Subordinate Hp functions

Orthogonal functions in H∞

“…The first equality is the definition of the H p norm, so we only have to prove the second. If g ∈ H p and f ∈ ᐁ 0 then by a result of Ryff [1966], g • f ∈ H p with smaller or equal norm. Thus |g| p is positive, continuous function on the disk which has nontangential boundary values almost everywhere, so Lemma 5.1 shows that…”

Subordination and 𝐻^{𝑝} functions

“…Since Fo W/FoB e H' by the proof of Theorem 4, then Woy/(FoWoy/)/(FoBoy/) e Hx'2 by [12]. Thus Roy/ is in Hl/2, and since it is positive almost everywhere on /, then it can be extended analytically across I [7, pp.…”

Restrictions of Analytic Functions. II

“…is in L2(-oo, oo). We note that by a theorem of Ryff [6], F(x) = F(x + ¿0) is the boundary function of F(z). This justifies the notation, but logically it is not needed here.…”