On uniformly lightness of one class of mappings
D. Romash, E. Sevost'yanov
TL;DR
The paper analyzes mappings that satisfy an inverse $p$-modulus inequality (Poletsky-type) and proves that, under domain regularity and appropriate $Q$-function restrictions, the family is uniformly light: the image of any continuum with a lower-bounded chordal diameter cannot collapse to zero diameter uniformly across the class. A central result, the Main Lemma, gives a uniform lower bound on the image diameter $h(f(C))$ for continua with $h(C)\geoldsymbol{ varepsilon}$ under $p$-uniformity and either $Q ext{ in }FMO$ or a divergence-type condition, with a proof relying on modulus estimates and inversion techniques. The work extends to Nakki-type theorems for $p$-modulus and develops corollaries and theorems (including th2 and th3) that link Loewner space structure and Poincaré inequalities to explicit modulus bounds and diameter distortion, supplemented by sharpness constructions that demonstrate the optimality of the assumptions. Overall, the results provide a rigorous framework for lower distance estimates and uniform diameter control in generalized quasiregular mappings, with implications for stability under local uniform limits.
Abstract
We consider mappings satisfying a certain estimate of the distortion of the modulus of families of paths, similar to the geometric definition of quasiconformal mappings. Under appropriate restrictions, we show that the class of such mappings is uniformly light, i.e., the chordal diameter of the image of continua whose diameter is bounded below is also bounded below uniformly over the class. Under some even greater restrictions, we establish some more explicit estimates of the distortion of the diameters of these continua.