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By El-Fallah O., Kellay K., Mashreghi J., Ransford T.

ISBN-10: 1107047528

ISBN-13: 9781107047525

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We remark that it is also possible to represent a function f ∈ D as a Cauchy-type transform of a function in L2 (D). Indeed, if we write (Cg)(z) := 1 π D g(w) dA(w) 1 − wz (z ∈ D), then f (z) = f (0) + zC( f ). 6. 2 Beurling’s theorem (for radial limits) appeared in [16]. Our approach follows Borichev [20]. 3 The weak-type inequality for capacity is due to Beurling [16]. 2 in his thesis [15], showing that, if f (0) = 0 and D( f ) ≤ 1, then for each 2 t > 0 we have |{| f ∗ | > t}| ≤ 2πe1−t . This was later strengthened by Chang and Marshall [34] (see also [47] and [75]), who proved that sup T exp(| f ∗ |2 ) dθ : f (0) = 0, D( f ) ≤ 1 < ∞.

Also, every sequence (μn ) in P(X), contains a weak*-convergent subsequence. 7 Let (μn ) be a sequence in P(X), and suppose that μn is weak*convergent to μ ∈ P(X). Then lim inf n→∞ IK (μn ) ≥ IK (μ). Proof We first claim that, if f : X × X → R is a continuous function, then f dμn dμn → f dμ dμ as n → ∞. Indeed, this is clearly true if f has the form f (x, y) = g(x)h(y) and, using the Stone–Weierstrass theorem, a general continuous f may be uniformly approximated by finite sums of functions of this special form.

Let En , νn and fνn be exactly as before. In addition, we choose a sequence rn increasing to 1 such that Re fνn (rn ζ) ≥ 1/2c(En ) (ζ ∈ E). 10). Set fn (z) := fνn (rn z) and f := n c(En ) fn . The Dirichlet norm of fn is no larger than that of fνn so, just as before, f ∈ D. Also, for every N, we have N lim inf Re f (z) ≥ z→ζ N c(En ) Re fn (ζ) ≥ n=1 1/2 = N/2 (ζ ∈ E), n=1 so limz→ζ Re f (z) = ∞ for all ζ ∈ E. 2, | fn (z)| = | fνn (rn z)| ≤ π/2 + log(2/ dist(rn z, En )) (z ∈ D). Therefore f is continuous on D \ E.

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A Primer on the Dirichlet Space by El-Fallah O., Kellay K., Mashreghi J., Ransford T.


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