diploma Thesis, Charles University, Prague, 2001. |
| Reference:
|
[JS] Jerrard R. L., Soner H. M.: Functions of bounded higher variation.Indiana Univ. Math. J. 51 (2002), 645–677. MR 2003e:49069. Zbl 1057.49036, MR 1911049 |
| Reference:
|
[KKM] Kauhanen J., Koskela P., Malý J.: On functions with derivatives in a Lorentz space.Manuscripta Math. 100 (1999), 87–101. Zbl 0976.26004, MR 2000j:46064. Zbl 0976.26004, MR 1714456 |
| Reference:
|
[KM] Kilpeläinen T., Malý J.: The Wiener test and potential estimates for quasilinear elliptic equations.Acta Math. 172 (1994), 137–161. Zbl 0820.35063, MR 95a:35050. MR 1264000 |
| Reference:
|
[KL] Kinnunen J., Latvala V.: Lebesgue points for Sobolev functions on metric spaces.Rev. Mat. Iberoamericana 18 (2002), 685–700. MR 1 954 868. Zbl 1037.46031, MR 1954868 |
| Reference:
|
[M1] Malý J.: Sufficient conditions for change of variables in integral.In: Proceedings on analysis and geometry. International conference in honor of the 70th birthday of Professor Yu. G. Reshetnyak, Novosibirsk, Russia, August 30–September 3, 1999 (S. K. Vodop’yanov, ed.). Izdatel’stvo Instituta Matematiki Im. S. L. Soboleva SO RAN, Novosibirsk, 2000, pp. 370–386. Zbl 0988.26011, MR 2002m:26013. MR 1847527 |
| Reference:
|
[M2] Malý J.: Wolff potential estimates of superminimizers of Orlicz type Dirichlet integrals.Manuscripta Math. 110 (2003), 513–525. Zbl 1098.35061, MR 1975101 |
| Reference:
|
[M3] Malý J.: Coarea properties of Sobolev functions.In: Function Spaces, Differential Operators, Nonlinear Analysis. The Hans Triebel Anniversary Volume (D. Haroske, T. Runst, H.-J. Schmeisser, eds.). Birkhäuser, Basel, 2003, pp. 371–381. Zbl 1036.46025, MR 1984185 |
| Reference:
|
[MM] Malý J., Martio O.: Lusin’s condition (N) and of the class $W^{1,n}$.J. Reine Angew. Math. 458 (1995), 19–36. Zbl 0812.30007, MR 95m:26024. MR 1310951 |
| Reference:
|
[MMo] Malý J.Mosco U.: Remarks on measure-valued Lagrangians on homogeneous spaces.(Italian). Papers in memory of Ennio De Giorgi. Ricerche Mat. 47 suppl. (1999), 217–231. Zbl 0957.46027, MR 2002e:31005. Zbl 0957.46027, MR 1765686 |
| Reference:
|
[MP] Malý J., Pick L.: The sharp Riesz potential estimates in metric spaces.Indiana Univ. Math. J. 51 (2002), 251–268. Zbl pre01780940, MR 2003d:46045. Zbl 1038.46027, MR 1909289 |
| Reference:
|
[MSZ1] Malý J., Swanson D., Ziemer W. P.: The co-area formula for Sobolev mappings.Trans. Amer. Math. Soc. 355 (2003), 477–492. Zbl pre01821246,MR 1 932 709. Zbl 1034.46032, MR 1932709 |
| Reference:
|
[MSZ2] Malý J., Swanson D., Ziemer W. P.: Fine behavior of functions with gradients in a Lorentz space.In preparation. |
| Reference:
|
[MZ] Malý J., Ziemer W..P.: Fine regularity of solutions of elliptic differential equations.Mathematical Surveys and Monographs 51. Amer. Math. Soc., Providence, R.I., 1997. Zbl 0882.35001, MR 98h:35080. MR 1461542 |
| Reference:
|
[MMi] Marcus M., Mizel V. J.: Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems.Bull. Amer. Math. Soc. 79 (1973), 790–795. Zbl 0275.49041, MR 48 #1013. Zbl 0275.49041, MR 0322651 |
| Reference:
|
[Mt] Mattila P.: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability.Cambridge Studies in Advanced Mathematics 44. Cambridge University Press, Cambridge, 1995. Zbl 0819.28004, MR 96h:28006. Zbl 0819.28004, MR 1333890 |
| Reference:
|
[MH] Maz’ya V. G., Havin V. P.: Nonlinear potential theory.Uspekhi Mat. Nauk 27 (1972), 67–138. English transl. in Russian Math. Surveys, 27 (1972), 71–148. MR 53 #13610. MR 0409858 |
| Reference:
|
[Mi] Mikkonen P.: On the Wolff potential and quasilinear elliptic equations involving measures.Ann. Acad. Sci. Fenn. Math. Diss. 104 (1996). Zbl 0860.35041, MR 97e:35069. Zbl 0860.35041, MR 1386213 |
| Reference:
|
[MW] Muckenhoupt B., Wheeden R. L.: Weighted norm inequalities for fractional integrals.Trans. Amer. Math. Soc. 192 (1974), 261–274. Zbl 0289.26010, MR 49 #5275. MR 0340523 |
| Reference:
|
[N] Nieminen E.: Hausdorff measures, capacities and Sobolev spaces with weights.Ann. Acad. Sci. Fenn. Math. Diss. 81 (1991). Zbl 0723.46024, MR 92i:46039. Zbl 0723.46024, MR 1108686 |
| Reference:
|
[R] Reshetnyak, Yu. G.: On the concept of capacity in the theory of functions with generalized derivatives.(Russian). Sibirsk. Mat. Zh. 10 (1969), 1109–1138. English transl. Siberian Math. J. 10 (1969), 818–842. Zbl 0199.20701. MR 0276487 |
| Reference:
|
[VP] Van der Putten R.: On the critical-values lemma and the coarea formula.(Italian). Boll. Un. Mat. Ital., VII. Ser. B 6 (1992), 561–578. Zbl 0762.46019. Zbl 0762.46019, MR 1191953 |
| Reference:
|
[Se] Semmes S.: Finding curves on general spaces through quantitative topology, with application to Sobolev and Poincaré inequalities.Selecta Math. (N. S.) 2 (1996), 155–295. Zbl 0870.54031, MR 97j:46033. MR 1414889 |
| Reference:
|
[Sh] Shanmugalingam N.: Newtonian spaces: An extension of Sobolev spaces to metric spaces.Rev. Mat. Iberoamericana 16 (2000), 243–279. Zbl 0974.46038, MR 2002b:46059. MR 1809341 |
| Reference:
|
[Sj] Sjödin T.: A note on capacity and Hausdorff measure in homogeneous spaces.Potential Anal. 6 (1997), 87–97. Zbl 0873.31013, MR 98e:31007. Zbl 0873.31013, MR 1436823 |
| Reference:
|
[T] Turesson B. O.: Nonlinear potential theory and weighted Sobolev spaces.Lecture Notes Math. 1736. Springer-Verlag, Berlin, 2000. Zbl 0949.31006, MR 2002f:31027. Zbl 0949.31006, MR 1774162 |
| Reference:
|
[V] Vodop’yanov S. K.: $L_p$-theory of potential for generalized kernels and its applications.(Russian). Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat. Novosibirsk, 1990. MR 1090089 |
| Reference:
|
[Z] Ziemer W. P.: Weakly differentiable functions. Sobolev spaces and function of bounded variation.Graduate Texts in Mathematics 120. Springer-Verlag, New York, 1989. Zbl 0692.46022, MR 91e:46046. MR 1014685 |
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