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 C<sub>h</sub> 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 < âÂÂ) 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 < âÂÂ
This inequality also holds for 0 < p < 1, although in this case there is no operator interpretation.
To prove the result for H<sup>2</sup> 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) = a<sub>0</sub>, 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 A<sup>2</sup> and
If f is in Hardy space H<sup>p</sup>, then it has a factorization
with f<sub>i</sub> an inner function and f<sub>o</sub> an outer function.
Then
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.