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Integration of Hölder forms and currents in snowflake spaces. (English) Zbl 1219.49036
Summary: For an oriented \(n\)-dimensional Lipschitz manifold \(M\) we give meaning to the integral \(\int_M f \, dg_1 \wedge \cdots \wedge dg_n\) in case the functions \(f, g_1, \dots, g_n\) are merely Hölder continuous of a certain order by extending the construction of the Riemann-Stieltjes integral to higher dimensions. More generally, we show that for \(\alpha \in (\frac{n}{n+1},1]\) the \(n\)-dimensional locally normal currents in a locally compact metric space \((X, d)\) represent a subspace of the \(n\)-dimensional currents in \((X, d^\alpha)\). On the other hand, for \({n \geq 1}\) and \(\alpha \leq \frac{n}{n+1}\) the vector space of \(n\)-dimensional currents in \((X, d^\alpha)\) is zero.

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)
Full Text: DOI arXiv
[1] Ambrosio L., Kirchheim B.: Currents in metric spaces. Acta Math. 185, 1–80 (2000) · Zbl 0984.49025 · doi:10.1007/BF02392711
[2] Assouad P.: Plongements lipschitziens dans R n . Bull. Soc. Math. France 111, 429–448 (1983) · Zbl 0597.54015
[3] Federer H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York (1969)
[4] Federer H., Fleming W.H.: Normal and integral currents. Ann. Math. 72(2), 458–520 (1960) · Zbl 0187.31301 · doi:10.2307/1970227
[5] Gromov, M.: Metric Structures for Riemannian and non-Riemannian Spaces. Progress in Mathematics, vol. 152. Birkhäuser, Boston, MA (1999) (with appendices by M. Katz, P. Pansu and S. Semmes) · Zbl 0953.53002
[6] Lang, U.: Local currents in metric spaces, Preprint 2008. http://www.math.ethz.ch/\(\sim\)lang/loc.pdf
[7] Lang U., Plaut C.: Bilipschitz embeddings of metric spaces into space forms. Geom. Dedicata 87, 285–307 (2001) · Zbl 1024.54013 · doi:10.1023/A:1012093209450
[8] Spivak M.: Calculus on Manifolds. A Modern Approach to Classical Theorems of Advanced Calculus. W. A. Benjamin, Inc., New York (1965) · Zbl 0141.05403
[9] Tukia P.: A quasiconformal group not isomorphic to a Möbius group. Ann. Acad. Sci. Fenn. Ser. A I Math. 6, 149–160 (1981) · Zbl 0443.30026
[10] Wenger S.: Flat convergence for integral currents in metric spaces. Calc. Var. Partial Differ. Equ. 28, 139–160 (2007) · Zbl 1110.53030 · doi:10.1007/s00526-006-0034-0
[11] Young L.C.: An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67, 251–282 (1936) · Zbl 0016.10404 · doi:10.1007/BF02401743
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