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Kalendarium

21

March

Analysis Seminar with Hervé Queffélec (Université de Lille)

Tid: 2023-03-21 13:15 till 14:15 Seminarium

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 ?, ? (?), 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.



Om händelsen
Tid: 2023-03-21 13:15 till 14:15

Plats
MH:332A

Kontakt
eskil [dot] rydhe [at] math [dot] lu [dot] se

Spara händelsen till din kalender

Sidansvarig: webbansvarig@math.lu.se | 2017-05-23