# Normal families of mappings with controlled $p$-module

Seminar
Speaker
Prof. A. Golberg, Holon Institute of Technology
Date
09/12/2013 - 14:00Add to Calendar 2013-12-09 14:00:00 2013-12-09 14:00:00 Normal families of mappings with controlled $p$-module ~We consider the generic discrete open mappings in ${\mathbb R}^n$ under which the perturbation of extremal lengths of curve collections is controlled integrally via $\int Q(x)\eta^p(|x-x_0|) dm(x)$ with $n-1<p<n$, where $Q$ is a measurable function on ${\mathbb R}^n$ and $\int\limits_{r_1}^{r_2} \eta(r) dr \ge 1$ for any $\eta$ on a given interval $[r_1,r_2].$ The main results state that the family of all open discrete mappings of above type is normal under appropriate restrictions on the majorant $Q.$ We also provide conditions ensuring the local H\"older continuity of such mappings with respect to euclidian distances (in the general case with respect to their logarithms). The inequalities defining the continuity are sharp with respect to the order. This is a joint work with R. Salimov and E. Sevost'yanov. אוניברסיטת בר-אילן - המחלקה למתמטיקה mathoffice@math.biu.ac.il Asia/Jerusalem public
Abstract

~We consider the generic discrete open mappings in ${\mathbb R}^n$ under which the perturbation of extremal lengths of curve collections is controlled integrally via $\int Q(x)\eta^p(|x-x_0|) dm(x)$ with $n-1<p<n$, where $Q$ is a measurable function on ${\mathbb R}^n$ and $\int\limits_{r_1}^{r_2} \eta(r) dr \ge 1$ for any $\eta$ on a given interval $[r_1,r_2].$ The main results state that the family of all open discrete mappings of above type is normal under appropriate restrictions on the majorant $Q.$

We also provide conditions ensuring the local H\"older continuity
of such mappings with respect to euclidian distances (in the general case with respect to their
logarithms). The inequalities defining the continuity are sharp with respect to
the order.

This is a joint work with R. Salimov and E. Sevost'yanov.

תאריך עדכון אחרון : 03/12/2013