BMO and Asymptotic Homogeneity

Abstract

First, we prove that the BMO condition by John–Nirenberg leads in the natural way to the asymptotic homogeneity at the origin of regular homeomorphic solutions of the degenerate Beltrami equations. Then, on this basis we establish a series of criteria for the existence of regular homeomorphic solutions of the degenerate Beltrami equations in the whole complex plane with asymptotic homogeneity at infinity. These results can be applied to the fluid mechanics in strongly anisotropic and inhomogeneous media because the Beltrami equation is a complex form of the main equation of hydromechanics.

Approximation Theory and Related Applications

“…In [1], the authors discussed the well-known BMO class of functions of bounded mean oscillation by John-Nirenberg, which, long ago, became one of the most important concepts in harmonic analysis, partial differential equations, and related areas. Specifically, they investigated its applications to modern mapping theory.…”