By Dan Pedoe

*Mathematics Gazette*

*n*dimensions; the projective new release of conics and quadrics; Moebius tetrahedra; the tetrahedral complicated; the twisted cubic curve; the cubic floor; orientated circles; and creation to algebraic geometry.

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N=1 The foregoing estimates on Int1 and Int2 give µTa,b f,p ∈ CMp . 2, we can estimate the distance of a Bloch function to the Qp space. 1. 3. For p ∈ (0, 2) and f ∈ B let distB (f, Qp ) = inf{ f − g Qp }. Then distB (f, Qp ) ≈ inf > 0 : 1Ω (f ) (z)(1 B : g∈ − |z|2 )p−2 dm(z) ∈ CMp , where Ω (f ) = {z ∈ D : (1 − |z|2)|f (z)| ≥ } and 1E stands for the characteristic function of a set E. Proof. Because of f ∈ B, this function has the following integral representation: f (z) = f (0) + 1 π D (1 − |w|2 )f (w) dm(w) = f1 (z) + f2 (z), w(1 ¯ − wz) ¯ 2 where f1 (z) = f (0) + 1 π Ω (f ) and 1 π f2 (z) = D\Ω (f ) Note that |f1 (z)| ≤ f B D (1 − |w|2 )f (w) dm(w) w(1 ¯ − wz) ¯ 2 (1 − |w|2 )f (w) dm(w).

If q ∈ [0, p + 2), then D |F (z)|p dm(z) ≈ D |F (z)|p−q |F (z)|q (1 − |z|2 )q dm(z). Proof. For any F ∈ Ap,0 , let I(F ; p, q) = D |F (z)|p−q |F (z)|q (1 − |z|2 )q dm(z). First of all, we recall two basic facts for f ∈ H, fr (z) = f (rz), r ∈ (0, 1) and p ∈ (0, ∞). The ﬁrst one is the Hardy–Stein identity which reads as: fr p Hp = 2π|f (0)|p + p2 D |fr (z)|p−2 |fr (z)|2 (− log |z|)dm(z). The second one is the following Littlewood–Paley inequalities: fr p Hp |f (0)|p + I(fr ; p, p), p ∈ (0, 2] and |f (0)|p + I(fr ; p, p) fr p Hp , p ∈ [2, ∞).

Since F ∈ Hp , this function can be written as F = BG where G has no zeros with G Hp = F Hp and B is a Blaschke product. Accordingly, |F |p−q |F |q ≤ 2q−1 (|G|p |B|p−q |B |q + |B|p |G|p−q |G |q ). p Since G = 0, letting h = G q yields h ∈ Hq , |h |q = pq −1 |G|p−q |G |q and |B(z)|p |G(z)|p−q |G (z)|q (1 − |z|2 )q−1 dm(z) D |h (z)|q (1 − |z|2 )q−1 dm(z) D q Hq h ≈ F p Hp . For the other estimate we use the Carleson embedding for Hp — see [Ga, pp. 238-239] to get D |G(z)|p |B(z)|p−q |B (z)|q (1 − |z|2 )q−1 dm(z) sup C(a) G a∈D where C(a) = D p Hp , 1 − |a|2 |B(z)|p−q |B (z)|q (1 − |z|2 )q−1 dm(z).