Conformal invariants and function-theoretic null-sets

The exponential transform: a renormalized Riesz potential at critical exponent

“…Thus ·, · is an inner product on a suitable space. In terms of that Since G(z) can be characterized as the largest negative subharmonic function in C \ K with the singularity − log |z| at infinity, and 1 2 log(1 − E Ω (z, z)) by (8.3) and (8.8) is a competing function in this respect we have Assuming |∂Ω| = 0, so that |Ω| = |K|, this is the Ahlfors-Beurling estimate [1].…”

The arc-length variation of analytic capacity and a conformal geometry

“…This lemma plays an important role to study φ(z; Q); φ(z; Q) 2 is called the Garabedian function of Ω c [11, p. 19]. The Ahlfors function f(z; Ω) of Ω c (i.e., /( β) e ff~(β), \\f(-Ω)\\ H o» = 1, f'(oo Ω) = γ(Ω c )) is expressed as g(e; fl)/^(e; β) [11, pp.…”

“…The quantitative properties of analytic capacity are important in the study of conformal mappings, the 2-dimensional fluid dynamics and singular integrals ( [16], [17], [23]). Vitushkin [34], Gamelin [7], Garnett [11], Zalcman [35] show that γ( ) is applicable to study approximation problems, and Ahlfors-Beurling [2], Pommerenke [22], Suita [30], [31], [32] study ϊ(') from the point of view of a conformal invariant. The author studied γ(-) in terms of integral geometry [17], [18] and fluid dynamics [19].…”

“…Let H\Ω) denote the ίf 2 -space of analytic functions in Ω such that where the orientation of dz is chosen so that Ω lies to the left [8]. For any ζ e Ω -{oo}, there exists uniquely a pair (K(-, ζ; β), L( , ζ; Ω)) of functions such that K( , ζ; β), ( -ζ)L( , ζ; β) e iϊ 2 (β), K(oo, ζ; β) = L(oo, ζ; β) = 0, C;β) = l and (4) ~L(z, ζ; Ω)dz = K(z, ζ; Ω)\dz\ a.e.…”

Some Note on Exceptional Values of Meromorphic Functions

“…1 'DΏ n for every n, 2) for each n, the boundary BΩ n of Ω n consists of a finite number of closed analytic curves, …”