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 $Q^{beta}_{p}(mathbb{D})$.