We study the density of set SNA$(M,Y)$ those Lipschitz maps from a (complete pointed) metric space $M$ to Banach $Y$ which strongly attain their norm (i.e., supremum defining is actually maximum). present new and somehow counterintuitive examples, we give some applications. First, show that SNA$(\mathbb T,Y)$ not dense in Lip$\_0(\mathbb for any $Y$, where $\mathbb T$ denotes unit circle Euclid...

We prove a basic inequality involving anticommutators in noncommutative $$L_p$$ -spaces. use it to complete our study of the Mazur maps from $$L_q$$ showing that they are Lipschitz on balls when $$0<q<p<\infty $$ .

We give examples of analytic circle maps with singularities of break type with the same rotation number and the same size of the break for which no conjugacy is Lipschitz continuous. In the second part of the paper, we discuss a class of rotation numbers for which a conjugacy is C1-smooth, although the numbers can be strongly non-Diophantine (Liouville). For the rotation numbers in this class, ...

The problem considered in the paper can be described as follows. We are given a continuous mapping from one metric space into another which is regular (in the sense of metric regularity or, equivalently, controllability at a linear rate) near a certain point. How small may be an additive perturbation of the mapping which destroys regularity? The paper contains a new proof of a recent theorem of...

We obtain an explicit uniform upper bound for the derivative of a conformal mapping unit disk onto convex domain. This estimate depends only on outer and inner radii domain, minimum curvature radius its boundary. Its proof is based Möbius invariant metric hyperbolic type, introduced by Kulkarni Pinkall (Math Z 216(1):89–129, 1994).