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A) Suppose (X, Σ, μ), (Y, Σ , ν) are two measure spaces, A, B ∈ Σ, A , B ∈ Σ , and μ(A) = ν(A ), μ(B) = ν(B ). Suppose also either B ⊂ A or A ⊂ B . Prove that μ(A ∩ B) ≤ ν(A ∩ B ). ) 17. (a) If A, B are balls in Rν centered at 0, prove that x → |(A + x) ∩ B| is spherically symmetric in x and decreasing as |x| increases. 85) implies |x| = |y| ⇒ f (x) = f (y). Prove that f ∗ g is spherically symmetric and decreasing. ) 18. 87) Licensed to AMS. 3. The Hardy–Littlewood Maximal Inequality 41 19. 89) (c) (f + g)∗ (t1 + t2 ) ≤ f ∗ (t1 ) + g ∗ (t2 ) (d) mf (f ∗ (t)) ≤ t; when does equality fail?

So, in some sense, the approach in this chapter is the only one. e. ω is to first prove a maximal inequality! Stein [775] (see also Burkholder [122] and de Guzmán [205]) has even shown that if X = Lp (Ω, dμ), Tn bounded from Lp to Lp , μ(X) < ∞, and 1 ≤ p ≤ 2, a pointwise convergence result implies a maximal function inequality of the form μ({α | (T ∗ f )(ω) > λ}) ≤ Cλ−p f p p For these results, μ(X) < ∞ is critical. If Tn : L1 (R, dx) → L1 (R, dx) by (Tn f )(x) = χ[n,n+1] (x) f (y) dy then (T ∗ f )(x) = χ[1,∞) (x) f (y) dy and the measures of sets where (T ∗ f )(ω) > λ can be infinite.

1. Jensen’s Formula and the Poisson–Jensen Formula. 1) The proof is easy. 2) b(z, w) = |w| w−z w 1−wz ¯ , otherwise This is analytic in a neighborhood of D, vanishes to order 1 precisely at z = w and obeys |b(eiθ , w)| = 1. 2, the additional fact that b(0, w) ≥ 0 and > 0 unless w = 0 and in that case b (0, w) > 0. 1), one considers only R = 1 which suffices by scaling. 3) j=1 If f has no zeros on ∂D, g is analytic and nonvanishing in a neighborhood of D, so log g(z) is analytic there. 3). If f has a zero on ∂D, one proves the result for R = 1 − ε and takes ε ↓ 0.

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