Densities non-realizable as the Jacobian of a 2-dimensional bi-Lipschitz map are generic

In this work, positive functions defined on the plane are considered from a generic viewpoint, both in the continuous and bounded setting. By pursuing on constructions of Burago–Kleiner and McMullen, we show that, generically, such a function cannot be written as the Jacobian of a bi-Lipschitz homeomorphism.

How Lipschitz Functions Characterize the Underlying Metric Spaces

Let X and Y be metric spaces and E, F be Banach spaces. Suppose that both X and Y are realcompact, or both E, F are realcompact. The zero set of a vector-valued function ƒ is denoted by z(ƒ). A linear bijection T between local or generalized Lipschitz vector-valued function spaces is said to preserve zero-set containments or nonvanishing functions ifrespectively. Every zero-set containment preserver, and every nonvanishing function preserver when dim E = dim F <+∞, is a weighted composition operator . We show that the map τ : Y → X is a locally (little) Lipschitz homeomorphism.

Generalized weak peripheral multiplicativity in algebras of Lipschitz functions

Let (X, d X) and (Y,d Y) be pointed compact metric spaces with distinguished base points e X and e Y. The Banach algebra of all $\mathbb{K}$-valued Lipschitz functions on X — where $\mathbb{K}$ is either‒or ℝ — that map the base point e X to 0 is denoted by Lip0(X). The peripheral range of a function f ∈ Lip0(X) is the set Ranµ(f) = {f(x): |f(x)| = ‖f‖∞} of range values of maximum modulus. We prove that if T 1, T 2: Lip0(X) → Lip0(Y) and S 1, S 2: Lip0(X) → Lip0(X) are surjective mappings such that $Ran_\pi (T_1 (f)T_2 (g)) \cap Ran_\pi (S_1 (f)S_2 (g)) \ne \emptyset $ for all f, g ∈ Lip0(X), then there are mappings φ1φ2: Y → $\mathbb{K}$ with φ1(y)φ2(y) = 1 for all y ∈ Y and a base point-preserving Lipschitz homeomorphism ψ: Y → X such that T j(f)(y) = φ j(y)S j(f)(ψ(y)) for all f ∈ Lip0(X), y ∈ Y, and j = 1, 2. In particular, if S 1 and S 2 are identity functions, then T 1 and T 2 are weighted composition operators.