Smirnov–Sobolev spaces and their embeddings

Let $G$ be a bounded simply connected domain with rectifiable boundary $\partial G$ and assume that $0<p<\infty$. Let $E_p(G)$ be the Smirnov space of analytic functions $f$ in $G$. The Smirnov–Sobolev space $E_p^s(G)$, $s\in\mathbb N$, consists of analytic functions $f$ in $G$ such that $f^{(s)}\in E_p(G)$. If $G$ is a disc, then $E_p(G)$ is the Hardy space and $E_p^s(G)$ is the Hardy–Sobolev space. In that case the following Hardy–Littlewood embedding is known:
$$ E_\sigma^s(G)\subset E_p(G), \qquad 1/\sigma=s+1/p. $$

The author has recently extended this embedding to Smirnov–Sobolev spaces in Lavrent'ev domains. A further generalization of the Hardy–Littlewood embedding is obtained in the present paper. Namely, it is shown that such an embedding holds if the domain $G$ satisfies the following condition: for all points $\xi$ and $\eta$ in $\partial G$,
$$ |\Gamma(\xi,\eta)|\leqslant \chi\rho^+(\xi,\eta), \qquad \chi=\chi(G)\geqslant 1. $$

Here $|\Gamma(\xi,\eta)|$ is the length of the smallest of the two arcs of $\partial G$ connecting $\xi$ and $\eta$; $\rho^+(\xi,\eta)$ is the interior distance (with respect to $G$) between the points $\xi$ and $\eta$.