Differentiability of mappings of the Sobolev space $W^1_{n-1}$ with conditions on the distortion function

We define two scales of the mappings that depend on two real parameters $p$ and $q$, with $n-1\leq q\leq p<\infty$, as well as a weight function $\theta$. The case $q=p=n$ and $\theta\equiv1$ yields the well-known mappings with bounded distortion. The mappings of a two-index scale are applied to solve a series of problems of global analysis and applications. The main result of the article is the a.e. differentiability of mappings of two-index scales.