Boundedness of fractional integral operators on Musielak–Orlicz Hardy spaces

Let α∈(0,n)$\alpha \in (0,n)$ and let φ1,φ2:Rn×[0,∞)→[0,∞)$\varphi _1,\varphi _2:\mathbb {R}^n\times [0,\infty )\rightarrow [0,\infty )$ be Musielak–Orlicz functions such that φ1(x,·)$\varphi _1(x,\cdot )$, φ2(x,·)$\varphi _2(x,\cdot )$ are Orlicz functions and φ1(·,t)$\varphi _1(\cdot ,t)$, φ2(·,t)$\varphi _2(\cdot ,t)$ are Muckenhoupt A∞(Rn)$A_\infty (\mathbb {R}^n)$ weights. In this paper, we give the necessary and sufficient condition for the boundedness of the fractional integral operator Iα$I_\alpha$ from the Musielak–Orlicz Hardy space Hφ1(Rn)$H^{\varphi _1}(\mathbb {R}^n)$ into the Musielak–Orlicz Hardy space Hφ2(Rn)$H^{\varphi _2}(\mathbb {R}^n)$. We also give the necessary and sufficient condition for the boundedness of Iα$I_\alpha$ from Hφ1(Rn)$H^{\varphi _1}(\mathbb {R}^n)$ into the Musielak–Orlicz space Lφ2(Rn)$L^{\varphi _2}(\mathbb {R}^n)$. Our results generalize the main results in [5, 20, 21].