Littlewood subordination theorem explained

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 l{D}(D) .

The norms on these spaces are defined by:

\|f\| ^{p} _{H ^{p}} = \sup _{r} { 1 \over 2 \pi}\int ^{2 \pi} _{0} |f(re ^{i\theta} )| ^{p} d\theta \|f\| ^{p} _{A ^{p}} = { 1 \over \pi} \iint _{D} |f(z)| ^{p} dxdy \|f\| ^{2} _{l D} = { 1 \over \pi} \iint _{D} |f ^{\prime(z)|} ^{2} dxdy = { 1 \over 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. Thenif 0 < r < 1 and 1 ≤ p < ∞

\int ^{2 \pi} _{0} |f(h(re ^{i\theta} ))| ^{p} d\theta \le \int ^{2 \pi} _{0} |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

\displaystyle{\|C _{h} f\| ^{2} \le \|f\| ^{2,}}

Let U be the unilateral shift defined by

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

This has adjoint U* given by

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

Since f(0) = a₀, this gives

f = a _{0} + zU ^{*f}

and hence

C _{h} f = a _{0} + h C ^{*f.} _{hU}

Thus

\|C _{h} f\| ^{2} = |a ^{2} _{0 |} + \|hC ^{*f\|} _{hU} ^{2} \le |a ^{2 | +} _{0} \|C _{h} U ^{*f\|} ^{2.}

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

\|C _{h} U ^{*f\|} ^{2} \le \|U ^{*f\|} ^{2} = \|f\| ^{2} - |a ^{2 ,} _{0 |}

and hence

\|C _{h} f\| ^{2} \le \|f\| ^{2.}

The same method of proof works for A² and

l 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

\|C _{h} f\| _{H ^{p}} \le \|(C _{hf} _{i)} (C _{h} f _{o)\|} ^{H ^{p}} \le \|C _{h} f _{o\|} ^{H ^{p}} \le \|C _{h} f ^{p/ 2} _{o} \| ^{2 /p} _{H ^{2}} \le \|f\| _{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, using subharmonic functions. The inequaties in turn immediately imply the subordination theorem for general Bergman spaces.