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

We study the ambient-domain Hardy–Littlewood maximal operator $${\mathcal{M}}_{\mathrm{\Omega }}f(x):=\underset{B\ni x}{\sup }\frac{1}{\mathrm{\mid }B\mathrm{\mid }}{\int }_{B\cap \mathrm{\Omega }}\mathrm{\mid }f(y)\mathrm{\mid }dy,x\in \mathrm{\Omega }$$ on weighted Musielak–Orlicz spaces over a general open set $\mathrm{\Omega }\subset {\mathbb{R}}^n$, where the supremum is taken over all Euclidean balls $B\subset {\mathbb{R}}^n$. For a Musielak–Orlicz function $\phi$, we use the pointwise lower Matuszewska–Orlicz index $p_{\phi }(\cdot )$ and the lower-index normalization $${\psi }_{\phi }(x,t)=\phi (x,t{)}^{1\mathrm{/}{p}_{\phi }(x)}\mathrm{.}$$ This factorizes the modular as a weighted variable-exponent modular applied to ${\psi }_{\phi }(x,\mathrm{\mid }f\mathrm{\mid })$. 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 $${\mathcal{M}}_{\mathrm{\Omega }}:L_{\omega }^{\phi (\cdot )}(\mathrm{\Omega })\to L_{\omega }^{\phi (\cdot )}(\mathrm{\Omega })\mathrm{.}$$ For the converse direction we use the natural Köthe-associate ambient ball condition $A_{\phi }(\mathrm{\Omega })$. Under the local characteristic-function hypothesis, boundedness of ${\mathcal{M}}_{\mathrm{\Omega }}$ implies $\omega \in A_{\phi }(\mathrm{\Omega })$. On the whole space ${\mathbb{R}}^n$, this framework provides a weighted characterization conditional on an associate-to-lower-index product reduction. In the present paper this reduction is verified for uniformly lower-index-power-equivalent models; it remains open for genuinely two-phase growth such as $t^p+a(x)t^q$. As an application, we prove density of $C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ in weighted Musielak–Orlicz–Sobolev spaces.