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DOI: 10.46698/w5793-5981-8894-o

On Poletsky-type Modulus Inequalities for Some Classes of Mappings

Vodopyanov, S. K.
Vladikavkaz Mathematical Journal 2022. Vol. 24. Issue 4.
Abstract:
It is well-known that the theory of mappings with bounded distortion was laid by Yu. G. Reshetnyak in 60-th of the last century [1]. In papers [2, 3], there was introduced the two-index scale of mappings with weighted bounded \((q, p)\)-distortion. This scale of mappings includes, in particular, mappings with bounded distortion mentioned above (under \(q=p=n\) and the trivial weight function). In paper [4], for the two-index scale of mappings with weighted bounded \((q, p)\)-distortion, the Poletsky-type modulus inequality was proved under minimal regularity; many examples of mappings were given to which the results of [4] can be applied. In this paper we show how to apply results of [4] to one such class. Another goal of this paper is to exhibit a new class of mappings in which Poletsky-type modulus inequalities is valid. To this end, for \(n=2\), we extend the validity of the assertions in [4] to the limiting exponents of summability: \(1 < q\leq p \leq \infty\). This generalization contains, as a special case, the results of recently published papers. As a consequence of our results, we also obtain estimates for the change in capacit of condensers.
Keywords: quasiconformal analysis, Sobolev space, modulus of a family of curves, modulus estimate
Language: English Download the full text  
For citation: Vodopyanov, S. K. On Poletsky-type Modulus Inequalities for Some Classes of Mappings, Vladikavkaz Math. J., 2022, vol. 24, no. 4, pp. 58-69. DOI 10.46698/w5793-5981-8894-o
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