  We introduce and discuss some basic properties of some integral transforms in
the framework of specific functional Hilbert spaces, the holomorphic
Bargmann-Fock spaces on $\mathbb{C}$ and $\mathbb{C}^2$ and the slice
hyperholomorphic Bargmann-Fock space on $\mathbb{H}$. The first one is a
natural integral transform mapping isometrically the standard Hilbert space on
the real line into the two-dimensional Bargmann-Fock space. It is obtained as
composition of the one and two dimensional Segal-Bargmann transforms and
reduces further to an extremely integral operator that looks like a composition
operator of the one-dimensional Segal-Bargmann transform with a specific
symbol. We study its basic properties, including the identification of its
image and the determination of a like-left inverse defined on the whole
two-dimensional Bargmann-Fock space. We also examine their combination with the
Fourier transform which lead to special integral transforms connecting the
two-dimensional Bargmann-Fock space and its analogue on the complex plane. We
also investigate the relationship between special subspaces of the
two-dimensional Bargmann-Fock space and the slice-hyperholomorphic one on the
quaternions by introducing appropriate integral transforms. We identify their
image and their action on the reproducing kernel.

Complex Variables