By Pavlovic M.
This article comprises a few evidence, rules, and methods that may aid or inspire the reader to learn books and papers on numerous periods of capabilities at the disk and the circle. The reader will locate a number of popular, primary theorems in addition to many of the author's effects, and new proofs or extensions of identified effects. such a lot of assertions are proved, even though occasionally in a slightly concise approach. a couple of assertions are named via workout, whereas sure assertions are gathered in Miscellaneous or comments: such a lot of them will be handled by means of the reader as exercises.The reader is believed to have stable starting place in Lebesgue integration, complicated research, sensible research, and Fourier sequence, this means that specifically that he/she had an outstanding education via those components. it truly is of a few significance that the reader can settle for the following:Throughout this article, constants are frequently given with no computing their distinctive values. during an explanation, the price of a continuing С may perhaps swap from one incidence to the subsequent. hence, the inequality 2C<= С is correct no matter if С > zero.
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W ∈ T. The function T g belongs to L1 because g ∈ L2 and T is of strong type (p, p) for p ∈ (1, 2). It follows that if φ ∈ L∞ , then ζ −n (T g)(ζw)φ(w) |dw| (T g)(w)φ(w) |dw| = T T for every ζ ∈ T. Integrating this with respect to ζ and using Fubini’s theorem, we get (T g)(w)φ(w) |dw| = T = 1 2π T 1 2π T ζ −n (T g)(ζw) |dζ| φ(w) |dw| T ζ −n (T g)(ζ) |dζ|. e. 27). 28) can then be deduced from the Weierstrass theorem that the trigonometric polynomials are dense in Lp . It remains to prove that T is of strong type (q, q) for q 2.
Namely, replacing Eλ by Ω and applying the formula {t ∈ [0, 1] : T gt (ω) λ} dµ(ω) = dµ(ω) Ω Ω 1 = 1 dt 0 dt T gt (ω) λ µ{ω ∈ Ω : T gt (ω) dµ(ω) = T gt (ω) λ 0 λ} dt, 34 2 Interpolation and maximal functions we get 1 µ{ω ∈ Ω : T gt (ω) µ(Eλ )/2 λ} dt. 3. 24) in this special case. Let X be a space of type p. 20) it follows that {t : gt λ} K/λp . 25) In order to exploit this fact, we start from the inequality √ {t : T gt (ω) λ} {t : gt λ} + {t : T gt √ gt λ} . 25) and Fubini’s theorem as above, we get {t : T gt (ω) µ(Eλ )/2 λ} dµ(ω) Ω √ 1 Kλ−p/2 µ(Ω) + µ ω : T gt (ω) gt λ dt.
Let BV [a, b] denote the set of functions of bounded variation on [a, b]. The Poisson/Stieltjes integral of a function γ ∈ BV = BV [−π, π] is defined as to be the harmonic function P S[γ](reiθ ) = 1 2π π P (r, θ − t) dγ(t). 10) −π In view to the well known connection between Borel measures and functions of bounded variation, the above assertions can be stated as follows. 10 Theorem A function f ∈ h(D) belongs to h1 iff f is equal to the Poisson/Stieltjes integral of a function γ ∈ BV . A positive harmonic function is equal to the Poisson/Stieltjes integral of an increasing function.