It is well-known that
H∞+ C on the unit circle is a closed subalgebra of L∞(T), and Rudin proved the
(H∞+ C)(T2) is a closed subspace of L∞(T2) but it is not an algebra. The
multiplier algebra ℳ of (H∞+ C)(T2) is determined. Using this charaterization, we
study Bourgain algebras of type H∞+ C on the torus T2 and the polydisk U2. Both
Bourgain algebras of H∞+ C and ℳ on the torus coincide with ℳ. We denote by
ℳ the space pf Poisson integral of functions in ℳ and CT2(Ū2) the space
of continuous functions on Ū2 which vanish on T2. It is proved that all
higher Bourgain algebras of H(U2) + C(Ū2) and H(U2) + CT2(Ū2) are all
distinct respectively, but every higher Bourgain algebra of H(U2) + C0(U2)
coincides with H(U2) + C0(U2). It is also proved that all higher Bourgain
algebras of ℳ and ℳ+ C0(U2) are all distinct respectively, but every higher
Bourgain algebra of ℳ+ CT2(Ū2) coincides with the first Bourgain algebra of
ℳ+ CT2(Ū2).
