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Continuity of Hausdorff measure distortion under planar quasiconformal mappings

Michael T. Lacey
Georgia Institute of Technology

1 Απριλίου 2009, Ζ301, 18:15

It is well-known that a K-quasiconformal measure can distort Hausdorff dimension. Astala famously proved sharp Hausdorff dimension distortion inequalities for planar $K$-quasiconformal mapping $f$: If $E$ had H-dimension $d$, the image has H-dimension at most $d'$, where $d'=2Kd/(2+k-1)d$. We answer in the positive Astala's question if a set of zero Hausdorff $d$-measure is carried into Hausdorff $d'$-measure. The ingredients of the proof come from Astala's original approach, geometric measure theory, and some new weighted norm inequalities for Calderón-Zygmund singular integral operators which cannot be deduced from the classical Muckenhoupt $A_p$ theory.

Joint work with Ignacio Uriate-Tuero and Eric Sawyer.

http://fourier.math.uoc.gr/~ seminar