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

Abstract

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 K\"othe-associate ambient ball condition \(A_\varphi(\Omega)\). Under the local characteristic-function hypothesis, boundedness of \(\mathcal M_\Omega\) implies \(\omega\in A_\varphi(\Omega)\). 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 \(C_c^\infty(\mathbb R^n)\) in weighted Musielak--Orlicz--Sobolev spaces.