2026/07/15 by Tan Duc Do
#math.FA
We study the ambient-domain Hardy--Littlewood maximal operator \mathcal MΩf(x) := supB\ni x\frac1|B|∫B∩Ω|f(y)| dy, x∈Ω on weighted Musielak--Orlicz spaces over a general open set \(Ω⊂\mathbb Rn\), where the supremum is taken over all Euclidean balls \(B⊂\mathbb Rn\). For a Musielak--Orlicz function \(φ\), we use the pointwise lower Matuszewska--Orlicz index \(pφ(⋅)\) and the lower-index normalization ψφ(x,t)=φ(x,t)1/pφ(x). This factorizes the modular as a weighted variable-exponent modular applied to \(ψφ(x,|f|)\). 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 MΩ:Lφ(⋅)ω(Ω)→ Lφ(⋅)ω(Ω). For the converse direction we use the natural Köthe-associate ambient ball condition \(Aφ(Ω)\). Under the local characteristic-function hypothesis, boundedness of \(\mathcal MΩ\) implies \(ω∈ Aφ(Ω)\). On the whole space \(\mathbb Rn\), 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 \(tp+a(x)tq\). As an application, we prove density of \(Cc^∞(\mathbb Rn)\) in weighted Musielak--Orlicz--Sobolev spaces.