Generalized Homogeneous Littlewood–Paley g-Function on Some Function Spaces

Abstract

Let $({{\mathcal {X}}},d,\mu )$ 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 $\lambda$ satisfies weak reverse doubling conditions, in this paper, the authors prove that the generalized homogeneous Littlewood–Paley g-function ${\dot{g}}_{r} (r\in [2,\infty ))$ is bounded from the Lipschitz spaces ${\mathrm{Lip}}_{\beta }(\mu )$ into the Lipschitz spaces ${\mathrm{Lip}}_{\beta }(\mu )$ for $\beta \in (0,1)$, and the commutator ${\dot{g}}_{r,b}$ generated by the $b\in {\mathrm{Lip}}_{\beta }(\mu )$ and the ${\dot{g}}_{r}$ is bounded on the Lebesgue space $L^{p}(\mu )$ with $p\in (1,+\infty )$. Furthermore, the boundedness of the ${\dot{g}}_{r}$ and the commutator ${\dot{g}}_{r,b}$ on generalized Morrey space $L^{p,\phi }(\mu )$ is also obtained, respectively.