By John Garnett

ISBN-10: 0387336214

ISBN-13: 9780387336213

The publication is a bit turse. the writer may have integrated extra info within the proofs.

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**Extra info for Bounded Analytic Functions**

**Sample text**

Proof. The convexity of ϕ means that ϕ(t) is the supremum of the linear functions lying below ϕ: ϕ(t0 ) = sup{at0 + b : at + b ≤ ϕ(t), t ∈ }ޒ. Whenever at + b ≤ ϕ(t), we have a v dμ + b = (av + b) dμ ≤ ϕ(v) dμ, and the supremum of the left sides of these inequalities is ϕ( v dμ). Jensen’s inequality is also true if v dμ = −∞, provided that ϕ is defined at t = −∞ and increasing on [−∞, ∞). The proof is trivial in that case. 2. Let v(z) be a subharmonic function on , and let ϕ(t) be an increasing convex function on [−∞, ∞), continuous at t = −∞.

If 1 2π v(z 0 + r eiθ ) dθ = −∞, then since v is bounded above and since Poisson kernels are bounded and positive, we have 1 2π Pz (θ)v(z 0 + r eiθ ) dθ = −∞, Consequently Un (z) → −∞ for each z ∈ −∞ on (z 0 , r ). The nonempty set {z ∈ |z| < 1. 3 v = : v(z) ≡ −∞ on a neighborhood of z} is then open and closed, and we again have a contradiction. 5. Let v(z) be a subharmonic function in the unit disc D. Assume v(z) ≡ −∞. For 0 < r < 1, let ⎧ |z| ≤ r, ⎨v(z), vr (z) = 1 ⎩ Pz/r (θ)v(r eiθ ) dθ, |z| < r.

We take α = 1. The proof for a different α is similar. Assume σ is a Carleson measure. The open set {t : u ∗ (t) > λ} is the union of a disjoint sequence of open intervals {I j }, with centers c(I j ). Let T j be the tent T j = {z : |x − c(I j )| + y < |I j |/2}, an isosceles right triangle with hypotenuse I j . If |u(z)| > λ, then u ∗ (t) > λ on the interval {|t − x| < y} and this interval is contained in some I j . 5. Consequently, ∞ {z : |u(z)| > λ} ⊂ Tj . 6) holds. 5. Conversely, let I be an interval {x0 < t < x0 + h} and let u(z) = Py ∗ f (x) with f (x) = 4λχ I (x).