The Ubiquitous Quasidisk

Abstract

For additional information and updates on this book, visit www.ams.org/bookpages/surv-184 Library of Congress Cataloging-in-Publication Data Gehring, Frederick W. The ubiquitous quasidisk / Frederick W. Gehring, Kari Hag ; with contributions by Ole Jacob Broch. pages cm.-(Mathematical surveys and monographs ; volume 184) Includes bibliographical references and index. ISBN 978-0-8218-9086-8 (alk. paper) 1. Quasiconformal mappings. 2. Geometric function theory. 3. Functions of a complex variable. I. Hag,

The boundary Harnack inequality for variable exponent $$p$$ p -Laplacian, Carleson estimates, barrier functions and $${p(\cdot )}$$ p ( · ) -harmonic measures

“…We note that in fact the (interior) corkscrew condition is implied by a uniform domain, see Bennewitz and Lewis [17] and Gehring [30]. Among examples of NTA domains, we mention quasidisks, bounded Lipschitz domains and domains with fractal boundary such as the von Koch snowflake.…”

Apollonian metric, uniformity and Gromov hyperbolicity

“…It is natural to study whether or not these metrics are comparable in some sense. It turns out that the comparison properties of metrics imply geometric properties of the domain: this idea was used by Gehring and Osgood [7] to characterise so called uniform domains, by Gehring and Hag [5] to study quasidisks, by Vuorinen [28] to define ϕ-uniform domains, by Hästö [8] to study comparison properties of so called Apollonian metric. Seittenranta [16] defined a Möbius invariant metric on subdomains of R n and, comparing this metric to Ferrand's metric, defined a Möbius invariant class of domains.…”

The Apollonian inner metric and uniform domains

“…out that, in fact, the constant d can be choosen to zero by possibly increasing the constant c. See also [6,10]. The domain satisfying k D ≈ j D had been applied in many areas of analysis (see [5,20]). We refer to the standard book by Vuorinen [28] where one can find results about some of these metrics.…”