Littlewood’s subordination principle for weighted Hardy spaces

Speaker: Hervé Queffélec (Université de Lille)

Abstract: Let ? = (?_?)_?≥? with ?_? ≻ ? and lim inf_?⭢∞ (?_?)^?/? ≥ 1. The weighted Hardy space ?^?(?) ⊂ Hol (?) (with ? the unit disk) is the Hilbertian space of those analytic functions ? (?) = ∑?_??^? such that

(?)                        ‖? ‖_?^?≔∑|?_?|^??_? < ∞.                        

? is the set of analytic self-maps ? of ?, called symbols. Let also 

                             ?_? = {?∈? : ?(?) = ?},     ?_? = Aut ?.

Given ?, ? ∈ ℋ(?), we say that ? is subordinate to ? (formally ? ≺ ? ) if ? = ?∘? for some ?∈?. This amounts to ?(?) ⊂ ? (?) when ? is injective. And ? should be “smaller” than ? . The composition operator ?_? with symbol ?∈? is formally defined by

(?)                        ?_?(? ) = ?∘?, ?_?∘? = ?_??_?.                   

Each ?∈? writes ? = ?_?∘?_?, ?_? ∈ ?_?, ?_? ∈ ?_?. So that ?_? = ?_???_?? . If ?_? : ?^?(?) ⭢?^?(?), we say that ?_? is bounded. The Littlewood subordination principle (in the case ?_? ≡ ?, ?^?(?) = ?^?) tells that ?_? is a contraction if ?∈?_?. And ?_? is bounded as well for ?∈?_?. The cases ?∈?_?, ?∈?_? appear as significantly different, but exhaust everything.

Goluzin observed that the ?_?-case persists when ?_?⭣. But not the ?_?-case ! (cf. ?_? = e^−√?). The general situation remained poorly understood. In this talk, we shall sketch a proof of three theorems :

1. ?_? is bounded for all (some, non-trivial) ?∈?_? if and only if ? is “slowly
   oscillating”.
2. ?_? is bounded for all ?∈? if and only if ? is slowly oscillating and “essen-
   tially decreasing”.
3. If ? is “weakly decreasing”, all the ?_? with ?∈?_? are bounded.

This is recent joint work with P. Lefèvre, D. Li, L. Rodríguez-Piazza.