We show that there exists a strong uniform embedding from any proper metric space into any Banach space without cotype. Then we prove a result concerning the Lipschitz embedding of locally finite subsets of Lp-spaces. We use this locally finite result to construct a coarse bi-Lipschitz embedding for proper subsets of any Lp-space into any Banach space X containing the l n p ’s. Finally using an...

Let f be a bounded function on the real line IF!. One may characterize the structural properties off by the modulus of smoothness w(t,f) = sup{lf (4 -f( y)l; x, y E 08, I x y I < t>, and, if w(t) is a continuous nondecreasing function of t > 0 such that w(O) = 0, by the Lipschitz class Lip(w) which is the set of all continuous functions such that su~~<~<i w(t, f)/o(t) < 00. It is possible to ex...

For a metric space (K, d) the Banach space Lip(K) consists of all scalar-valued bounded Lipschitz functions on K with the norm ‖f‖L = max(‖f‖∞, L(f)), where L(f) is the Lipschitz constant of f . The closed subspace lip(K) of Lip(K) contains all elements of Lip(K) satisfying the lip-condition lim0<d(x,y)→0 |f(x) − f(y)|/d(x, y) = 0. For K = ([0, 1], | · |), 0 < α < 1, we prove that lip(K) is a p...

In [11] the author proved that every quasiconformal harmonic mapping between two Jordan domains with C, 0 < α ≤ 1, boundary is biLipschitz, providing that the domain is convex. In this paper we avoid the restriction of convexity. More precisely we prove: any quasiconformal harmonic mapping between two Jordan domains Ωj , j = 1, 2, with C, j = 1, 2 boundary is bi-Lipschitz.

We study functions on topometric spaces which are both (metrically) Lipschitz and (topologically) continuous, using them in contexts where, in classical topology, ordinary continuous functions are used. We study the relations of such functions with topometric versions of classical separation axioms, namely, normality and complete regularity, as well as with completions of topometric spaces. We ...

In this paper we study removable singularities for holomorphic functions such that supz∈Ω |f (z)|dist(z, ∂Ω) < ∞. Spaces of this type include spaces of holomorphic functions in Campanato classes, BMO and locally Lipschitz classes. Dolzhenko (1963), Král (1976) and Nguyen (1979) characterized removable singularities for some of these spaces. However, they used a different removability concept th...

We show that an analytic subset of the finite dimensional Euclidean space R is purely unrectifiable if and only if the image of any of its compact subsets under every local Lipschitz quotient function is a Lebesgue null. We also construct purely unrectifiable compact sets of Hausdorff dimension greater than 1 which are necessarily sent to finite sets by local Lipschitz quotient functions.

The Lipschitz and C harmonic capacities κ and κc in R n can be considered as high-dimensional versions of the so-called analytic and continuous analytic capacities γ and α (respectively). In this paper we provide a dual characterization of κc in the spirit of the classical one for the capacity α by means of the Garabedian function. Using this new characterization, we show that κ(E) = κ(∂oE) for...