Weighted estimates for diffeomorphic extensions of homeomorphisms

Let $\mathrm{\Omega }\subset {\text{mbr}}^2$ be an internal chord-arc domain and $\varphi : \mbs^1 \rightarrow \partial \Omega$ be a homeomorphism. Then there is a diffeomorphic extension $h:\text{mbd}\to \mathrm{\Omega }$ of $\phi \mathrm{.}$ We study the relationship between weighted integrability of the derivatives of $h$ and double integrals of $\phi$ and of ${\phi }^{-1}\mathrm{.}$