Ambient Hardy--Littlewood Maximal Functions on Weighted Musielak--Orlicz Spaces over Domains

We study the ambient-domain Hardy--Littlewood maximal operator
\[
\mathcal M_\Omega f(x)
:=
\sup_{B\ni x}\frac1{|B|}\int_{B\cap\Omega}|f(y)|\,dy,
\qquad x\in\Omega
\]
on weighted Musielak--Orlicz spaces over a general open set \(\Omega\subset\mathbb R^n\), where the supremum is taken over all Euclidean balls \(B\subset\mathbb R^n\). For a Musielak--Orlicz function \(\varphi\), we use the pointwise lower Matuszewska--Orlicz index \(p_\varphi(\cdot)\) and the lower-index normalization
\[
\psi_\varphi(x,t)=\varphi(x,t)^{1/p_\varphi(x)}.
\]
This factorizes the modular as a weighted variable-exponent modular applied to \(\psi_\varphi(x,|f|)\). Under endpoint lower growth, normalized weighted generalized Orlicz \((A0)\)--\((A2)\) assumptions and an admissible whole-space extension hypothesis for the weight at the lower-index exponent, we prove the boundedness of
\[
\mathcal M_\Omega:L^{\varphi(\cdot)}_\omega(\Omega)\to L^{\varphi(\cdot)}_\omega(\Omega).
\]
For the converse direction we use the natural Köthe-associate ambient ball condition \(A_\varphi(\Omega)\). Boundedness of \(\mathcal M_\Omega\) implies \(\omega\in A_\varphi(\Omega)\).
On the whole space $\mathbb{R}^n$, this framework recovers the classical characterizations for both variable exponent and scalar Orlicz models.
As an application, we prove density of \(C_c^\infty(\mathbb R^n)\) in weighted Musielak--Orlicz--Sobolev spaces.