Kalendarium
21
March
Analysis Seminar with Hervé Queffélec (Université de Lille)
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 :
- ?? is bounded for all (some, non-trivial) ?∈?? if and only if ? is “slowly
oscillating”. - ?? is bounded for all ?∈? if and only if ? is slowly oscillating and “essen-
tially decreasing”. - If ? is “weakly decreasing”, all the ?? with ?∈?? are bounded.
This is recent joint work with P. Lefèvre, D. Li, L. Rodríguez-Piazza.
Om händelsen
Tid:
2023-03-21 13:15
till
14:15
Plats
MH:332A
Kontakt
eskil [dot] rydhe [at] math [dot] lu [dot] se