Page 149 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts

RENT TOPICS IN COMPLEX ANALYSIS (MS-32)

Asymptotic estimates for one class of homeomorphisms

Bogdan Klishchuk, kban1988@gmail.com
Institute of Mathematics of National Academy of Sciences of Ukraine, Ukraine

Coauthor: Ruslan Salimov

Let Γ be a family of curves γ in Rn, n 2. A Borel measurable function ρ : Rn → [0, ∞] is
called admissible for Γ, (abbr. ρ ∈ adm Γ), if

ρ(x) ds 1

γ

for any curve γ ∈ Γ. Let p ∈ (1, ∞).
The quantity

Mp(Γ) = inf ρp(x) dm(x)
ρ∈adm Γ
Rn

is called p–modulus of the family Γ.
For arbitrary sets E, F and G of Rn we denote by ∆(E, F, G) a set of all continuous curves

γ : [a, b] → Rn that connect E and F in G, i.å., such that γ(a) ∈ E, γ(b) ∈ F and γ(t) ∈ G for
a < t < b.

Let D be a domain in Rn, n 2, x0 ∈ D and d0 = dist(x0, ∂D). Set

A(x0, r1, r2) = {x ∈ Rn : r1 < |x − x0| < r2} ,

Si = S(x0, ri) = {x ∈ Rn : |x − x0| = ri} , i = 1, 2 .

Let a function Q : D → [0, ∞] be Lebesgue measurable. We say that a homeomorphism
f : D → Rn is ring Q-homeomorphism with respect to p-modulus at x0 ∈ D if the relation

Mp(∆(f S1, f S2, f D)) Q(x) ηp(|x − x0|) dm(x)

A

holds for any ring A = A(x0, r1, r2) , 0 < r1 < r2 < d0, d0 = dist(x0, ∂D) and for any
measurable function η : (r1, r2) → [0, ∞] such that

r2

η(r) dr = 1 .

r1

Let

L(x0, f, R) = sup |f (x) − f (x0)| .

|x−x0| R

Theorem. Suppose that f : Rn → Rn is a ring Q-homeomorphism with respect to p-
modulus at a point x0 with p > n where x0 is some point in Rn and for some numbers c > 0,
κ p, r0 > 0 the condition

Q(x) ψp(|x − x0|) dm(x) c Iκ(r0, R) ∀ R > r0 ,

A(x0,r0,R)

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