Littlewood subordination theorem

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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

defines a linear operator with operator norm less than 1 on the Hardy spaces , the Bergman spaces . (1 ≤ p < ∞) and the Dirichlet space .

The norms on these spaces are defined by:

Littlewood's inequalities[edit]

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 < ∞

This inequality also holds for 0 < p < 1, although in this case there is no operator interpretation.

Proofs[edit]

Case p = 2[edit]

To prove the result for H2 it suffices to show that for f a polynomial[1]

Let U be the unilateral shift defined by

This has adjoint U* given by

Since f(0) = a0, this gives

and hence

Thus

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

and hence

The same method of proof works for A2 and

General Hardy spaces[edit]

If f is in Hardy space Hp, then it has a factorization[2]

with fi an inner function and fo an outer function.

Then

Inequalities[edit]

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

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[edit]

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

References[edit]

  • 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