This talk continues our research of the generic properties of mappings with integrallybounded distortions

Golberg, А.

Salіmov, R. R.

Sevost’yanov, Е. А.

Zhytomyr State University Library

ON REMOVABILITY OF SINGULARITIES FOR DISCRETE OPENMAPPINGS WITH CONTROLLED p-MODULEANATOLY GOLBERG, RUSLAN SALIMOV AND EVGENY SEVOST'YANOVThis talk continues our research of the generic properties of mappings with integrallybounded distortions. We consider discrete open mappings f : D ! Rn, n  2, of domainsD  Rn, satisfying the inequality controlling the distortion of p-module (p > 1) byMp(f( (S1; S2; A))) ZAQ(x) p(jx  x0j) dm(x); (1)when A  D are spherical rings A = A(r1; r2; x0) = fx 2 D : r1 < jx   x0j < r2g,0 < r1 < r2 < r0 := dist (x0; @D), and  is arbitrary measurable function  : (r1; r2)! [0;1]such thatr2Zr1(r)dr  1;while Q : D ! [0;1] in (1) is a given measurable function. The point x0 is xed in D.The purpose of the talk is to discuss the removability of isolated boundary points whenthe mappings are subject to (1). It was established by the third author, under rathergeneral assumptions, that if a discrete open mapping f : D n fx0g ! Rn; n  2; omitsthe values which range over some set of positive p-capacity and obeys (1), then f has a(nite or innite) limit at x0 when x approaches x0. In this case, one has to assume thatcapp Rn n f(D n fx0g)> 0.We are focussed to the case n   1 < p < n; and show, rst, that the same removabil-ity phenomenon holds also for such p, i.e. an essential singularity also cannot occur, andsecondly, that the assumption capp Rn n f(D n fx0g)> 0 now can be omitted. This factreveals an essential dierence between the cases n   1 < p < n and p = n. In the lastcase, for example, the holomorphic mapping f(z) = e1=z; z 2 C; has nonremovable singu-larity at the origin and satises the inequality (1) at every point of the complex plane withQ(x)  1. So we establish a removability of singularities without an expected requirementcapp Rn n f(D n fx0g)> 0:1

Onremovability of singularities for discrete open mappings with controlled p-Module

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