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By Hampton N. Shirer

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In the case where X is Lp (T) or Lp (D), we put fζ (z) = f (ζz). Proof. Let dµ(ζ) = |dζ|/2π. 2 guarantees that there is a set B so p that µ(B) > 0 and µ Aλ ∩ B C ( f /λ) , where Aλ = {ω ∈ T : |(T f )(ω)| > λ}, and C is independent of f and λ. If we put fζ (ζ ∈ T) instead of f and apply the p hypotheses of the theorem, we get µ (ζ −1 Aλ ) ∩ B C ( f /λ) . 6 Nikishin and Stein’s theorem 35 p C f , which was to be proved. 26), we write the left-hand side as we get µ(Aλ ) µ ζ −1 A ∩ B dµ(ζ) = T χζ −1 A (ω) dµ(ω) dµ(ζ) B T (χ is the characteristic function), and then apply Fubini’s theorem together with the relation χζ −1 A (ω) = χω−1 A (ζ).

2, the required conditions are satisfied by the functions h1 = P [g1 ] and h2 = P [g2 ]. 4 Lemma Let u > 0 belong to hp , 1 < p < ∞. Then u p p = |u(0)|p + p(p − 1) 2 up−2 |∇u(z)|2 log D 1 dA. |z| Proof. This is easily deduced from Green’s formula and the formula ∆(up ) = p(p − 1)up−2 |∇u|2 . 1(a). It is enough to consider real-valued functions. 3, we can suppose that u is positive. 2 give u p p p2 − p 2 |∇u|2 22−p |∇u|p−2 (1 − |z|)p−1 dA, D which implies the desired conclusion. ” Let Xp be the (real) subspace of hp consisting of real-valued functions.

4)(q = 1) by taking g = |f |/(1 + |f |). An operator T that maps a quasi-normed space X to the set of nonnegative measurable functions is said to be sublinear if (almost everywhere): (a) T (f + g) T f + T g for f, g ∈ X; (b) T (λf ) = |λ| T f for f ∈ X and λ ∈ C. e. for all f , then we can treat it as an operator from X to L0 . Since (a) and (b) imply |T f − T g| T (f − g), we see that continuity of T at the origin implies continuity of T on all of X. 20) that rj (t) = sign sin(2j tπ). Every p-Banach space (0 < p with K = 1.

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