Volume 8, Issue 1

The aim of this note is to prove a representation theorem for left–invariant functionals in Carnot groups. As a direct consequence, we can also provide a Г -convergence result for a smaller class of functionals.

This paper studies the Hausdorff dimension of the intersection of isotropic projections of subsets of ℝ 2 n , as well as dimension of intersections of sets with isotropic planes. It is shown that if A and B are Borel subsets of ℝ 2 n of dimension greater than m, then for a positive measure set of isotropic m-planes, the intersection of the images of A and B under orthogonal projections onto these planes have positive Hausdorff m -measure. In addition, if A is a measurable set of Hausdorff dimension greater than m , then there is a set B ⊂ ℝ 2 n with dim B ⩽ m such that for all x ∈ ℝ 2 n \ B there is a positive measure set of isotropic m-planes for which the translate by x of the orthogonal complement of each such plane, intersects A on a set of dimension dim A – m . These results are then applied to obtain analogous results on the n th Heisenberg group.

We study Hausdorff convergence (and related topics) in the chordalization of a metric space to better understand pointed Gromov-Hausdorff convergence of quasihyperbolic distances (and other conformal distances).

The deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.

We show that the combination of doubling and (1, p )-Poincaré inequality is equivalent to a version of the A p -condition on rooted K -ary trees.

Hare, Mendivil, and Zuberman have recently shown that if X ⊂ ℝ is compact and of non-zero Assouad dimension dim A X , then for all s > dim A X , X supports measures with Assouad dimension s . We generalize this result to arbitrary complete metric spaces.

Gromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.

We prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.

In this paper, using the idea of ultrametrization of metric spaces we introduce ultradiversification of diversities. We show that every diversity has an ultradiversification which is the greatest nonexpansive ultra-diversity image of it. We also investigate a Hausdorff-Bayod type problem in the setting of diversities, namely, determining what diversities admit a subdominant ultradiversity. This gives a description of all diversities which can be mapped onto ultradiversities by an injective nonexpansive map. The given results generalize similar results in the setting of metric spaces.

Let (𝒳, d , μ ) be a space of homogeneous type, in the sense of Coifman and Weiss, with the upper dimension ω . Assume that η ∈(0, 1) is the smoothness index of the wavelets on 𝒳 constructed by Auscher and Hytönen. In this article, via grand maximal functions, the authors introduce the Hardy–Lorentz spaces H*p,q(𝒳) H_*^{p,q}\left( \mathcal{X} \right) with the optimal range p∈(ωω+η,∞) p \in \left( {{\omega \over {\omega + \eta }},\infty } \right) and q ∈ (0, ∞]. When and p∈(ωω+η,1] p \in ({\omega \over {\omega + \eta }},1] q ∈ (0, ∞], the authors establish its real-variable characterizations, respectively, in terms of radial maximal functions, non-tangential maximal functions, atoms, molecules, and various Littlewood–Paley functions. The authors also obtain its finite atomic characterization. As applications, the authors establish a real interpolation theorem on Hardy–Lorentz spaces, and also obtain the boundedness of Calderón–Zygmund operators on them including the critical cases. The novelty of this article lies in getting rid of the reverse doubling assumption of μ by fully using the geometrical properties of 𝒳 expressed via its dyadic reference points and dyadic cubes and, moreover, the results in the case q ∈ (0, 1) of this article are also new even when 𝒳 satisfies the reverse doubling condition.

The aim of this paper is to prove the weak type vector-valued inequality for the modified Hardy– Littlewood maximal operator for general Radon measure on ℝ n . Earlier, the strong type vector-valued inequality for the same operator and the weak type vector-valued inequality for the dyadic maximal operator were obtained. This paper will supplement these existing results by proving a weak type counterpart.

In this article the authors study complex interpolation of Sobolev-Morrey spaces and their generalizations, Lizorkin-Triebel-Morrey spaces. Both scales are considered on bounded domains. Under certain conditions on the parameters the outcome belongs to the scale of the so-called diamond spaces.

In this paper, we study the boundedness and compactness of the commutator of Calderón– Zygmund operators T on spaces of homogeneous type ( X , d , µ ) in the sense of Coifman and Weiss. More precisely, we show that the commutator [ b , T ] is bounded on the weighted Morrey space Lωp,k(X) L_\omega ^{p,k}\left( X \right) with κ ∈ (0, 1) and ω ∈ A p ( X ), 1 < p < ∞, if and only if b is in the BMO space. We also prove that the commutator [ b , T ] is compact on the same weighted Morrey space if and only if b belongs to the VMO space. We note that there is no extra assumptions on the quasimetric d and the doubling measure µ .

We study the space BMO 𝒢 (𝕏) in the general setting of a measure space 𝕏 with a fixed collection 𝒢 of measurable sets of positive and finite measure, consisting of functions of bounded mean oscillation on sets in 𝒢. The aim is to see how much of the familiar BMO machinery holds when metric notions have been replaced by measure-theoretic ones. In particular, three aspects of BMO are considered: its properties as a Banach space, its relation with Muckenhoupt weights, and the John-Nirenberg inequality. We give necessary and sufficient conditions on a decomposable measure space 𝕏 for BMO 𝒢 (𝕏) to be a Banach space modulo constants. We also develop the notion of a Denjoy family 𝒢, which guarantees that functions in BMO 𝒢 (𝕏) satisfy the John-Nirenberg inequality on the elements of 𝒢.

We consider generalized weak Morrey spaces with variable growth condition on spaces of homogeneous type and characterize the pointwise multipliers from a generalized weak Morrey space to another one. The set of all pointwise multipliers from a weak Lebesgue space to another one is also a weak Lebesgue space. However, we point out that the weak Morrey spaces do not always have this property just as the Morrey spaces not always.

In this paper, we present a construction of frames on the Heisenberg group without using the Fourier transform. Our methods are based on the Calderón-Zygmund operator theory and Coifman’s decomposition of the identity operator on the Heisenberg group. These methods are expected to be used in further studies of several complex variables.

We consider different notions of boundary traces for functions in Sobolev spaces defined on regular trees and show that the almost everywhere existence of these traces is independent of the chosen definition of a trace.

We consider the general framework of a metric measure space satisfying the doubling volume property, associated with a non-negative self-adjoint operator, whose heat kernel enjoys standard Gaussian localization. We prove embedding theorems between Triebel-Lizorkin spaces associated with operators. Embeddings for non-classical Triebel-Lizorkin and (both classical and non-classical) Besov spaces are proved as well. Our result generalize the Euclidean case and are new for many settings of independent interest such as the ball, the interval and Riemannian manifolds.