Littlewood’s subordination principle for weighted Hardy spaces

Speaker: Hervé Queffélec (Université de Lille)

Abstract: Let 𝛽 = (𝛽_𝑛)_𝑛≥𝟶 with 𝛽_𝑛 ≻ 𝟶 and lim inf_𝑛⭢∞ (𝛽_𝑛)^𝟷/𝑛 ≥ 1. The weighted Hardy space 𝐻^𝟸(𝛽) ⊂ Hol (𝔻) (with 𝔻 the unit disk) is the Hilbertian space of those analytic functions 𝑓 (𝑧) = ∑𝑐_𝑛𝑧^𝑛 such that

(𝟷)                        ‖𝑓 ‖_𝛽^𝟸≔∑|𝑐_𝑛|^𝟸𝛽_𝑛 < ∞.                        

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

                             𝑆_𝟶 = {𝜑∈𝑆 : 𝜑(𝟶) = 𝟶},     𝑆_𝟷 = Aut 𝔻.

Given 𝑓, 𝑔 ∈ ℋ(𝔻), we say that 𝑔 is subordinate to 𝑓 (formally 𝑔 ≺ 𝑓 ) 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 𝐶_𝜑 with symbol 𝜑∈𝑆 is formally defined by

(𝟸)                        𝐶_𝜑(𝑓 ) = 𝑓∘𝜑, 𝐶_𝜑∘𝜓 = 𝐶_𝜓𝐶_𝜑.                   

Each 𝜑∈𝑆 writes 𝜑 = 𝜑_𝟷∘𝜑_𝟶, 𝜑_𝟷 ∈ 𝑆_𝟷, 𝜑_𝟶 ∈ 𝑆_𝟶. So that 𝐶_𝜑 = 𝐶_𝜑𝟶𝐶_𝜑𝟷 . If 𝐶_𝜑 : 𝐻^𝟸(𝛽) ⭢𝐻^𝟸(𝛽), we say that 𝐶_𝜑 is bounded. The Littlewood subordination principle (in the case 𝛽_𝑛 ≡ 𝟷, 𝐻^𝟸(𝛽) = 𝐻^𝟸) tells that 𝐶_𝜑 is a contraction if 𝜑∈𝑆_𝟶. And 𝐶_𝜑 is bounded as well for 𝜑∈𝑆_𝟷. The cases 𝜑∈𝑆_𝟶, 𝜑∈𝑆_𝟷 appear as significantly different, but exhaust everything.

Goluzin observed that the 𝑆_𝟶-case persists when 𝛽_𝑛⭣. But not the 𝑆_𝟷-case ! (cf. 𝛽_𝑛 = e^−√𝑛). The general situation remained poorly understood. In this talk, we shall sketch a proof of three theorems :

1. 𝐶_𝜑 is bounded for all (some, non-trivial) 𝜑∈𝑆_𝟷 if and only if 𝜑 is “slowly
   oscillating”.
2. 𝐶_𝜑 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 𝐶_𝜑 with 𝜑∈𝑆_𝟶 are bounded.

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