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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 𝛽 = (𝛽𝑛)𝑛≥𝟶 with 𝛽𝑛𝟶 and lim inf𝑛⭢∞ (𝛽𝑛)𝟷/𝑛 1. The weighted Hardy space 𝐻𝟸(𝛽) Hol (𝔻) (with 𝔻 the unit disk) is the Hilbertian space of those analytic functions 𝑓 (𝑧) = ∑𝑐𝑛𝑧𝑛 such that

(𝟷)                        ‖𝑓 𝛽𝟸≔∑|𝑐𝑛|𝟸𝛽𝑛 < .                        

𝑆 is the set of analytic self-maps 𝜑 of 𝔻, called symbols. Let also 

                             𝑆𝟶 = {𝜑∈𝑆 : 𝜑(𝟶) = 𝟶},     𝑆𝟷 = Aut 𝔻.

Given 𝑓, 𝑔 (𝔻), we say that 𝑔 is subordinate to 𝑓 (formally 𝑔𝑓 ) if 𝑔 = 𝑓∘𝜑 for some 𝜑∈𝑆. This amounts to 𝑔(𝔻) 𝑓 (𝔻) when 𝑓 is injective. And 𝑔 should be “smaller” than 𝑓 . The composition operator 𝐶𝜑 with symbol 𝜑∈𝑆 is formally defined by

(𝟸)                        𝐶𝜑(𝑓 ) = 𝑓∘𝜑, 𝐶𝜑∘𝜓 = 𝐶𝜓𝐶𝜑.                   

Each 𝜑∈𝑆 writes 𝜑 = 𝜑𝟷∘𝜑𝟶, 𝜑𝟷 𝑆𝟷, 𝜑𝟶 𝑆𝟶. So that 𝐶𝜑 = 𝐶𝜑𝟶𝐶𝜑𝟷 . If 𝐶𝜑 : 𝐻𝟸(𝛽) ⭢𝐻𝟸(𝛽), we say that 𝐶𝜑 is bounded. The Littlewood subordination principle (in the case 𝛽𝑛 ≡ 𝟷, 𝐻𝟸(𝛽) = 𝐻𝟸) tells that 𝐶𝜑 is a contraction if 𝜑∈𝑆𝟶. And 𝐶𝜑 is bounded as well for 𝜑∈𝑆𝟷. The cases 𝜑∈𝑆𝟶, 𝜑∈𝑆𝟷 appear as significantly different, but exhaust everything.

Goluzin observed that the 𝑆𝟶-case persists when 𝛽𝑛. But not the 𝑆𝟷-case ! (cf. 𝛽𝑛 = e−√𝑛). The general situation remained poorly understood. In this talk, we shall sketch a proof of three theorems :

  1. 𝐶𝜑 is bounded for all (some, non-trivial) 𝜑∈𝑆𝟷 if and only if 𝜑 is “slowly
  2. 𝐶𝜑 is bounded for all 𝜑∈𝑆 if and only if 𝛽 is slowly oscillating and “essen-
    tially decreasing”.
  3. If 𝛽 is “weakly decreasing”, all the 𝐶𝜑 with 𝜑∈𝑆𝟶 are bounded.

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

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


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