By John Garnett

The booklet, which covers a variety of attractive issues in research, is very good geared up and good written, with based, targeted proofs. The publication has informed an entire new release of mathematicians with backgrounds in complicated research and serve as algebras. It has had a superb impression at the early careers of many best analysts and has been broadly followed as a textbook for graduate classes and studying seminars in either the U.S. and abroad.

- From the quotation for the 2003 Leroy P. Steele Prize for Exposition

The writer has now not tried to supply a compendium. really, he has chosen a number of subject matters in a many-faceted thought and, inside that diversity, penetrated to massive depth...the writer has succeeded in bringing out the great thing about a idea which, regardless of its fairly complex age---now coming near near eighty years---continues to shock and to please its practitioners. the writer has left his mark at the subject.

- Donald Sarason, Mathematical Reviews

Garnett's ** Bounded Analytic Functions** is to operate thought as Zygmund's

**is to Fourier research.**

*Trigonometric Series***is largely considered as a vintage textbook used world wide to coach present day practioners within the box, and is the first resource for the specialists. it's superbly written, yet deliberately can't be learn as a singular. fairly it provides simply the fitting point of aspect in order that the stimulated pupil develops the needful abilities of the exchange within the means of learning the great thing about the combo of actual, advanced and useful analysis.**

*Bounded Analytic Functions*- Donald E. Marshall, collage of Washington

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**Extra resources for Bounded Analytic Functions**

**Sample text**

Proof. 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).

Moreover, the inequalities 1 − | f (xn )| 1 + xn |1 − f (xn )| 1 − | f (xn )| 1 − xn2 ≤ 1 − xn 1 + | f (xn )| 1 + | f (xn )| 1 − | f (xn )| (1 − xn )2 |1 − f (xn )|2 1 − xn2 ≤ 1 − | f (xn )|2 (1 − xn )2 show that lim 1 − | f (xn )| ≤ B. 1 − xn So when z n = xn , the hypotheses of Julia’s lemma hold with A = B. 1), so that |1 − f (xn )| → 1. 1 − | f (xn )| 1 − f (xn ) → B, 1 − xn It follows that arg(1 − f (x)) → 0 as x → 1 and consequently that lim x↑1 1 − | f (x)| |1 − f (x)| 1 − f (x) = lim = lim = B.

The nontangential maximal function u ∗ will be more important to us than the Hardy–Littlewood maximal function Mf. 1. 2. If u(z) is harmonic on H and if p > 1, then sup |u(x + i y)| p d x ≤ B p sup y y |u(x + i y)| p d x. 2 is false at p = 1. Take u(x, y) = Py ∗ f (x), f ∈ L 1 , f > 0. Then sup y |u(x, y)| ≥ M f (x) and M f ∈ L 1 . 3 (Fatou). assume Let u(z) be harmonic on H , let 1 ≤ p ≤ ∞ and sup u(x + i y) L p (d x) y < ∞. Then for almost all t the nontangential limit lim α (t) z→t u(z) = f (t) exists.