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 \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 {\|C_{h}f\|^{2}\leq \|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}+hC_{h}U^{*}f.

Thus

\|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

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

and hence

\|C_{h}f\|^{2}\leq \|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^[2]

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}}}\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

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.

Notes

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

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