Calculus One & Several Variables 8e instructor's solutions by S.L. Salas

By S.L. Salas

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Since 5ftF > 0, then F = \F\ei4>, where -TT/2 < < TT/2. Also observe that since F has no zeros in the disk, Fp is also analytic on D and Fp = |F| p (cos(p0) + ism(P(j))). Forpe (0,1), K(F p ) = \F\P cos{p Prom this we conclude \F\pcos(P7r/2). / \F(r()\p dm{Q < Ap [ M(Fp(rQ) dm{() = AP^R(F^ (0)). JT JT The last equality follows from the fact that 3ft(Fp) is harmonic. The desired inequality follows from the observation 3ft(Fp(0)) < |F(0)| P . Finally, the constant Ap above is equal to l/cos(p7r/2) = 0 ( ( 1 — p)"1).

192. 3. P O I S S O N AN D C O N J U G A T E P O I S S ON I N T E G R A L S 49 3. Similar to the iiP(D) case, functions / belonging to Hp also have nontangential limits almost everywhere. Here a typical "non-tangential approach region" is of the form {x + iy G C + : \x — t\ < a y} for some a > 0, which is a cone with vertex t G R and angle equal to 2arctana. 4. Though not needed in our further discussion, there is an analogous factorization theorem similar to the Nevanlinna factorization theorem for Hp(3).

2. Moreover g G PA 1 / p _ 1 and £/ie norm of the linear functional in eq. 32) is equivalent to the A 1 / p _ 1 norm of g. Before proceeding to the next section, we make some remarks about the complexity of the topology of the Hp (0 < p < 1) spaces. As mentioned previously, iJ p , with the metric d(f,g) = \\f-9\\p is an F-space (a complete topological vector space whose topology is given by a translation invariant metric) and many of the results from Banach space theory such as the uniform boundedness principle, the open mapping theorem, and the closed graph theorem are still valid [30], Chapter II.

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