An extension of the Uchiyama theorem and its application to composition operators which preserve BMO

Uchiyama constructed certain extremal BMO functions on Euclidean spaces which show that the John-Nirenberg estimate is in some sense remarkably precise. We extend his result to the case of general subdomains of Euclidean spaces. Also, as an application, we characterize Lebesgue measurable maps between general subdomains of Euclidean spaces which preserve BMO. To the memory of Professor Nobuyuki Suita     

Instability for harmonic foliations on compact homogeneous spaces

In this paper we discuss the instability of harmonic foliations on compact submanifolds immersed in Euclidean spaces and compact homogeneous spaces. We obtain a sufficient condition for a harmonic foliation to be unstable on compact submanifolds in a Euclidean space and on compact isotropy irreducible homogeneous spaces. We also classify compact symmetric spaces which have no non-trivial stable harmonic foliation.     

A characterization of Lévy probability distribution functions on Euclidean spaces

In 1937, Paul Lévy proved two theorems that characterize one-dimensional distribution functions of class L. In 1972, Urbanik generalized Lévy's first theorem. In this note, we generalize Lévy's second theorem and obtain a new characterization of Lévy probability distribution functions on Euclidean spaces. This result is used to obtain a new characterization of operator stable distribution functions on Euclidean spaces and to show that symmetric Lévy distribution functions on Euclidean spaces need not be symmetric unimodal.     

On the representation of biharmonic functions with singularities in ℝ n

Biharmonic functions are defined on Euclidean spaces, Riemannian manifolds, infinite trees, and more generally on abstract harmonic spaces. In this note, we consider biharmonic functions b defined on annular sets Ω \ K and obtain Laurent-type decompositions for b in the Euclidean spaces and in infinite trees. Particular importance is given to the investigation when b extends as a distribution on Ω.     

Nonlinear A-Dirac Equations

This paper is a study of solutions to nonlinear Dirac equations, in domains in Euclidean space, which are generalizations of the Clifford Laplacian as well as elliptic equations in divergence form. A Caccioppoli estimate is used to prove a global integrability theorem for the image of a solution under the Euclidean Dirac operator. Oscillation spaces for Clifford valued functions are used which generalize the usual spaces of bounded mean oscillation, local Lipschitz continuity or local order of growth of real-valued functions.     

Globular lattices

We study “globular lattices”, i.e. compact sublattices of Euclidean spaces for which the sets of primes agrees with the set of co-primes. These lattices are a generalization of homogeneous sublattices of Euclidean spaces. For dimension n?4, a full classification of globular lattices is obtained.     

Isosceles triangles in Riemannian geometry – a characterization of the <Emphasis Type="Italic">n</Emphasis>-sphere

New characterization theorems for the linear Euclidean spaces and Euclidean spheres are given in terms of convexity and isosceles triangles in Riemannian manifolds.     

A Schwarz Lemma for Harmonic Mappings Between the Unit Balls in Real Euclidean Spaces

The authors prove a Schwarz lemma for harmonic mappings between the unit balls in real Euclidean spaces. Roughly speaking, this result says that under a harmonic mapping between the unit balls in real Euclidean spaces, the image of a smaller ball centered at origin can be controlled. This extends the related result proved by Chen in complex plane.     

Erratum to: “Embeddings of Spaces of Functions of Positive Smoothness on Irregular Domains in Lebesgue Spaces”

Mathematical Notes - An embedding theorem of weighted spaces of functions of positive smoothness defined on irregular domains of n-dimensional Euclidean space in weighted Lebesgue spaces is...     

Singularities of Solution Surfaces for Quasilinear First-Order Partial Differential Equations

For a closed curve in a CAT(K) space with given circumradius and upper bound on curvature, a basic lower bound on the length is established. The inequality is sharp, assumed only when the curve is the boundary of an isometric copy of a racetrack (the convex hull of two congruent circles) from a plane of constant curvature K. Previously such a theorem was proved for Euclidean plane curves by G.D.Chakerian, H.H. Johnson, and A. Vogt, and for curves in higher dimensional Euclidean spaces by A.D. Milka. A similar theorem is proved for nonclosed curves, with a notion of breadth replacing circumradius. Thus we illustrate how singular methods can extend classical Euclidean theorems to a large class of new spaces (including Riemannian manifolds of curvature bounded above) and also give significant strengthenings even in Euclidean space.     

A Remark on Lower Bounds for the Chromatic Numbers of Spaces of Small Dimension with Metrics ℓ1 and ℓ2

Mathematical Notes - A particular class of estimates related to the Nelson–Erd?s–Hadwiger problem is studied. For two types of spaces, Euclidean and spaces with metric ?1,...     

On the Riesz potential operator of variable order from variable exponent Morrey space to variable exponent Campanato space

For the Riesz potential of variable order over bounded domains in Euclidean space, we prove the boundedness result from variable exponent Morrey spaces to variable exponent Campanato spaces. A special attention is paid to weaken assumptions on variability of the Riesz potential.     

Wavelet para-bases and sampling numbers in function spaces on domains

This paper deals with wavelet frames (para-bases), local polynomial reproducing formulas, and sampling numbers in function spaces on arbitrary and on E-thick domains in Euclidean n-space. In an Appendix we collect some recent instruments for corresponding function spaces on Euclidean n-space.     

On the spaces Fs p.q of iiardy-soblow type: equivalent quasi-norys,multipliers, spaces on doilains,regular elliptic botjndary vaiue prblems

Smary. The paper is the continuation of [22] . It deals with Hardy-Sobolev spaces in the Euclidean nspace and in domains, as well as with geaeral regular elliptic boundary value problems in these spaces.     

Embeddings of Spaces of Functions of Positive Smoothness on Irregular Domains

Doklady Mathematics - For spaces of functions of positive smoothness defined on irregular domains of n-dimensional Euclidean space, an embedding theorem into spaces of the same type is proved and...     

A pointwise characterization of functions of bounded variation on metric spaces

We give a new characterization of the space of functions of bounded variation in terms of a pointwise inequality connected to the maximal function of a measure. The characterization is new even in Euclidean spaces and it holds also in general metric spaces.     

On a problem of Kuiper

We construct explicit Morse functions on Grassmannian manifolds, and use them to find explicit taut embeddings of the quadratic hypersurfaces in complex projective spaces into Euclidean spaces.     

Sobolev spaces and symbolic calculus

We use elementary theory of distributions and geometry of Euclidean spaces to obtain a theorem on the symbolic calculus of several variables in spaces of Fourier series with weights.     

On a problem of Kuiper

We construct explicit Morse functions on Grassmannian manifolds, and use them to find explicit taut embeddings of the quadratic hypersurfaces in complex projective spaces into Euclidean spaces.     

On Embeddings of Tori in Euclidean Spaces

Using the relation between the set of embeddings of tori into Euclidean spaces modulo ambient isotopies and the homotopy groups of Stiefel manifolds, we prove new results on embeddings of tori into Euclidean spaces.