By John Garnett

ISBN-10: 0387336214

ISBN-13: 9780387336213

The booklet is a bit turse. the writer may have incorporated extra information within the proofs.

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T Let K be a compact subset of E λ and cover K by finitely many intervals I1 , . . 4). Applying the lemma to {I1 , . . , In } gives us pairwise disjoint intervals J1 , J2 , . . 4) such that n m |J j |. Ij ≤ 2 j=1 j=1 Sect. 4 hardy–littlewood maximal function 25 Then n |K | ≤ Ij ≤ 2 j i=1 1 λ | f | ds ≤ Jj 2 λ | f | ds. Letting |K | increase to |E λ | gives us part (a). The proof of part (b) depends on the interpolation theorem of Marcinkiewicz. 5. Let (X, μ) and (Y, ν) be measure spaces, and let 1 < p1 ≤ ∞.

Fix r < 1. 6, |gn (r eiθ )| p dθ ≤ lim R→1 2π | f (Reiθ )| p dθ . |Bn (Reiθ )| p 2π Hp 54 Chap. II spaces If 1 − R is small, then |Bn (Reiθ )| > 1 − ε, so that |gn (r eiθ )| p dθ ≤ lim R→1 2π | f (Reiθ )| p dθ = f 2π Since |gn | increases to |g|, and since |g| ≥ | f |, this gives g p Hp p Hp. = F p Hp. 8 shows that sup y | f (x + i y)| p d x = lim y→0 | f (x + i y)| p d x. Blaschke products have a simple characterization in terms of harmonic majorants. 4. Let f (z) ∈ H ∞ (D) f alent. ∞ ≤ 1. Then the following are equiv- (a) f (z) = λB(z), where λ is constant, |λ| = 1, and B(z) is a Blaschke product.

T|≥|x| If ψ(x) ∈ L 1 , then the operator Mϕ f (x) = sup y>0 1 y ϕ x −t y f (t) dt 45 exercises and further results is weak-type 1–1, |{x : Mϕ f (x) > λ}| ≤ C f |1 , λ and Mϕ is bounded on L p , 1 < P ≤ ∞, Mϕ f ≤ Cp f p p, 1 < p ≤ ∞. 2) 1 y f (x) = lim y→0 ϕ x −t y f (t) dt almost everywhere. 2) holds at every point of the Lebesgue set of f , which is independent of ϕ. (c) Formulate and prove a similar result about nontangential convergence and nontangential maximal functions. 12. If f (x) has support a bounded interval I, then only if I | f | log+ | f | d x < ∞ (Stein [1969]).

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Bounded Analytic Functions by John Garnett

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