Under a condition of the Calderon type on $\varphi$, we show that a homeomorphism $f$ of finite distortion in $W^{ 1,\varphi}_{ \rm loc}$ and, in particular, $f\in W^{ 1,q}_{ \rm loc}$ for $q>n-1$ in ${ \Bbb R}^n$, $n\ge 3$, is a lower $Q$-homeomorphisms with respect to the $p$-modulus with $\left[ K_{ I,\alpha}(x,f)\right]^{ \beta}$, $p>n-1$ and a ring $Q_{ *}$-homeomorphism with respect to the $\frac{ p}{ p-n+1}$-modulus with $Q_{ *}(x)=K_{ I,\alpha}(x,f)$ where $ K_{ I,\alpha}(x,f)$ is its inner $\alpha$-dilatation and $\alpha=\frac{ p}{ p-n+1}$, $\beta=\frac{ p-n+1}{ n-1}$.
Categories: math.CV