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

[math]\displaystyle{ C_h(f) = f\circ h }[/math]

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

The norms on these spaces are defined by:

[math]\displaystyle{ \|f\|_{H^p}^p = \sup_r {1\over 2\pi}\int_0^{2\pi} |f(re^{i\theta})|^p \, d\theta }[/math]

[math]\displaystyle{ \|f\|_{A^p}^p = {1\over \pi} \iint_D |f(z)|^p\, dx\,dy }[/math]

[math]\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 }[/math]

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 < ∞

[math]\displaystyle{ \int_0^{2\pi} |f(h(re^{i\theta}))|^p \, d\theta \le \int_0^{2\pi} |f(re^{i\theta})|^p \, d\theta. }[/math]

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

[math]\displaystyle{ \displaystyle{\|C_h f\|^2 \le \|f\|^2,} }[/math]

Let U be the unilateral shift defined by

[math]\displaystyle{ \displaystyle{Uf(z)= zf(z)}. }[/math]

This has adjoint U* given by

[math]\displaystyle{ U^*f(z) ={f(z)-f(0)\over z}. }[/math]

Since f(0) = a₀, this gives

[math]\displaystyle{ f= a_0 + zU^*f }[/math]

and hence

[math]\displaystyle{ C_h f = a_0 + h C_hU^*f. }[/math]

Thus

[math]\displaystyle{ \|C_h f\|^2 = |a_0|^2 + \|hC_hU^*f\|^2 \le |a_0^2|+ \|C_h U^*f\|^2. }[/math]

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

[math]\displaystyle{ \|C_h U^*f\|^2 \le \|U^*f\|^2 = \|f\|^2 - |a_0|^2, }[/math]

and hence

[math]\displaystyle{ \|C_h f\|^2 \le \|f\|^2. }[/math]

The same method of proof works for A² and [math]\displaystyle{ \mathcal D. }[/math]

General Hardy spaces

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

[math]\displaystyle{ f(z) = f_i(z)f_o(z) }[/math]

with f_i an inner function and f_o an outer function.

Then

[math]\displaystyle{ \|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}. }[/math]

Inequalities

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

[math]\displaystyle{ f_r(z)=f(rz). }[/math]

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.

References

• Duren, P. L. (1970), Theory of H ^p spaces, Pure and Applied Mathematics, 38, Academic Press 
• Littlewood, J. E. (1925), "On inequalities in the theory of functions", Proc. London Math. Soc. 23: 481–519, doi:10.1112/plms/s2-23.1.481 
• Nikolski, N. K. (2002), Operators, functions, and systems: an easy reading. Vol. 1. Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs, 92, American Mathematical Society, ISBN 0-8218-1083-9 
• Riesz, F. (1925), "Sur une inégalite de M. Littlewood dans la théorie des fonctions", Proc. London Math. Soc. 23: 36–39, doi:10.1112/plms/s2-23.1.1-s 
• Shapiro, J. H. (1993), Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, ISBN 0-387-94067-7