Analytic version of critical $Q$ spaces and their properties

In this paper, we establish an analytic version of critical spaces $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$ on unit disc $\mathbb{D}$, denoted by $Q^{\beta}_{p}(\mathbb{D})$. Further we prove a relation between $Q^{\beta}_{p}(\mathbb{D})$ and Morrey spaces. By the boundedness of two integral operators, we give the multiplier spaces of $Q^{\beta}_{p}(\mathbb{D})$.