Friday January 23, 2009
4:00 - 5:00 pm
Milner Hall 317
4:00 - 5:00 pm
Milner Hall 317
Michael Lacey, Georgia Institute of Technology
Continuity of Hausdorff measure distortion under planar quasiconformal mappings
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 Ap theory. Joint work with Ignacio Uriate-Tuero and Eric Sawyer.
Continuity of Hausdorff measure distortion under planar quasiconformal mappings
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 Ap theory. Joint work with Ignacio Uriate-Tuero and Eric Sawyer.