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 $\mathcal{D}(D)$.

The norms on these spaces are defined by:

$\|f\|_{H^p}^p = \sup_r {1\over 2\pi}\int_0^{2\pi} |f(re^{i\theta})|^p \, d\theta$

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

$\|f\|_{\mathcal D}^2 = {1\over \pi} \iint_D |f^\prime(z)|^2\, dx\,dy= {1\over 4 \pi} \iint_D |\partial_x f|^2 + |\partial_y f|^2\, dx\,dy$

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 < ∞

$\int_0^{2\pi} |f(h(re^{i\theta}))|^p \, d\theta \le \int_0^{2\pi} |f(re^{i\theta})|^p \, d\theta.$

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

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \displaystyle{\|C_h f\|^2 \le \|f\|^2,}

Let U be the unilateral shift defined by

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \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

$C_h f = a_0 + h C_hU^*f.$

Thus

$\|C_h f\|^2 = |a_0|^2 + \|hC_hU^*f\|^2 \le |a_0^2|+ \|C_h U^*f\|^2.$

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

$\|C_h U^*f\|^2 \le \|U^*f\|^2 = \|f\|^2 - |a_0|^2,$

and hence

$\|C_h f\|^2 \le \|f\|^2.$

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

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

$\|C_h f\|_{H^p} \le \|(C_hf_i) (C_h f_o)\|_{H^p} \le \|C_h f_o\|_{H^p} \le \|C_h f_o^{p/2}\|_{H^2}^{2/p} \le \|f\|_{H^p}.$

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 Riesz (1925), using subharmonic functions. The inequaties in turn immediately imply the subordination theorem for general Bergman spaces.