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 $\mathcal(D)$.

The norms on these spaces are defined by:

:$\, f\, _^p = \sup_r \int_0^ , f(re^), ^p \, d\theta$

:$\, f\, _^p = \iint_D , f(z), ^p\, dx\,dy$

:$\, f\, _^2 = \iint_D , f^\prime(z), ^2\, dx\,dy= \iint_D , \partial_x f, ^2 + , \partial_y f, ^2\, dx\,dy$

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'' < ∞

:$\int_0^ , f(h(re^)), ^p \, d\theta \le \int_0^ , f(re^), ^p \, d\theta.$

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 ''H''² it suffices to show that for ''f'' a polynomial

:$\displaystyle$

Let ''U'' be the unilateral shift defined by

:$\displaystyle.$

This has adjoint ''U''* given by

:$U^*f(z) =.$

Since ''f''(0) = ''a''₀, 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 ''A''² 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

:$\, C_h f\, _ \le \, (C_hf_i) (C_h f_o)\, _ \le \, C_h f_o\, _ \le \, C_h f_o^\, _^ \le \, f\, _.$

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.

Notes

References

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*{{citation, last=Shapiro, first=J. H., title=Composition operators and classical function theory, series=Universitext: Tracts in Mathematics, publisher= Springer-Verlag, year= 1993, isbn=0-387-94067-7

Operator theory
Theorems in complex analysis