Geodesic
Remarks on the regularity of quasislits
Lukas Schoug1 ; Atul Shekhar2 ; Fredrik Viklund3
1 University of Cambridge, Faculty of Mathematics
2 Université Lyon 1, Département Mathématiques
3 KTH Royal Institute of Technology, Department of Mathematics
Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 355-370.

Voir la notice de l'article provenant de la source Journal.fi

A quasislit is the image of a vertical line segment $[0, iy]$, $y > 0$, under a quasiconformal homeomorphism of the upper half-plane fixing $\infty$. Quasislits correspond precisely to curves generated by the Loewner equation with a driving function in the Lip-$\frac{1}{2}$ class. It is known that a quasislit is contained in a cone depending only on its Loewner driving function Lip-$\frac{1}{2}$ seminorm, $\sigma$. In this note we use the Loewner equation to give quantitative estimates on the opening angle of this cone in the maximal range $\sigma <4$. The estimate is shown to be sharp for small $\sigma$. As consequences, we derive explicit Hölder exponents for $\sigma < 4$ as well as estimates on winding rates. We also relate quantitatively the Lip-$\frac{1}{2}$ seminorm with the quasiconformal dilatation and discuss the optimal regularity of quasislits achievable through reparametrization.  
Keywords: Loewner differential equation, quasicircles, quasislits

Lukas Schoug 1 ; Atul Shekhar 2 ; Fredrik Viklund 3

1 University of Cambridge, Faculty of Mathematics
2 Université Lyon 1, Département Mathématiques
3 KTH Royal Institute of Technology, Department of Mathematics 
@article{AFM_2021_46_1_a19,
     author = {Lukas Schoug and Atul Shekhar and Fredrik Viklund},
     title = {Remarks on the regularity of quasislits},
     journal = {Annales Fennici Mathematici},
     pages = {355--370},
     publisher = {mathdoc},
     volume = {46},
     number = {1},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a19/}
}
TY  - JOUR
AU  - Lukas Schoug
AU  - Atul Shekhar
AU  - Fredrik Viklund
TI  - Remarks on the regularity of quasislits
JO  - Annales Fennici Mathematici
PY  - 2021
SP  - 355
EP  - 370
VL  - 46
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a19/
LA  - en
ID  - AFM_2021_46_1_a19
ER  - 
%0 Journal Article
%A Lukas Schoug
%A Atul Shekhar
%A Fredrik Viklund
%T Remarks on the regularity of quasislits
%J Annales Fennici Mathematici
%D 2021
%P 355-370
%V 46
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a19/
%G en
%F AFM_2021_46_1_a19
Lukas Schoug; Atul Shekhar; Fredrik Viklund. Remarks on the regularity of quasislits. Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 355-370. http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a19/