in this paper, a complete description concerning linear operators of banach spaces with range in lipschitz algebras $lip_al(x)$ is provided. necessary and sufficient conditions are established to ensure boundedness and (weak) compactness of these operators. finally, a lower bound for the essential norm of such operators is obtained.

We present a proof of a Ramsey-type theorem for infinite metric spaces due to Matoušek. Then we show that for every K > 1 every uncountable Polish space has a perfect subset that K-bi-Lipschitz embeds into the real line. Finally we study decompositions of infinite separable metric spaces into subsets that, for some K > 1, K-bi-Lipschitz embed into the real line.

In this paper we address some of the most fundamental questions regarding the diierentiability structure of locally Lipschitz functions deened on Banach spaces. For example, we examine the relationship between inte-grability, D-representability and strict diierentiability. In addition to this, we show that on a large class of Banach spaces there is a signiicant family of locally Lipschitz funct...

We show that a Lipschitz domain can be expanded solely near a part of its boundary, assuming that the part is enclosed by a piecewise C curve. The expanded domain as well as the extended part are both Lipschitz. We apply this result to prove a regular decomposition of standard vector Sobolev spaces with vanishing traces only on part of the boundary. Another application in the construction of lo...

A leaf of a compact foliated space has a well defined quasi-isometry type and it is a natural question to ask which quasi-isometry types of (intrinsic) metric spaces can appear as leaves of foliated spaces. There are two more or less related concepts of quasi-isometry. The first one is that used in Riemannian geometry, namely, two (Lipschitz) manifolds are quasi-isometric if there is a Lipschit...

We prove that the Lipschitz free spaces over certain types of discrete metric have Radon–Nikodým property. also show space a complete, locally compact has Schur or approximation property when

We study p-harmonic functions on metric measure spaces, which are formulated as minimizers to certain energy functionals. For spaces supporting a p-Poincaré inequality, we show that such functions satisfy an infinitesmal Lipschitz condition almost everywhere. This result is essentially sharp, since there are examples of metric spaces and p-harmonic functions that fail to be locally Lipschitz co...