Function spaces and geometry of mappings

The Sobolev space $L_p^1(D), p \in[1, \infty)$, on the domain $D \subset \mathbb{R}^n, n \geq 2$, consists of locally integrable functions on $D$ having the first generalized derivatives that are summable to the power $p$. The seminorm of a function $v \in L_p^1(D)$ equals the norm in $L_p(D)$ of its generalized gradient $\nabla v$. If $\varphi: D \rightarrow D^{\prime}$ is a homeomorphism of two domains $D, D^{\prime} \subset \mathbb{R}^n$, then a natural question arises: under what conditions the composition operator $\varphi^*: L_p^1\left(D^{\prime}\right) \rightarrow$ $L_q^1(D), 1 \leq q \leq p<\infty$, where $u=\varphi^*(v)=v \circ \varphi$, will be bounded. We get a more general problem if instead of the space $L_p^1\left(D^{\prime}\right)$ we shall consider the weighted Sobolev space $L_p^1\left(D^{\prime}, \omega\right)$, where $\omega$ is a locally summable weight function. We present a solution to the problem in a generalized setting and show that for some particular summability exponents $q$ and $p$, and particular weight function $\omega$ the resulting classes of mappings coincide with the mappings studied in earlier papers.
Within the framework of the generalized theory, we obtained results that are new even for the classical theory of quasiconformal mappings. For example, the norm of the composition operator $\varphi^*: L_n^1\left(D^{\prime}\right) \rightarrow L_n^1(D)$ is equal to $K^{1 / n}$, where $K^{1 / n}$, где $K=\underset{x \in D}{\operatorname{ess} \sup } \frac{|D \varphi(x)|}{|\operatorname{det} D \varphi(x)|^{1 / n}}$ is the quasi-conformity coefficient.
It will also be shown new questions appearing in the study of the connection "Sobolev space — the geometry of mappings" on a special class of nilpotent Lee groups: Carnot groups. Here we define a new object on Carnot groups: mappings on Carnot groups of Sobolev class $W_{\nu, \text{loc}}^1(D)$ with finite distortion. These are mappings $D \rightarrow \mathbb{G}$ of the Sobolev class $\varphi: W_{\nu, \text{loc}}^1(D)$, where $D$ is a domain on the Carnot group $\mathbb{G}$, with non-negative Jacobian such that $D \varphi(x)=0$ a.e. on the set of zeros of the Jacobian (here $\nu$ is the Hausdorff dimension of the group $\mathbb{G}$ ). It is established that such mappings are continuous, $\mathcal{P}$-differentiable almost everywhere, and have the Luzin property $\mathcal{N}$.