До теорії модулів поверхонь

V. Ryazanov; Ye. Sevost’ yanov

doi:10.3842/umzh.v75i9.7651

V. Ryazanov Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine, Slovyansk, Donetsk region
Ye. Sevost’ yanov Ivan Franko Zhytomyr State University; Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine, Slovyansk, Donetsk region

DOI: https://doi.org/10.3842/umzh.v75i9.7651

Keywords: Moduli of surfaces in Euclidean spaces, Hausdorff measures, measurable (Borel) hulls of sets

Abstract

UDC 517.5

We continue the development of the theory of moduli of the families of surfaces, in particular, strings of various dimensions $m=1,2,\ldots,n-1$ in Euclidean spaces $\mathbb{R}^n,$ $n\geq 2.$ On the basis of the proof of Lemma 1 on the relationships between the moduli and Lebesgue measures, we prove the corresponding analog of the Fubini theorem in terms of moduli that extends the known Väisälä theorem for families of curves to the families of surfaces of arbitrary dimensions. It should be emphasized that the crucial place in the proof of Lemma 1 is Proposition 1 on measurable (Borel) hulls of sets in Euclidean spaces. In addition, we prove similar Lemma 2 and Proposition 2 for the families of concenteric spheres.

References

B. T. Rushing, Topological embeddings, Pure and Appl. Math., vol. 52, Academic Press, New York and London (1973).

L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Math., CRC Press, Boca Raton, FL (1992).

H. Federer, Geometric measure theory, Springer-Verlag, Berlin (1969).

S. Saks, Theory of the integral, Dover, New York (1964).

T. Rado, P. V. Reichelderfer, Continuous trasformations in analysis, Springer-Verlag, Berlin (1955). DOI: https://doi.org/10.1007/978-3-642-85989-2

B. Fuglede, Extremal length and functional completion, Acta Math., 98, 171–219 (1957). DOI: https://doi.org/10.1007/BF02404474

J. Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lect. Notes Math., vol. 229, Springer-Verlag, Berlin etc. (1971). DOI: https://doi.org/10.1007/BFb0061216

D. Kovtonyk, V. Ryazanov, On the theory of mappings with finite area distortion, J. Anal. Math., 104, 291–306 (2008). DOI: https://doi.org/10.1007/s11854-008-0025-5

O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in modern mapping theory, Springer Sci.- Business Media, LLC, New York (2009).

O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Mappings with finite length distortion, J. Anal. Math., 93, 215–236 (2004). DOI: https://doi.org/10.1007/BF02789308

B. Bojarski, V. Gutlyanskii, O. Martio, V. Ryazanov, Infinitesimal geometry of quasiconformal and bi-Lipschitz mappings in the plane, EMS Tracts Math., vol. 19, Eur. Math. Soc., Zürich (2013). DOI: https://doi.org/10.4171/122

V. Gutlyanskii, V. Ryazanov, U. Srebro, E. Yakubov, The Beltrami equation. A geometric approach, Dev. Math., 26, Springer, New York (2012). DOI: https://doi.org/10.1007/978-1-4614-3191-6

E. A. Sevost'yanov, On the boundary and global behavior of mappings of Riemannian surfaces, Filomat, 36, 1295–1327 (2022). DOI: https://doi.org/10.2298/FIL2204295S

P. R. Halmos, Measure theory, Springer, New York etc. (1974).

B. R. Gelbaum, J. M. H. Olmsted, Counterexamples in analysis, Corrected reprint of the second (1965) edition, Dover Publ., Inc. Mineola, NY (2003).

K. Kuratowski, Topology, vol. 1, Acad. Press, New York, London (1968). DOI: https://doi.org/10.1016/B978-0-12-429202-4.50005-4

Д. Ковтонюк, Р. Салимов, Е. Севостьянов, К теории отображений классов Соболева и Орлича–Соболева, Наук. думка, Киев (2013).

E. A. Sevost'yanov, On the local and boundary behaviour of mappings of factor spaces, Complex Var. and Elliptic Equat., 67, № 2, 284–314 (2022). DOI: https://doi.org/10.1080/17476933.2020.1825392

On the theory of moduli of surfaces

Abstract

References