  Positive energy unitary irreducible representations of $SU(2,2)$ can be
constructed with the aid of bosonic oscillators in (anti)fundamental
representation of $SU(2)_L\times SU(2)_R$ that are closely related to Penrose
twistors. Starting with the correspondence between the doubleton
representations, homogeneous functions on projective twistor space and on-shell
$SL(2,\mathbb C)$ generalized Weyl curvature spinors and their low-spin
counterparts, we study in the similar way the correspondence between the
massless representations, homogeneous functions on ambitwistor space and, via
the Penrose transform, with the gauge fields on Minkowski boundary of $AdS_5$.
The possibilities of reconstructing massless fields on $AdS_5$ and some
applications are also discussed.
