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0 Then H is continuous on T × T and sup |ζ|=1, |ω|=1 Let ϕ(h) = | H(ζ, ω) | = || h || C . ∫ H(ζ, ω) d(µ × υ) (ζ, ω) for h 0 C. Then ϕ is linear and T ×T bounded since | ϕ(h) | ≤ || h ||C || µ || || υ || . By the Riesz representation theorem there exists λ 0 M with ϕ(h) = ∫ h ( x ) dλ ( x ) T © 2006 by Taylor & Francis Group, LLC 24 Fractional Cauchy Transforms and ||λ|| < ||µ|| || υ ||. In particular if h(x) = x n for x 0 T and n = 0, 1, 2, … , then ∫ T 1 x n dλ ( x ) = ∫ ∫ (tζ + (1 − t) ω) n dρ( t ) d (µ × υ) (ζ, ω) .

1) ∫ 1 π −π ∫ π −π ⎞ | f ( tre iθ ) | dθ ⎟⎟ (1 − t ) α − 2 dt ⎠ ⎞ | f ( te iθ ) | dθ ⎟⎟ (1 − t ) α − 2 dt ⎠ α −1 A. 20) imply that f 0 Fα and the measures representing g in F1 are the same as the measures representing f in Fα. Therefore || f || Fα ≤ (α–1)A/2π. Therefore g 0 H1. It follows that g 0 F1 and || g ||F1 ≤ || g ||H1 ≤ Next we relate Fα to a certain Besov space. For α > 0, let Bα denote the set of functions f that are analytic in D and satisfy 1 π 0 −π ∫ ∫ | f ′(re iθ ) | (1 − r ) α −1 dθ dr < ∞.

We recall that hp (0 < p < ∞) is defined as the set of complex-valued functions u that are harmonic in D and satisfy sup ∫ π 0 < r <1 − π | u (reiθ ) |p dθ < ∞ . 12) where µ 0 M. Suppose that the function f is analytic in D and u = Re ( f ) 0 h1. 12) holds where µ is a real-valued measure. 13) for |z| < 1 and hence f 0 F1. If, in addition, f (0) is real then f can be represented in F1 using a real measure. The converse also holds, that is, if f 0 F1 can be represented by a real measure then Re ( f ) 0 h1 and f (0) is real.

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