Abstract
We define Hardy spaces {mathcal {H}}^p for quasiregular mappings in the plane, and show that for a particular class of these mappings many of the classical properties that hold in the classical setting of analytic mappings still hold. This particular class of quasiregular mappings can be characterised in terms of composition operators when the symbol is quasiconformal. Relations between Carleson measures and Hardy spaces play an important role in the discussion. This program was initiated and developed for Hardy spaces of quasiconformal mappings by Astala and Koskela in 2011 in their paper {mathcal {H}}^p-theory for Quasiconformal Mappings (Pure Appl Math Q 7(1):19–50, 2011).
Highlights
Let D denote the unit disc {z ∈ C; |z| < 1} and T = ∂D
For a given point ξ ∈ T and c > 1, let us denote the cone with vertex at ξ ∈ T as follows (ξ ) = {z ∈ D; |z − ξ | < c (1 − |z|)}
Jerison and Weitsman constructed in [9] an example of an analytic function g ∈ H2 and a quasiconformal mapping φ from the disc onto itself such that the quasiregular map f = g ◦ φ ∈/ Hqpr for any p > 0
Summary
Jerison and Weitsman constructed in [9] an example of an analytic function g ∈ H2 and a quasiconformal mapping φ from the disc onto itself such that the quasiregular map f = g ◦ φ ∈/ Hqpr for any p > 0. Theorem 1 Let φ : D → D be a quasiconformal mapping and 0 < p < ∞. An example is provided by any bounded analytic function precomposed with a quasiconformal mapping whose inverse is not Lipschitz.
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