Abstract
We prove Poletskii-type moduli inequalities for the two-index scale of weighted bounded -distortion under minimal regularity. This implies, in particular, a positive solution to a question formulated in a Tengval's paper on the validity of Poletskii-type moduli inequalities for nonspherical condensers, for mappings of Sobolev classes with the least possible summability exponent.
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Acknowledgments
I greatly appreciate the anonymous reviewers for critically reading and comments, which helped improve and clarify the initial manuscript. The author thanks also an anonymous reviewer for his recommendation to add papers [Citation22,Citation23,Citation25–27] in the list of references.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notes
1 In [Citation2], there is also some improvement in this inequality in normal domains, see [Citation2, Theorem 2]. The useful interpretation of the latter found by Väisälä [Citation7, 3.1] is now called Väisälä's inequality in the literature. A little earlier, similar estimates were established for capacity, see [Citation8–10].
2 Let where
is a measurable set. We say that f enjoys Luzin's
-property if
,
, implies
.
3 Here is the area factor [Citation44, Section 3.2, Theorems 3, 4; Section 3.3, Theorem 1] of the Jacobi matrix
,
at m = n; and
is the Banach indicatrix.
4 Ahlfors and Beurling [Citation50] introduced the concept of modulus of a family of curves on the plane in 1950. Fuglede [Citation47] and Shabat [Citation51] extended it to higher-dimensional spaces. The modulus of a family of curves has been applied not only to obtain an equivalent geometric description of quasiconformal mappings, but also to develop a method for studying the properties of this class of mappings.
5 The formulas below do the job at almost all points provided conditions
(
) is a point of differentiability of
(
),
is a Lebesgue point for both integrals under consideration, and
. The derivative
can be found by the chain rule similar to (Equation45
(46)
(46) ).
6 By the condition of Theorem 4.3 we conclude by [Citation5, Theorem 26] that the given mapping
is differentiable a.e. Thence from the very beginning the mapping
is differentiable at all points of
,
, and its differential
is non-degenerate. Therefore, f is locally homeomorphic around x and
.