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Littlewood subordination theorem

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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 Ch defined on holomorphic functions f on D by

C h ( f ) = f ∘ h {\displaystyle C_{h}(f)=f\circ h}

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

The norms on these spaces are defined by:

‖ f ‖ H p p = sup r 1 2 π ∫ 0 2 π | f ( r e i θ ) | p d θ {\displaystyle \|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 π ∬ D | f ( z ) | p d x d y {\displaystyle \|f\|_{A^{p}}^{p}={1 \over \pi }\iint _{D}|f(z)|^{p}\,dx\,dy}
‖ f ‖ D 2 = 1 π ∬ D | f ′ ( z ) | 2 d x d y = 1 4 π ∬ D | ∂ x f | 2 + | ∂ y f | 2 d x d y {\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}

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 < ∞

∫ 0 2 π | f ( h ( r e i θ ) ) | p d θ ≤ ∫ 0 2 π | f ( r e i θ ) | p d θ . {\displaystyle \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 H it suffices to show that for f a polynomial

‖ C h f ‖ 2 ≤ ‖ f ‖ 2 , {\displaystyle \displaystyle {\|C_{h}f\|^{2}\leq \|f\|^{2},}}

Let U be the unilateral shift defined by

U f ( z ) = z f ( z ) . {\displaystyle \displaystyle {Uf(z)=zf(z)}.}

This has adjoint U* given by

U ∗ f ( z ) = f ( z ) − f ( 0 ) z . {\displaystyle U^{*}f(z)={f(z)-f(0) \over z}.}

Since f(0) = a0, this gives

f = a 0 + z U ∗ f {\displaystyle f=a_{0}+zU^{*}f}

and hence

C h f = a 0 + h C h U ∗ f . {\displaystyle C_{h}f=a_{0}+hC_{h}U^{*}f.}

Thus

‖ C h f ‖ 2 = | a 0 | 2 + ‖ h C h U ∗ f ‖ 2 ≤ | a 0 2 | + ‖ C h U ∗ f ‖ 2 . {\displaystyle \|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 ≤ ‖ U ∗ f ‖ 2 = ‖ f ‖ 2 − | a 0 | 2 , {\displaystyle \|C_{h}U^{*}f\|^{2}\leq \|U^{*}f\|^{2}=\|f\|^{2}-|a_{0}|^{2},}

and hence

‖ C h f ‖ 2 ≤ ‖ f ‖ 2 . {\displaystyle \|C_{h}f\|^{2}\leq \|f\|^{2}.}

The same method of proof works for A and D . {\displaystyle {\mathcal {D}}.}

General Hardy spaces

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

f ( z ) = f i ( z ) f o ( z ) {\displaystyle f(z)=f_{i}(z)f_{o}(z)}

with fi an inner function and fo an outer function.

Then

‖ C h f ‖ H p ≤ ‖ ( C h f i ) ( C h f o ) ‖ H p ≤ ‖ C h f o ‖ H p ≤ ‖ C h f o p / 2 ‖ H 2 2 / p ≤ ‖ f ‖ H p . {\displaystyle \|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 ( r z ) . {\displaystyle 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.

Notes

  1. Nikolski 2002, pp. 56–57
  2. Nikolski 2002, p. 57
  3. Duren 1970
  4. Shapiro 1993, p. 19

References

  • Duren, P. L. (1970), Theory of H spaces, Pure and Applied Mathematics, vol. 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, vol. 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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