By N. I. Akhiezer

This vintage textbook via mathematicians from the USSR's prestigious Kharkov arithmetic Institute introduces linear operators in Hilbert house, and offers intimately the geometry of Hilbert area and the spectral idea of unitary and self-adjoint operators. it truly is directed to scholars at graduate and complicated undergraduate degrees, yet as a result of the remarkable readability of its theoretical presentation and the inclusion of effects bought through Soviet mathematicians, it's going to end up useful for each mathematician and physicist.

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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).