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By John Garnett

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In their proof Hardy and Littlewood used rearrangements of functions. 4 is from Garsia’s book [1970], where it is credited to W. H. Young. Also see Stein [1970] for another covering lemma, which is valid in ‫ޒ‬n , and for a more general discussion of maximal functions and approximate identities. The books by Zygmund [1968] and by Stein and Weiss [1971] contain more information on the Marcinkiewicz theorem and other theorems on interpolation of operators. Fatou’s theorem is from his classic paper [1906], which was written not long after the introduction of the Lebesgue integral.

8 shows that |F(x + i y)| p d x is a decreasing function of y. Hence, F p Hp = lim y→0 |F(x + i y)| p d x and F p Hp = −c + lim u((1 + y)i) = y→0 1 π dμ(t) ≤ u(i). 2 shows that if g(w) ∈ H p (D), then F(z) = π −1/ p g(w(z)) ∈ H p (dt), (z + i)2/ p where w(z) = τ −1 (z), and F H p ≤ g H p . 2 is true and that F H p = g H p . Notice that because (z + i)−2/ p has no zeros on H , the family of zero sets of H p functions is invariant under z = τ (w). For p ≥ 1, H p is a normed linear space. For p < 1, the inequality |z 1 + z 2 | p ≤ |z 1 | p + |z 2 | p shows that H p is a metric space with metric d( f, g) = f − g p Hp.

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.

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