Estimates of Moduli of Curve Families for Mappings with Weighted Bounded \((p,q)\)-Distortion

For citation: Tryamkin M. V. Estimates of Moduli of Curve Families for Mappings with Weighted Bounded \((p,q)\)-Distortion. Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol.17, no. 3, pp.65-74. DOI 10.23671/VNC.2017.3.7266

1. Ahlfors L. and Beurling A. Conformal invariants and
function-theoretic null-sets. Acta Math., 1950, vol. 83, pp.
101-129.

2. Fuglede B. Extremal length and functional completion. Acta Math.,
1957, vol. 98, pp. 171-219.

3. Shabat B. V. The modulus method in space. Dokl. Akad. Nauk SSSR,
1960, vol. 130, pp. 1210-1213; English transl.: Soviet Math. Dokl.,
1960, vol. 1, pp. 165-168.

4. Reshetnyak Yu. G. Space Mappings with Bounded Distortion,
Providence (R.I.), American Math. Soc., 1989 (Translation of Math.
Monogr.; vol. 73).

5. Poletskii E. A. The modulus method for nonhomeomorphic
quasiconformal mappings. Mat. Sb., 1970, vol. 83 (125), pp. 261-272;
English transl.: Math. USSR Sb., 1970, vol. 12.

6. Vaisala J. Modulus and capacity inequalities for quasiregular
mappings. Ann. Acad. Sci. Fenn. Ser. A I, 1972, vol. 509, pp. 1-14.

7. Martio O., Rickman S. and Vaisala J. Definitions for quasiregular
mappings. Ann. Acad. Sci. Fenn. Ser. A I, 1969, vol. 448, pp. 5-40.

8.Martio O. A capacity inequality for quasiregular mappings. Ann.
Sc. Fenn. Ser. A I, 1970, vol. 474, pp. 1-18.

9. Poletskii E. A. On the removal of singularities of quasiconformal
mappings. Mat. Sb., 1973, vol. 92 (134), pp. 242-256; English
transl.: Math. USSR Sb., 1973, vol. 21.

10. Rickman S. Quasiregular Mappings, Berlin a. o., Springer-Verlag,
1993.

11. Heinonen J. and Koskela P. Quasiconformal maps in metric spaces
with controlled geometry. Acta Math., 1998, vol. 181, pp. 1-41.

12.  Martio O., Ryazanov V., Srebro U. and Yakubov E. Moduli in
Modern Mapping Theory, N.Y., Springer, 2009.

13.  Troyanov M. and Vodop'yanov S. K. Liouville type theorems for
mappings with bounded (co)-distortion. Ann. Inst. Fourier, 2002,
vol. 52, no. 6, pp. 1753-1784.

14.  Vodop'yanov S. K. and Ukhlov A. Mappings with bounded
(P,Q)-distortion on Carnot groups. Bull. Sci. Math., 2010, vol. 134,
no. 6, pp. 605-634.

15.  Koskela P. and Onninen J. Mappings of finite distortion:
Capacity and modulus inequalities. J. Reine Angew. Math., 2006, vol.
599, pp. 1-26.

16.  Salimov R. and Sevost'yanov E. The Poletskii and Vaisala
inequalities for the mappings with (p,q)-distortion. Complex
Variables and Elliptic Equations: An International J., 2014, vol.
59, no. 2, pp. 217-231.

17.  Sevost'yanov E. About One Modulus Inequality of the Vaisala
Type, 2013, URL: http://arxiv.org/abs/1204.3810v4 (Preprint).

18.  Baykin A. N. and Vodop'yanov S. K. Capacity estimates,
Liouville's theorem, and singularity removal for mappings with
bounded (p,q)-distortion. Siberian Math. J., 2015, vol. 56, no. 2,
pp. 237-261.

19.  Vodop'yanov S. K. On the regularity of the Poletskii function
under weak analytic assumptions on the given mapping. Dokl. Ross.
Akad. Nauk., 2014, vol. 455, no. 2, pp. 130-134; English transl.:
Dokl. Math., 2014, vol. 89, no. 2, pp. 157-161.

20.  Vodop'yanov S. K. On regularity of mappings inverse to Sobolev
mappings. Dokl. Ross. Akad. Nauk., 2008, vol. 423, no. 5, pp.
592-596; English transl.: Dokl. Math., 2008, vol. 78, no. 3, pp.
891-895.

21.  Vodop'yanov S. K. Mappings of finite distortion and Sobolev
classes of functions. Dokl. Ross. Akad. Nauk, 2011, vol. 440, no. 3,
pp. 301-305; English transl.: Dokl. Math., 2011, vol. 84, no. 2, pp.
640-644.

22.  Vodop'yanov S.  K. Regularity of mappings inverse to Sobolev
mappings. Mathematics, 2012, vol. 203, no. 10, pp. 1383-1410.

23.  Ponomarev S. P. On the N-property of homeomorphisms of the
class W^1_p. Sibirsk. Mat. Zh., 1987, vol. 28, no. 2, pp. 140-148.

24.  Tryamkin M. V. Modulus inequalities for mappings with weighted
bounded (p,q)-distortion. Siberian Math. J., 2015, vol. 56, no. 6
(to appear).

25.  Vaisala J. Lectures on n-Dimensional Quasiconformal Mappings,
Berlin, Springer-Verlag, 1971.