Littlewood subordination theorem

In mathematics, the Littlewood subordination theorem, proved by J. E. Littlewood in 1925, is a theorem in operator theory and complex analysis. It states that any holomorphic univalent self-mapping of the unit disk in the complex numbers that fixes 0 induces a contractive composition operator on various function spaces of holomorphic functions on the disk. These spaces include the Hardy spaces, the Bergman spaces and Dirichlet space.

Subordination theorem

Let h be a holomorphic univalent mapping of the unit disk D into itself such that h = 0. Then the composition operator C_h defined on holomorphic functions f on D by
defines a linear operator with operator norm less than 1 on the Hardy spaces, the Bergman spaces.
and the Dirichlet space.
The norms on these spaces are defined by:

Littlewood's inequalities

Let f be a holomorphic function on the unit disk D and let h be a holomorphic univalent mapping of D into itself with h = 0. Then
if 0 < r < 1 and 1 ≤ p < ∞
This inequality also holds for 0 < p < 1, although in this case there is no operator interpretation.

Proofs

Case ''p'' = 2

To prove the result for H² it suffices to show that for f a polynomial
Let U be the unilateral shift defined by
This has adjoint U* given by
Since f = a₀, this gives
and hence
Thus
Since U*''f has degree less than f'', it follows by induction that
and hence
The same method of proof works for A² and

General Hardy spaces

If f is in Hardy space H^p, then it has a factorization
with f_i an inner function and f_o an outer function.
Then

Inequalities

Taking 0 < r < 1, Littlewood's inequalities follow by applying the Hardy space inequalities to the function
The inequalities can also be deduced, following, using subharmonic functions. The inequaties in turn immediately imply the subordination theorem for general Bergman spaces.