On homeomorphisms with integral constraints acting in a domain with Poincaré inequalities

Abstract

We study homeomorphisms with one normalization condition that transform some domain of the Euclidean space onto a domain which is Ahlfors regular and supports Poincare inequality. We assume that, the homeomorphisms satisfy the weighted modulus Poletsky condition. If the majorant in this condition satisfies some relation written in terms of singular parameters, then the specified homeomorphisms satisfy the corresponding distortion estimate at the boundary points written in their terms. In more detail, the manuscript is written in the vein of research on mappings with bounded and finite distortion, which have been actively studied recently. Observe that, for mappings with direct and inverse Poletsky inequality, the estimates of the distortion under the mappings at the inner points of a domain may be considered known, since such estimates have been obtained by various authors over the last 10-15 years. We could, in particular, point to the results of V. Ryazanov, O. Martio, U. Srebro, E. Yakubov, M. Cristea, E. Sevost'yanov, S. Skvortsov, and O. Dovhopiatyi in this regard. It should be noted that, under certain additional assumptions, mappings with a direct Poletsky inequalities are included in the class of mappings with finite distortion by Iwaniec-Martin. At the same time, the above-mentioned estimates of distance distortion at the boundary points of the domain have not been sufficiently studied. Our manuscript is dedicated to obtaining such estimates. Of course, we require that the majorant in Poletsky's inequality satisfy certain requirements for obtaining them. The conditions written in terms of singular parameters are the most convenient and cover important partial cases. In particular, functions of finite and bounded mean oscillation by John-Nirenberg, as well as functions with the Lehto integral divergence condition can be described in terms of singular parameters, and this will be taken into account in our future research. In addition to singular parameters and the use of modulus techniques, the key point of our manuscript is spaces with Poincaré inequalities. Note that one of the characteristic features of such spaces is their fulfillment of Loewner-type inequalities, i.e., metric lower bounds of the modulus of families of curves through the diameter of the sets they join. This inequality is crucial for obtaining the result of the paper.