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
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
Case p = 2
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
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.
Duren, P. L. (1970), Theory of H p 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, ISBN0-8218-1083-9
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, ISBN0-387-94067-7
January 31, 2023
littlewood, subordination, theorem, mathematics, proved, littlewood, 1925, theorem, operator, theory, complex, analysis, states, that, holomorphic, univalent, self, mapping, unit, disk, complex, numbers, that, fixes, induces, contractive, composition, operator. 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 Contents 1 Subordination theorem 2 Littlewood s inequalities 3 Proofs 3 1 Case p 2 3 2 General Hardy spaces 3 3 Inequalities 4 Notes 5 ReferencesSubordination theorem EditLet 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 lt 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 p 0 2 p f r e i 8 p d 8 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 p 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 p D f z 2 d x d y 1 4 p 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 EditLet 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 lt r lt 1 and 1 p lt 0 2 p f h r e i 8 p d 8 0 2 p f r e i 8 p d 8 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 lt p lt 1 although in this case there is no operator interpretation Proofs EditCase p 2 Edit To prove the result for H2 it suffices to show that for f a polynomial 1 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 A2 and D displaystyle mathcal D General Hardy spaces Edit If f is in Hardy space Hp then it has a factorization 2 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 Edit Taking 0 lt r lt 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 3 4 The inequaties in turn immediately imply the subordination theorem for general Bergman spaces Notes Edit Nikolski 2002 pp 56 57 Nikolski 2002 p 57 Duren 1970 Shapiro 1993 p 19References EditDuren P L 1970 Theory of H p 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 inegalite de M Littlewood dans la theorie 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 Retrieved from https en wikipedia org w index php title Littlewood subordination theorem amp oldid 695749203, wikipedia, wiki, book, books, library,