Bicomplex Bloch and little Bloch Spaces

Throughout this paper, we will denote the set of bicomlplex numbers as $ \mathbb{BC}. $ In fact, the theory of bicomplex holomorphic functions has enhanced its development, see \cite{alpay2014,colombo13,elizarraras2015,Lu-Sh-Va12, reyes2019, segre1892} and the references therein. The classical theory of holomorphic functions, where we deal with the unit disk and here we deal with the bidisk. Let $ \mathbb{U}_{\mathbb{BC}} = \D \times \D$ denote the bidisk in $ \mathbb{BC}.$ A bidisk $ \mathbb{U}_{\mathbb{BC}} $ with centered $ (a_{1}, a_{2})$ and associated radii $( r_{1},r_{2})$ is defined as
\begin{equation}\label{a1}
\mathbb{U}_{\mathbb{BC}}= \{ Z \in \mathbb{BC}~:~ Z = e \eta_{1} + e^{\dagger} \eta_{2},~ \| \eta_{1}-a_{1}\|_{k} < r_{1},~ \| \eta_{2}-a_{2}\|_{k} < r_{2} \}.
\end{equation}
The bicomplex Bloch spaces were first introduces by Res$\acute{e}$ndis and Tovar in \cite{resendis2020}. They studied the bicomplex Bergman Projection onto the bicomplex Bloch space and also proved the decomposition
$$ \mathfrak{B}_{\mathbb{BC}} = e \mathfrak{B}+ e^{\dagger} \mathfrak{B}.$$In this paper, we wish to extend their work and further we define the little Bloch space in bicomplex setting. We denote the bicomplex Bloch and little Bloch spaces by $ \mathfrak{B}_{\mathbb{BC}}$ and $\mathfrak{B}_{0,\mathbb{BC}}$ respectively. We also discuss the M$\ddot{o}$bius invariance of the bicomplex Bloch space which is obvious from the idempotent decomposition of the space. We define the bicomplex little-Bloch space $ \mathfrak{B}_{0,\mathbb{BC}}$ in the bidisk $ \mathbb{U}_{\mathbb{BC}}$ and we will see that that $ \mathfrak{B}_{0,\mathbb{BC}} $ can be splitted into two classical little Bloch spaces on the unit disk. We also study the bicomplex Bergman projection onto the little Bloch space.