×

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.

MSC:

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.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.