arXiv subjects

Ignacio Uriarte-Tuero

Publications and source records attributed to Ignacio Uriarte-Tuero.

40 records · Page 3Linked to original sources

arXiv2008

Quasiconformal mappings and singularity of boundary distortion

We extend a well-known theorem by Jones and Makarov [JM] on the singularity of boundary distortion of planar conformal mappings. We use a different technique to recover the previous result and, moreover, generalize the result for quasiconformal mappings of the unit ball $\B^n\subset \mathbb{R}^n$, $n\ge 2$. We also establish an estimate on the Hausdorff (gauge) dimension of the boundary of the image domain outside an exceptional set of given size on the sphere $\partial \B^n$. Furthermore, we show that this estimate is essentially sharp. [JM] P. W. Jones and N. Makarov: Density properties of harmonic measure. Ann. Math. 142 (1995), 427--455.

math.CV↗

arXiv2007

Sharp nonremovability examples for Hölder continuous quasiregular mappings in the plane

Let $α\in(0,1)$, $K\geq 1$, and $d=2\frac{1+αK}{1+K}$. Given a compact set $E\subset\C$, it is known that if $\H^d(E)=0$ then $E$ is removable for $α$-Hölder continuous $K$-quasiregular mappings in the plane. The sharpness of the index $d$ is shown with the construction, for any $t>d$, of a set $E$ of Hausdorff dimension $\dim(E)=t$ which is not removable. In this paper, we improve this result and construct compact nonremovable sets $E$ such that $0<\H^d(E)<\infty$. For the proof, we give a precise planar $K$-quasiconformal mapping whose Hölder exponent is strictly bigger than $\frac{1}{K}$, and that exhibits extremal distortion properties.

math.CV↗

arXiv2007

Theory of dimension for large discrete sets and applications

We define two notions of discrete dimension based on the Minkowski and Hausdorff dimensions in the continuous setting. After proving some basic results illustrating these definitions, we apply this machinery to the study of connections between the Erdos and Falconer distance problems in geometric combinatorics and geometric measure theory, respectively.

math.CO↗

arXiv2006

Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings

The classical Painlevé theorem tells that sets of zero length are removable for bounded analytic functions, while (some) sets of positive length are not. For general $K$-quasiregular mappings in planar domains the corresponding critical dimension is $\frac{2}{K+1}$. We show that when $K>1$, unexpectedly one has improved removability. More precisely, we prove that sets $E$ of $σ$-finite Hausdorff $\frac{2}{K+1}$-measure are removable for bounded $K$-quasiregular mappings. On the other hand, $\dim(E) = \frac{2}{K+1}$ is not enough to guarantee this property. We also study absolute continuity properties of pull-backs of Hausdorff measures under $K$-quasiconformal mappings, in particular at the relevant dimensions 1 and $\frac{2}{K+1}$. For general Hausdorff measures ${\cal H}^t$, $0 < t < 2$, we reduce the absolute continuity properties to an open question on conformal mappings.

math.CV↗