Sobolev Peano cubes

Abstract

A classical theorem of Hahn [8] and Mazurkiewicz [19] states that X is a locally connected continuum if and only if there exists a continuous surjection f : [0,1] → X. Since any cube [0,1] is a continuous image of [0,1], an equivalent statement is: X is a locally connected continuum if and only if there exists a continuous surjection f : [0,1] → X. The purpose of this paper is to generalize the Hahn–Mazurkiewicz theorem to differentiable and weakly differentiable mappings. Not surprisingly, our assumptions on X will be stronger. Following Kirchheim [15], we say that a map f : → X from an open set ⊂ R to a metric space X is metrically differentiable at x ∈ if there is a seminorm ‖·‖x on R such that d(f(x), f(y))− ‖y − x‖x = o(|y − x|) for y ∈ . (1.1) The seminorm assumption means that ‖a + b‖x ≤ ‖a‖x + ‖b‖x and ‖ta‖x = |t |‖a‖x but ‖·‖x can vanish on a linear subspace on R, and (1.1) means that

Locations

• The Michigan Mathematical Journal - View - PDF