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.
Let h be a holomorphic univalent mapping of the unit disk D into itself such that h(0) = 0. Then the composition operator Ch 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 . (1 ≤ p < ∞) and the Dirichlet space .
The norms on these spaces are defined by:
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) = 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.
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(0) = a0, this gives
Since U*f has degree less than f, it follows by induction that
The same method of proof works for A2 and
General Hardy spaces
Taking 0 < r < 1, Littlewood's inequalities follow by applying the Hardy space inequalities to the function
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