On a power-type estimate of the area of the image of a disk
DOI:
https://doi.org/10.37069/3154-8229-2026-40-5Keywords:
modulus of a family of paths, \(Q\)-homeomorphism, ring \(Q\)-homeomorphism, finite mean \(l\)-oscillationAbstract
The problem of area distortion under quasiconformal mappings originates from the work of B. Boyarskii, see [1]. A number of results in this direction were obtained in [2–4]. An upper estimate for the area of the image of a disk under quasiconformal mappings first appeared in the monograph by M.O. Lavrentiev, see [5]. In the monograph [6] (see Theorem 3.7), a refinement of Lavrentiev's inequality in terms of angular dilatation is presented. In this paper, classes of ring \(Q\)-homeomorphisms in the complex plane are studied. Under the assumption that the function \(Q\) has finite mean \(l\)-oscillation at a point, upper estimates for the area of the image of a disk are obtained. In addition, asymptotic estimates of power-type growth of ring \(Q\)-homeomorphisms are established in terms of the limit inferior. Consequences of these results for Lebesgue points of the function \(Q\) are also derived. To illustrate the obtained results, examples confirming their effectiveness are provided. The proofs of the main results are based on auxiliary statements, in particular, on lemmas concerning the estimation of a singular integral for functions with finite mean \(l\)-oscillation and an upper estimate for the modulus of a family of curves connecting the boundary components of the image of a ring.
References
Bojarski, B.V. (1955). Homeomorphic solutions of Beltrami systems. Doklady Akademii Nauk SSSR, 102, 661–664. [In Russian].
Gehring, F. W., & Reich, E. (1966). Area distortion under quasiconformal mappings. Annales Academiae Scientiarum Fennicae. Series A I Mathematica, 388, 1–15. https://doi.org/10.5186/aasfm.1966.388
Astala, K. (1994). Area distortion of quasiconformal mappings. Acta Mathematica, 173, 37–60. https://doi.org/10.1007/BF02392568
Eremenko, A., & Hamilton, D. H. (1995). On the area distortion by quasiconformal mappings. Proceedings of the American Mathematical Society, 123, 2793–2797. https://doi.org/10.1090/S0002-9939-1995-1283548-8
Lavrentiev, M. A. (1962). Variational method in boundary value problems for systems of elliptic equations. AS USSR. [In Russian].
Bojarski, B., Gutlyanskii, V., Martio, O., & Ryazanov, V. (2013). Infinitesimal geometry of quasiconformal and bi-Lipschitz mappings in the plane (EMS Tracts in Mathematics, Vol. 19). European Mathematical Society. https://doi.org/10.4171/122
Väisälä, J. (1971). Lectures on n-dimensional quasiconformal mappings (Lecture Notes in Mathematics, Vol. 229). Springer. https://doi.org/10.1007/BFb0061216
Martio, O., Ryazanov, V., Srebro, U., & Yakubov, E. (2004). Mappings with finite length distortion. Journal d'Analyse Mathématique, 93, 215–236. https://doi.org/10.1007/BF02789308
Martio, O., Ryazanov, V., Srebro, U., & Yakubov, E. (2004). Q-homeomorphisms. Contemporary Mathematics, 364, 193–203. https://doi.org/10.1090/conm/364
Martio, O., Ryazanov, V., Srebro, U., & Yakubov, E. (2005). On Q-homeomorphisms. Annales Academiae Scientiarum Fennicae Mathematica, 30, 49–69.
Ryazanov, V., Srebro, U., & Yakubov, E. (2001). BMO-quasiconformal mappings. Journal d'Analyse Mathématique, 83, 1–20. https://doi.org/10.1007/BF02790254
Ignat'ev, A., & Ryazanov, V. (2005). Finite mean oscillation in the mapping theory. Ukrainian Mathematical Bulletin, 2(3), 403–424.
Ignat'ev, A., & Ryazanov, V. (2006). To the theory of the boundary behavior of space mappings. Ukrainian Mathematical Bulletin, 3(2), 189–201.
Ryazanov, V., Srebro, U., & Yakubov, E. (2005). On ring solutions of the Beltrami equation. Journal d'Analyse Mathématique, 96, 117–150. https://doi.org/10.1007/BF02787826
Ryazanov, V., Srebro, U., & Yakubov, E. (2006). Finite mean oscillation and the Beltrami equation. Israel Journal of Mathematics, 153, 247–266. https://doi.org/10.1007/BF02771785
Ryazanov, V., Srebro, U., & Yakubov, E. (2006). On the theory of the Beltrami equation. Ukrainian Mathematical Journal, 58(11), 1786–1798. https://doi.org/10.1007/s11253-006-0168-4
Ryazanov, V., Srebro, U., & Yakubov, E. (2007). The Beltrami equation and ring homeomorphisms. Ukrainian Mathematical Bulletin, 4(1), 79–115.
Gehring, F. W. (1962). Rings and quasiconformal mappings in space. Transactions of the American Mathematical Society, 103, 353–393. https://doi.org/10.1090/S0002-9947-1962-0139735-8
Ryazanov, V., Srebro, U., & Yakubov, E. (2003). Degenerate Beltrami equation and radial Q-homeomorphisms (Reports in Mathematics No. 369). University of Helsinki.
Martio, O., Rickman, S., & Väisälä, J. (1969). Definitions for quasiregular mappings. Annales Academiae Scientiarum Fennicae. Series A I Mathematica, 448, 1–40. https://doi.org/10.5186/aasfm.1969.448
Shlyk, V. A. (1993). The equality between p-capacity and p-modulus. Siberian Mathematical Journal, 34(6), 216–221. https://doi.org/10.1007/BF00973485
Maz'ya, V. (2003). Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces. Contemporary Mathematics, 338, 307–340. https://doi.org/10.1090/conm/338
Ryazanov, V., Srebro, U., & Yakubov, E. (1997). BMO-quasiconformal mappings (Reports of the Department of Mathematics, University of Helsinki, No. 155). University of Helsinki.
Martio, O., Ryazanov, V., Srebro, U., & Yakubov, E. (2001). BMO-quasiconformal mappings and Q-homeomorphisms in space (Reports of the Department of Mathematics, University of Helsinki, No. 288). University of Helsinki.
Martio, O., Ryazanov, V., Srebro, U., & Yakubov, E. (2002). On boundary behavior of Q-homeomorphisms (Reports of the Department of Mathematics, University of Helsinki, No. 318). University of Helsinki.
Ryazanov, V., & Salimov, R. (2007). Weakly flat spaces and boundaries in the mapping theory. Ukrainian Mathematical Bulletin, 4(2), 199–233.
Ryazanov, V., Srebro, U., & Yakubov, E. (2005). The Beltrami equation and FMO functions. Contemporary Mathematics, 382, 357–364. https://doi.org/10.1090/conm/382
Ikoma, K. (1965). On the distortion and correspondence under quasiconformal mappings in space. Nagoya Mathematical Journal, 25, 175–203. https://doi.org/10.1017/S0027763000011521
Astala, K., Iwaniec, T., & Martin, G. (2009). Elliptic partial differential equations and quasiconformal mappings in the plane (Princeton Mathematical Series, Vol. 48). Princeton University Press.
Salimov, R. (2010). On regular homeomorphisms in the plane. Annales Fennici Mathematici, 35(1), 285–289. https://doi.org/10.5186/aasfm.2010.3517
