\begin{align*}
W^{m ,\Phi}(\Omega)=
\Bigl\{
& u\in L^{\Phi}(\Omega);
  \text{ there exists } \{g_{\alpha}\}\in L^{\Phi}(\Omega),
  \alpha=(\alpha_1,\dots,\alpha_N), \text{ such that}
\\
&\int_{\Omega}
  u\frac{\partial^{\alpha}\varphi}
        {\partial^{\alpha_1}x_1\dots\partial^{\alpha_N}x_N}\,dx
=(-1)^{\|\alpha\|_{S}}\int_{\Omega}g_{\alpha}\varphi \,dx,
\\
&\text{for any } \varphi\in C^{\infty}_0(\Omega)
\text{ and } \|\alpha\|_{S}\leq m \Bigr\}
\end{align*}

We denote by $W^{m ,\Phi}(\Omega)$ the set consisting
of all $u\in L^{\Phi}(\Omega)$ for which there exists
$\{g_{\alpha}\}\in L^{\Phi}(\Omega)$, with
$\alpha=(\alpha_1,\dots,\alpha_N)$, such that
\[
\int_{\Omega}
  u\frac{\partial^{\alpha}\varphi}
        {\partial^{\alpha_1}x_1\dots\partial^{\alpha_N}x_N}\,dx
=(-1)^{\|\alpha\|_{S}}\int_{\Omega}g_{\alpha}\varphi \,dx,
\]
for any $\varphi\in C^{\infty}_0(\Omega)$ and $\|\alpha\|_{S}\leq m$.