The Integrability of the Derivative in Conformal Mapping

Fractional integration, differentiation, and weighted Bergman spaces

“…Thus Proposition 5·1 implies that D t f ∈ A q for all q < 2/(2 + t). Alternatively, this fact follows in the case t = 1 from (the less subtle part of) a wellknown result of Brennan [7] which says that if f ∈ S then |f | q is integrable on the unit disc for all −1 − < q < 2 3 as long as > 0 is suitably small; the case 0 < t < 1 then follows as before.…”

“…Finally, in Section 5, we briefly consider the action of fractional differentiation on univalent A p functions; in particular, we generalize part of a well-known result of Brennan [7].…”

Harmonic Measure and SLE

“…In particular, Brennan's conjecture [4] about integrability of |ψ ′ | where ψ is a conformal map from a domain to the unit disc is equivalent to B(−2) = 1, while Carleson-Jones conjecture [5] about the decay rate of coefficient of a univalent function and the growth rate of the length of the Green's lines is equivalent to B(1) = 1/4.…”

Aggregation in the Plane and Loewner's Equation

“…(according to Brennan's conjecture [4] the value q = 3 is critical.) What Theorem A actually states is that an HS(δ)-process can not spend much time in configurations satisfying (1.18).…”