  We introduce the class of Beurling-Fourier algebras on locally compact groups
and show that they are non-commutative analogs of classical Beurling algebras.
We obtain various results with regard to the operator amenability, operator
weak amenability and Arens regularity of Beurling-Fourier algebras on compact
groups and show that they behave very similarly to the classical Beurling
algebras of discrete groups. We then apply our results to study explicitly the
Beurling-Fourier algebras on SU(2), the 2 $\times$ 2 unitary group. We
demonstrate that how Beurling-Fourier algebras are closely connected to the
amenability of the Fourier algebra of SU(2). Another major consequence of our
results is that our investigation allows us to construct families of unital
infinite-dimensional closed Arens regular subalgebras of the Fourier algebra of
certain products of SU(2).
