The notions of Lipschitz and bilipschitz mappings provide classes of mappings connected to the geometry of metric spaces in certain ways. A notion between these two is given by regular mappings (reviewed in Section 1), in which some non-bilipschitz behavior is allowed, but with limitations on this, and in a quantitative way. In this paper we look at a class of mappings called (s, t)-regular mappings. These mappings are the same as ordinary regular mappings when s = t, but otherwise they behave somewhat...

We discuss differentiability properties of convex functions on Heisenberg groups. We show that the notions of horizontal convexity (h-convexity) and viscosity convexity (v-convexity) are equivalent and that h-convex functions are locally Lipschitz continuous. Finally we exhibit Weierstrass-type h-convex functions which are nowhere differentiable in the vertical direction on a dense set or on a Cantor set of vertical lines.

We study the Hardy-Littlewood maximal operator defined via an unconditional norm, acting on block decreasing functions. We show that the uncentered maximal operator maps block decreasing functions of special bounded variation to functions with integrable distributional derivatives, thus improving their regularity. In the special case of the maximal operator defined by the ${\ell }_{\infty }$-norm, that is, by averaging over cubes, the result extends to block decreasing functions of bounded variation, not necessarily...

This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincaré inequality. Such a functional is defined via relaxation, and it defines a Radon measure on the space. For the singular part of the functional, we get the expected integral representation with respect to the variation measure. A new feature is that in the representation for the absolutely continuous part, a constant appears already in the weighted Euclidean...

We study properties of the functional$$\begin{array}{c}\hfill {ℱ}_{\mathrm{loc}}(u,\Omega ):=\underset{\left(u_j\right)}{inf}\left\{\underset{j\to \infty }{lim\; inf}{\int }_{\Omega }f\left(\nabla u_j\right)x\left|\begin{array}{cc}& \left(u_j\right)\subset W_{\mathrm{loc}}^{1,r}\left(\Omega ,\right)\hfill \\ & u_{j}u\mathrm{in}\left(\Omega ,\right)\hfill \end{array}\right.\right\},\end{array}$$F loc ( u,Ω ) : = inf ( u j ) lim inf j → ∞ ∫ Ω f ( ∇ u j ) d x , whereu ∈ BV(Ω;RN), and f:RN × n → R is continuous and satisfies 0 ≤ f(ξ) ≤ L(1 + | ξ | r). For r ∈ [1,2), assuming f has linear growth in certain rank-one directions, we combine a result of [A. Braides and A. Coscia, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 737–756] with a new technique involving mollification to prove an upper bound for Floc. Then, for $r\in [1,\frac{n}{n-1})$ r ∈ [ 1 , n n − 1 ) , we prove that...

We study WDC sets, which form a substantial generalization of sets with positive reach and still admit the definition of curvature measures. Main results concern WDC sets $A\subset {ℝ}^2$. We prove that, for such $A$, the distance function $d_A=\mathrm{dist}(·,A)$ is a “DC aura” for $A$, which implies that each closed locally WDC set in ${ℝ}^2$ is a WDC set. Another consequence is that compact WDC subsets of ${ℝ}^2$ form a Borel subset of the space of all compact sets.

Let P(X,ℱ) denote the property: For every function f: X × ℝ → ℝ, if f(x,h(x)) is continuous for every h: X → ℝ from ℱ, then f is continuous. We investigate the assumptions of a theorem of Luzin, which states that P(ℝ,ℱ) holds for X = ℝ and ℱ being the class C(X) of all continuous functions from X to ℝ. The question for which topological spaces P(X,C(X)) holds was investigated by Dalbec. Here, we examine P(ℝⁿ,ℱ) for different families ℱ. In particular, we notice that P(ℝⁿ,"C¹") holds, where...

We first provide an approach to the conjecture of Bierstone-Milman-Pawłucki on Whitney’s problem on $C^d$ extendability of functions. For example, the conjecture is affirmative for classical fractal sets. Next, we give a sharpened form of Spallek’s theorem on flatness.