Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces

We use the heat equation to establish the Lipschitz continuity of Cheeger-harmonic functions in certain metric spaces. The metric spaces under consideration are those that are endowed with a doubling measure supporting a (1,2)-Poincaré inequality and in addition supporting a corresponding Sobolev-Poincaré-type inequality for the modification of the measure obtained via the heat kernel. Examples are given to illustrate the necessity of our assumptions on these spaces. We also provide an example to show that in the general setting the best possible regularity for the Cheeger-harmonic functions is Lipschitz continuity.     

Mappings of BMO–bounded distortion

Received: 24 November 1998 / Accepted: 20 October 1999 / Published online: 28 June 2000     

NMR chemical shift and J coupling parameterization and quantum mechanical reference spectrum simulation for selected nerve agent degradation products in aqueous conditions

The spectral parameters of selected nerve agent degradation products relevant to the Chemical Weapons Convention, namely, ethyl methylphosphonate, isopropyl methylphosphonate, pinacolyl methylphosphonate and methylphosphonic acid, were studied in wide range of pH conditions and selected temperatures. The pH and temperature dependence of chemical shifts and J couplings was parameterized using Henderson–Hasselbalch‐based functions. The obtained parameters allowed calculation of precise chemical shifts and J coupling constants in arbitrary pH conditions and typical measurement temperatures, thus facilitating quantum mechanical simulation of reference spectra in the chosen magnetic field strength for chemical verification. Copyright © 2017 John Wiley & Sons, Ltd.     

Isoperimetric inequality via Lipschitz regularity of Cheeger-harmonic functions

Let (X,d,μ)$(X,d,\mu )$ be a complete, locally doubling metric measure space that supports a local weak L²$L^2$-Poincaré inequality. We show that optimal gradient estimates for Cheeger-harmonic functions imply local isoperimetric inequalities.     

Pathwise Connectivity of a Conformal Boundary

This paper demonstrates that, in dimensions n 3, the metricboundary of a conformal deformation of the unit ball is pathwiseconnected, and even of bounded turning, provided that the conformalscaling factor satisfies a Harnack inequality and the volumegrowth of the deformed space is at most euclidean. 2000 MathematicsSubject Classification 30C65.     

Poincaré domains and quasiconformal mappings

The second author is partially supported by NSF (Grant DMS-9305742).     

Definitions of quasiconformality

Summary We establish that the infinitesimal H-definition for quasiconformal mappings on Carnot groups implies global quasisymmetry, and hence the absolute continuity on almost all lines. Our method is new even inR ⁿ where we obtain that the limsup condition in theH-definition can be replaced by a liminf condition. This leads to a new removability result for (quasi)conformal mappings in Euclidean spaces. An application to parametrizations of chord-arc surfaces is also given.Oblatum 25-IV-1994 & 9-VI-1994Both authors supported in part by NSF. The first author also acknowledges the support of the Academy of Finland and the Sloan Foundation.     

Continuity of Sobolev Functions and Dirichletg Finite Harmonic Measures

We investigate the existence of finely continuous Sobolev functions u which have traces equal to the characteristic function of a bounded set E . As an application we examine the existence of Dirichlet finite p-harmonic measures on the unit ball of R ⁿ.