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) = 0. Then the composition operator C_h defined on holomorphic functions f on D by

${\displaystyle C_{h}(f)=f\circ h}$

defines a linear operator with operator norm less than 1 on the Hardy spaces ${\displaystyle H^{p}(D)}$, the Bergman spaces ${\displaystyle A^{p}(D)}$. (1 ≤ p < ∞) and the Dirichlet space ${\displaystyle {\mathcal {D}}(D)}$.

The norms on these spaces are defined by:

${\displaystyle \|f\|_{H^{p}}^{p}=\sup _{r}{1 \over 2\pi }\int _{0}^{2\pi }|f(re^{i\theta })|^{p}\,d\theta }$

${\displaystyle \|f\|_{A^{p}}^{p}={1 \over \pi }\iint _{D}|f(z)|^{p}\,dx\,dy}$

${\displaystyle \|f\|_{\mathcal {D}}^{2}={1 \over \pi }\iint _{D}|f^{\prime }(z)|^{2}\,dx\,dy={1 \over 4\pi }\iint _{D}|\partial _{x}f|^{2}+|\partial _{y}f|^{2}\,dx\,dy}$

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) = 0. Then if 0 < r < 1 and 1 ≤ p < ∞

${\displaystyle \int _{0}^{2\pi }|f(h(re^{i\theta }))|^{p}\,d\theta \leq \int _{0}^{2\pi }|f(re^{i\theta })|^{p}\,d\theta .}$

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^[1]

${\displaystyle \displaystyle {\|C_{h}f\|^{2}\leq \|f\|^{2},}}$

Let U be the unilateral shift defined by

${\displaystyle \displaystyle {Uf(z)=zf(z)}.}$

This has adjoint U* given by

${\displaystyle U^{*}f(z)={f(z)-f(0) \over z}.}$

Since f(0) = a₀, this gives

${\displaystyle f=a_{0}+zU^{*}f}$

and hence

${\displaystyle C_{h}f=a_{0}+hC_{h}U^{*}f.}$

Thus

${\displaystyle \|C_{h}f\|^{2}=|a_{0}|^{2}+\|hC_{h}U^{*}f\|^{2}\leq |a_{0}^{2}|+\|C_{h}U^{*}f\|^{2}.}$

Since U*f has degree less than f, it follows by induction that

${\displaystyle \|C_{h}U^{*}f\|^{2}\leq \|U^{*}f\|^{2}=\|f\|^{2}-|a_{0}|^{2},}$

and hence

${\displaystyle \|C_{h}f\|^{2}\leq \|f\|^{2}.}$

The same method of proof works for A² and ${\displaystyle {\mathcal {D}}.}$

General Hardy spaces

If f is in Hardy space H^p, then it has a factorization^[2]

${\displaystyle f(z)=f_{i}(z)f_{o}(z)}$

with f_i an inner function and f_o an outer function.

Then

${\displaystyle \|C_{h}f\|_{H^{p}}\leq \|(C_{h}f_{i})(C_{h}f_{o})\|_{H^{p}}\leq \|C_{h}f_{o}\|_{H^{p}}\leq \|C_{h}f_{o}^{p/2}\|_{H^{2}}^{2/p}\leq \|f\|_{H^{p}}.}$

Inequalities

Taking 0 < r < 1, Littlewood's inequalities follow by applying the Hardy space inequalities to the function

${\displaystyle f_{r}(z)=f(rz).}$

The inequalities can also be deduced, following Riesz (1925), using subharmonic functions.^[3][4] The inequaties in turn immediately imply the subordination theorem for general Bergman spaces.