We consider the case of two real variables. A function g : R 2 ! R is blowanalytic if there exists a composition of simple blowings up, each centered at a point, = 1 n : X ! R 2 such that g is analytic ([7][4]). If a homeomorphism h : R 2 ! R 2 and its inverse, h 1 , both have blow-analytic components, we say h is blow-analytic. S. Koike([5]) was the rst to discover a blow-analytic homeomorphis...

We prove limiting imbeddings of spaces with dominating mixed derivatives into the spaces of almost Lipschitz continuous functions.

In this paper the universal properties of spaces of Lipschitz functions, defined over metric spaces, are investigated.

We construct an invariant of the bi-Lipschitz equivalence of analytic function germs (Rn,0) → (R,0) that varies continuously in many analytic families. This shows that the bi-Lipschitz equivalence of analytic function germs admit continuous moduli. For a germ f the invariant is given in terms of the leading coefficients of the asymptotic expansions of f along the sets where the size of |x||grad...

We study Lipschitz differentiability spaces, a class of metric measure spaces introduced by Cheeger in [8]. We show that if an Ahlfors regular Lipschitz differentiability space has charts of maximal dimension, then, at almost every point, all its tangents are uniformly rectifiable. In particular, at almost every point, such a space admits a tangent that is isometric to a finite-dimensional Bana...

We consider metric measure spaces satisfing a doubling condition and a Poincaré inequality in the upper gradient sense. We show that the results of [Che99] on differentiability of real valued Lipschitz functions and the resulting bi-Lipschitz nonembedding theorems for finite dimensional vector space targets extend to Banach space targets having what we term a good finite dimensional approximati...

We consider boundary value problems and transmission problems for strongly elliptic second-order systems with boundary conditions on a compact nonclosed Lipschitz surface S with Lipschitz boundary. The main goal is to find conditions for the unique solvability of these problems in the spaces H , the simplest L2-spaces of the Sobolev type, with the use of potential type operators on S . We also ...

We examine the relationship between infinity harmonic functions, absolutely minimizing Lipschitz extensions, strong absolutely minimizing Lipschitz extensions, and absolutely gradient minimizing extensions in CarnotCarathéodory spaces. Using the weak Fubini property we show that absolutely minimizing Lipschitz extensions are infinity harmonic in any sub-Riemannian manifold.

We compute the Lipschitz-free spaces of subsets of the real line and characterize subsets of metric trees by the fact that their Lipschitz-free space is isometric to a subspace of L1.

In this paper we take up the question of analyticity properties of Dirichlet–Neumann operators with respect to boundary deformations. In two separate results, we show that if the deformation is sufficiently small and lies either in the class of C1+α (any α > 0) or Lipschitz functions, then the Dirichlet–Neumann operator is analytic with respect to this deformation. The proofs of both results ut...