Bounded Analytic Functions by John Garnett

By John Garnett

The e-book, which covers quite a lot of appealing issues in research, is very good equipped and good written, with stylish, unique proofs. The publication has informed a complete iteration of mathematicians with backgrounds in advanced research and serve as algebras. It has had an outstanding effect at the early careers of many top analysts and has been extensively followed as a textbook for graduate classes and studying seminars in either the USA and abroad.

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

The writer has no longer tried to provide a compendium. relatively, he has chosen a number issues in a many-faceted concept and, inside of that variety, penetrated to substantial depth...the writer has succeeded in bringing out the great thing about a conception which, regardless of its quite complex age---now drawing close 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 idea as Zygmund's Trigonometric Series is to Fourier research. Bounded Analytic Functions is extensively considered as a vintage textbook used around the globe to teach modern day practioners within the box, and is the first resource for the specialists. it really is superbly written, yet deliberately can't be learn as a unique. fairly it offers simply the best point of aspect in order that the inspired scholar develops the needful abilities of the exchange within the strategy of learning the wonderful thing about the combo of actual, complicated and practical analysis.

- Donald E. Marshall, college of Washington

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Riesz [1926]). II H p Spaces The classical theory of the Hardy spaces H p is a mixture of real and complex analysis. This chapter is a short introduction to this theory, with special emphasis put on the results and techniques we will need later. The theory has three cornerstones: (i) nontangential maximal functions; (ii) the subharmonicity of | f | p and log | f | for an analytic function f (z); (iii) the use of Blaschke products to reduce problems to the case of a nonvanishing analytic function.

T) The value of u ∗ depends on the parameter α, but since α has been fixed we will ignore that distinction. 1. Let u(z) be harmonic on H and let 1 ≤ p < ∞. Assume |u(x + i y)| p d x < ∞. 1) p p ≤ B p sup y |u(x + i y| p d x. 2) |{t : u ∗ (t) > λ} ≤ B1 sup λ y |u(x + i y)| d x. The constants B p depend only on p and α. Proof. Let p > 1. Then u(z) is the Poisson integral of a function f (t) ∈ L p (‫)ޒ‬, and 1/ p f p ≤ sup y |u(x + i y)| p d x . 1). 28 Chap. I preliminaries If p = 1, we know only that u(z) is the Poisson integral of a finite measure μ on ‫ ޒ‬and |u(x + i y)| d x, |dμ| ≤ sup y because μ is a weak-star limit of the measures u(x + i y) d x, y → 0.

Letting n → ∞ now yields 1 2π |B(eiθ )|dθ = 1. so that |B(eiθ )| = 1 almost everywhere. The purpose of the convergence factors −¯z n /|z n | is to make arg bn (z) converge. To remember the convergence factors, note that they are chosen so that bn (0) > 0. 2, the analytic function f (z) has a factorization f (z) = B(z)g(z), z ∈ D, where B(z) is a Blaschke product and where g(z) has no zeros on D, if and only if the subharmonic function log | f (z)| has a harmonic majorant. 3) yn < ∞, 1 + |z n |2 z n = xn + i yn , and the Blaschke product with zeros {z n } is B(z) = z−i z+i m z n =i |z n2 + 1| z − z n .

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