LAUSR

Let $$({{\mathcal {X}}},d,\mu )$$ ( X , d , μ ) be a non-homogeneous metric measure space satisfying the so-called upper doubling and the geometrically doubling conditions in the sense of Hytönen. Under the assumption that the dominating function $$\lambda $$ λ satisfies weak reverse doubling conditions, in this paper, the authors prove that the generalized homogeneous Littlewood–Paley g -function $${\dot{g}}_{r} (r\in [2,\infty ))$$ g ˙ r ( r ∈ [ 2 , ∞ ) ) is bounded from the Lipschitz spaces $${\mathrm{Lip}}_{\beta }(\mu )$$ Lip β ( μ ) into the Lipschitz spaces $${\mathrm{Lip}}_{\beta }(\mu )$$ Lip β ( μ ) for $$\beta \in (0,1)$$ β ∈ ( 0 , 1 ) , and the commutator $${\dot{g}}_{r,b}$$ g ˙ r , b generated by the $$b\in {\mathrm{Lip}}_{\beta }(\mu )$$ b ∈ Lip β ( μ ) and the $${\dot{g}}_{r}$$ g ˙ r is bounded on the Lebesgue space $$L^{p}(\mu )$$ L p ( μ ) with $$p\in (1,+\infty )$$ p ∈ ( 1 , + ∞ ) . Furthermore, the boundedness of the $${\dot{g}}_{r}$$ g ˙ r and the commutator $${\dot{g}}_{r,b}$$ g ˙ r , b on generalized Morrey space $$L^{p,\phi }(\mu )$$ L p , ϕ ( μ ) is also obtained, respectively.