TY - JOUR

T1 - Mollifications of Contact Mappings of Engel Group

AU - Басалаев, Сергей Геннадьевич

PY - 2023

Y1 - 2023

N2 - The contact mappings belonging to the metric Sobolev classes are studied on an Engel group with a left-invariant sub-Riemannian metric. In the Euclidean space one of the main methods to handle non-smooth mappings is the mollification, i.e., the convolution with a smooth kernel. An extra difficulty arising with contact mappings of Carnot groups is that the mollification of a contact mapping is usually not contact. Nevertheless, in the case considered it is possible to estimate the magnitude of deviation of contactness sufficiently to obtain useful results. We obtain estimates on convergence (or sometimes divergence) of the components of the differential of the mollified mapping to the corresponding components of the Pansu differential of the contact mapping. As an application to the quasiconformal analysis, we present alternative proofs of the convergence of mollified horizontal exterior forms and the commutativity of the pull-back of the exterior form by the Pansu differential with the exterior differential in the weak sense. These results in turn allow us to obtain such basic properties of mappings with bounded distortion as Holder continuity, differentiability almost everywhere in the sense of Pansu, Luzin N-property.

AB - The contact mappings belonging to the metric Sobolev classes are studied on an Engel group with a left-invariant sub-Riemannian metric. In the Euclidean space one of the main methods to handle non-smooth mappings is the mollification, i.e., the convolution with a smooth kernel. An extra difficulty arising with contact mappings of Carnot groups is that the mollification of a contact mapping is usually not contact. Nevertheless, in the case considered it is possible to estimate the magnitude of deviation of contactness sufficiently to obtain useful results. We obtain estimates on convergence (or sometimes divergence) of the components of the differential of the mollified mapping to the corresponding components of the Pansu differential of the contact mapping. As an application to the quasiconformal analysis, we present alternative proofs of the convergence of mollified horizontal exterior forms and the commutativity of the pull-back of the exterior form by the Pansu differential with the exterior differential in the weak sense. These results in turn allow us to obtain such basic properties of mappings with bounded distortion as Holder continuity, differentiability almost everywhere in the sense of Pansu, Luzin N-property.

KW - CARNOT GROUP

KW - ENGEL GROUP

KW - QUASICONFORMAL MAPPINGS

KW - BOUNDED DISTORTION

KW - quasiconformal mappings

KW - bounded distortion

KW - Engel group

KW - Carnot group

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85154600593&origin=inward&txGid=1ec9a1c9e38bcf33ac9911c93ee8d4ed

UR - https://elibrary.ru/item.asp?id=50430411

UR - https://www.mendeley.com/catalogue/eb59bd85-a129-379e-8957-1f5efe378af6/

U2 - 10.46698/n0927-3994-6949-u

DO - 10.46698/n0927-3994-6949-u

M3 - Article

VL - 25

SP - 5

EP - 19

JO - Владикавказский математический журнал

JF - Владикавказский математический журнал

SN - 1683-3414

IS - 1

M1 - 1

ER -