By Pascal Lefevre, Daniel Li, Herve Queffelec, Luis Rodriguez-piazza

The authors examine composition operators on Hardy-Orlicz areas whilst the Orlicz functionality \Psi grows speedily: compactness, vulnerable compactness, to be p-summing, order bounded, \ldots, and convey how those notions behave in keeping with the expansion of \Psi. They introduce an tailored model of Carleson degree. They build numerous examples displaying that their effects are basically sharp. within the final half, they examine the case of Bergman-Orlicz areas

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**Example text**

Hence the canonical embedding H Ψ → LΨ (μ) is continuous. 3): 3. 12. Let μ be a positive ﬁnite measure on D. Assume that the canonical embedding jμ : H p → Lp (μ) is continuous for some 0 < p < ∞. Then jμ : H Ψ → LΨ (μ) is continuous. 8). 12) that the composition operator Cφ : H Ψ → H Ψ is always continuous. This can be read as the continuity of H Ψ → LΨ (μφ ). Hence condition (R) must be satisﬁed, for some A > 0. Note that for A ≤ 1, 1/χA (1/h) ≥ h, and so condition (R) is implied by the fact that μφ is a Carleson measure.

The following assertions are equivalent: 1) Cφ : H Ψp → H Ψp is order bounded into M Ψp (T); 2) Cφ : H Ψp → H Ψp is compact; 3) Cφ : H Ψp → H Ψp is weakly compact; 1 4) 1−|φ| ∈ Lr (T), ∀r ≥ 1; 5) ∀q ≥ 1 ∃Cq > 0: m(1 − |φ| < λ) ≤ Cq λq ; 6) sup Cφ (ua,r ) a∈T Ψp =o log(1 − r) −1/p as r → 1 7) ∀q ≥ 1 φn 2 =√o (n−q ); 8) φn Ψp = o (1/ p log n). Remark. Observe that conditions 4), 5) and 7) do not depend on p. Hence this is equivalent to the same properties of Cφ acting on H Ψs for another s > 0. 26.

In the same spirit, if the composition operator Cφ : H Ψ → H Ψ is compact and Ψ veriﬁes a very fast growth condition (such as Δ2 for instance and Cφ is even order bounded into M Ψ (T) in that situation), we could expect that Cφ : H Ψ → H Ψ is actually nuclear as this is the case when H Ψ is replaced by H ∞ . 27: the composition operator cannot be nuclear since it is not an absolutely summing operator. CHAPTER 4 Carleson measures 1. Introduction B. 12) has characterized compact composition operators on Hardy spaces H p (p < ∞) in term of Carleson measures.