In this paper, we give a characterization of compact sets in Lp-spaces on metric measure spaces, which is generalization the Kolmogorov-Riesz theorem. Using criterion, investigate topological type space consisting Lipschitz maps with bounded supports.

We prove that certain Lipschitz properties of the inverse F-1 of a set-valued map F are inherited by the map (f+F)~x when / has vanishing strict derivative. In this paper, we present an inverse mapping theorem for set-valued maps F acting from a complete metric space I toa linear space Y with a (translation) invariant metric. We prove that, for any function f: X -> Y with "vanishing strict deri...

In this paper, three key theorems (the open mapping theorem, the inverse function theorem, and the implicit function theorem) for continuously differentiable maps are shown to hold for nonsmooth continuous maps which are not necessarily Lipschitz continuous. The significance of these extensions is that they are given using generalized Jacobians, called approximate Jacobians. The approximate Jac...

In this paper, we develop dissipativity notions for dynamical systems with discontinuous vector fields. Specifically, we consider dynamical systems with Lebesgue measurable and locally essentially bounded vector fields characterized by differential inclusions involving Filippov set-valued maps specifying a set of directions for the system velocity and admitting Filippov solutions with absolutel...

Abstract In this paper, we prove the Lipschitz regularity of continuous harmonic maps from a finite-dimensional Alexandrov space to compact smooth Riemannian manifold. This solves conjecture F. H. Lin in [38]. The proof extends argument Huang-Wang [28].

The Lipschitz function algebras were first defined in the 1960s by some mathematicians, including Schubert. Initially, the Lipschitz real-value and complex-value functions are defined and quantitative properties of these algebras are investigated. Over time these algebras have been studied and generalized by many mathematicians such as Cao, Zhang, Xu, Weaver, and others. Let  be a non-emp...

our aim in this paper is to prove an analog of younis's theorem on the image under the jacobi transform of a class functions satisfying a generalized dini-lipschitz condition in the space $mathrm{l}_{(alpha,beta)}^{p}(mathbb{r}^{+})$, $(1< pleq 2)$. it is a version of titchmarsh's theorem on the description of the image under the fourier transform of a class of functions satisfying the dini-lip...

We study the quantitative properties of Lipschitz mappings from Euclidean spaces into metric spaces. prove that it is always possible to decompose domain such a mapping pieces on which “behaves like projection mapping” along with “garbage set” arbitrarily small in an appropriate sense. Moreover, our control quantitative, i.e., independent both particular and space maps into. This i...

we characterize compact composition operators on real banach spaces of complex-valued bounded lipschitz functions on metric spaces, not necessarily compact, with lipschitz involutions and determine their spectra.

We show that the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem with general demands has integrality gap (log n)Ω(1). This is achieved by exhibiting n-point metric spaces of negative type whose L1 distortion is (log n)Ω(1). Our result is based on quantitative bounds on the rate of degeneration of Lipschitz maps from the Heisenberg group to L1 when restricted to cosets of the...