Littlewood subordination theorem

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 \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

:\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

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_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

: 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

• {{citation|last=Duren|first= P. L.|title=Theory of H p spaces|series=Pure and Applied Mathematics|volume= 38| publisher= Academic Press|year=1970}}

References

1. {{harvnb. Nikolski. 2002
2. {{harvnb. Nikolski. 2002
3. {{harvnb. Duren. 1970
4. {{harvnb. Shapiro. 1993