Download Composition operators on Hardy-Orlicz spaces by Pascal Lefevre, Daniel Li, Herve Queffelec, Luis PDF

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

ISBN-10: 082184637X

ISBN-13: 9780821846377

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

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On the other hand, let us first point out that, for 0 ≤ t ≤ 1, F (t) = 1 + 1 4 π 2 /4 ≤ 2· 2 ≤ 1 + t2 t sin t Hence, for h small enough: 2π = F (bn ) − F (bn+1 ) = bn bn+1 F (t) dt ≤ 4 bn − bn+1 , bn+1 bn and we get: 2 (bn − bn+1 ). 5): b2n+1 ≤ bn+1 bn ≤ ∞ ∞ 4h (bn − bn+1 ) π n=0 n=0 √ 4h 4h √ 16 3/2 ≤ 24 h2 + b0 ≤ 24 h2 + 4 h ≤ 24 h + h ≤ 6 h3/2 , π π π for h small enough. (bn − an ) ≤ (b0 − a0 ) + 36 ` ´ QUEFFELEC, ´ PASCAL LEFEVRE, DANIEL LI, HERVE LUIS RODR´IGUEZ-PIAZZA In the same way, we have ∞ √ m({0 < t ≤ 3 h ; |α + t − cot t + 2πn| ≤ h}) ≤ 6 h3/2 .

Schwartz’s thesis [37]: see [42], page 471). Now, let: 1+z M (z) = exp − 1−z and φ2 (z) = φ1 (z) M (z). For simplicity, we shall write φ = φ2 , and we are going to show that Cφ is a compact operator on H 2 , using the criterion (MC). Let ξ = eiα ∈ T, with |α| ≤ π. We are going to prove that: μφ W (ξ, h) = O (h3/2 ). For θ ∈ (−π, π), one has |φ(eiθ )| = |φ1 (eiθ )| = cos(θ/2), and so the condition 1 − h < |φ(eiθ )| ≤ 1 is equivalent to 1 − h < cos(θ/2) < 1, which implies, since 1. e. θ 2 ≤ 20h and so |θ| ≤ 6 h.

Examples. It is immediately seen that the following functions satisfy ∇0 : Ψ(x) = α α − 1, Ψ(x) = ex − 1, α ≥ 1. e. Ψ(βx)/Ψ(x) is increasing for x large enough, then we have the dichotomy: either Ψ ∈ Δ2 , or Ψ ∈ Δ0 . 9) for x0 ≤ x ≤ y. 7. 1) Condition Δ2 implies condition ∇0 uniformly. 2) If Ψ ∈ ∇0 uniformly, then Ψ ∈ ∇1 . 9), ∇0 is satisfied with constant C = 1. We shall say that Ψ is κ-convex when κ is convex at infinity. Note that Ψ is κ-convex whenever Ψ is log-convex. In the above examples Ψ is κ-convex; it also the case of Ψ(x) = x2 / log x, x ≥ e; but, on the other hand, if Ψ(x) = x2 log x for x ≥ e, then Ψ is not κ-convex.

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