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

$C_h(f)=f\circ h$

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

The norms on these spaces are defined by:

$\Vert f{\Vert }_{H^p}^p=\underset{r}{\sup }\frac{1}{2\pi }{\int }_0^{2\pi }|f(r{e}^{i\theta }){|}^{p}d\theta$

$\Vert f{\Vert }_{A^p}^p=\frac{1}{\pi }{\iint }_D|f(z){|}^{p}dxdy$

$\Vert f{\Vert }_{𝒟}^2=\frac{1}{\pi }{\iint }_D|f^{\prime }(z){|}^{2}dxdy=\frac{1}{4\pi }{\iint }_D|{\partial }_{x}f{|}^2+|{\partial }_{y}f{|}^{2}dxdy$

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 < ∞

${\int }_0^{2\pi }|f(h(r{e}^{i\theta })){|}^{p}d\theta \le {\int }_0^{2\pi }|f(r{e}^{i\theta }){|}^{p}d\theta .$

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

$\Vert C_{h}f{\Vert }^2\le \Vert f{\Vert }^2,$

Let U be the unilateral shift defined by

$Uf(z)=zf(z).$

This has adjoint U* given by

$U^{\ast }f(z)=\frac{f(z)-f(0)}{z}.$

Since f(0) = a₀, this gives

$f=a_0+z{U}^{\ast }f$

and hence

$C_{h}f=a_0+h{C}_h{U}^{\ast }f.$

Thus

$\Vert C_{h}f{\Vert }^2=|a_0{|}^2+\Vert h{C}_h{U}^{\ast }f{\Vert }^2\le |a_0^2|+\Vert C_h{U}^{\ast }f{\Vert }^2.$

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

$\Vert C_h{U}^{\ast }f{\Vert }^2\le \Vert U^{\ast }f{\Vert }^2=\Vert f{\Vert }^2-|a_0{|}^2,$

and hence

$\Vert C_{h}f{\Vert }^2\le \Vert f{\Vert }^2.$

The same method of proof works for A² and $𝒟.$

General Hardy spaces

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

$f(z)=f_i(z)f_o(z)$

with f_i an inner function and f_o an outer function.

Then

$\Vert C_{h}f{\Vert }_{H^p}\le \Vert (C_h{f}_i)(C_h{f}_o){\Vert }_{H^p}\le \Vert C_h{f}_o{\Vert }_{H^p}\le \Vert C_h{f}_o^{p/2}{\Vert }_{H^2}^{2/p}\le \Vert f{\Vert }_{H^p}.$

Inequalities

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

$f_r(z)=f(rz).$

The inequalities can also be deduced, following Template:Harvtxt, using subharmonic functions. The inequaties in turn immediately imply the subordination theorem for general Bergman spaces.

References

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