On Poletsky-type Modulus Inequalities for Some Classes of Mappings

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 capacitу of condensers.