Hardy spaces via distribution spaces

Let ℐ(ℝⁿ) be the Schwartz class on ℝⁿ and ℐ_∞(ℝⁿ) be the collection of functions ϕ ∊ ℐ(ℝⁿ) with additional property that

On the subspace L((x ∧ y)m) of Sm(∧2?4)

We take the exterior power ℝ⁴ ∧ ℝ⁴ of the space ℝ⁴, its mth symmetric power V = S ^m (∧²ℝ⁴) = (ℝ⁴ ∧ ℝ⁴) ∨ (ℝ⁴ ∧ ℝ⁴) ∨ ... ∨(ℝ⁴ ∧ ℝ⁴), and put V ₀ = L((x ∧ y)∨ ... ∨(x ∧ y): x, y ∈ ℝ⁴). We find the dimension of V ₀ and an algorithm for distinguishing a basis for V ₀ efficiently. This problem arose in vector tomography for the purpose of reconstructing the solenoidal part of a symmetric tensor field. Original Russian Text Copyright ? 2009 Gubarev V. Yu. The author was supported by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-344.2008.1). __________ Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 3, pp. 503–514, May–June, 2009.     

A note on certain square functions on product spaces

We study certain square functions on product spaces Rn × Rm, whose integral kernels are obtained from kernels which are homogeneous in each factor Rn and Rm and locally in L(log L) away from Rn × {0} and {0} × Rm by means of polynomial distortions in the radial variable. As a model case, we obtain that the Marcinkiewicz integral operator is bounded on Lp(Rn × Rm)(P > 1) for Ω∈ e Llog L(Sn-1 × Sm-1) satisfying the cancellation condition.     

Darboux property of Gâteaux derivatives of functions on ℝ n

D. Preiss proved that the graph of the derivative of a continuous Gateaux-differentiable function f : ℝ² → ℝ is always connected. We show that this is no longer true in higher dimensions: we construct a continuous, Gateaux-differentiable function f : ℝ³ → ℝ for which the range of its gradient mapping {∇ f(x) : x ∈ ℝ³} is disconnected. We also give an example of an approximately differentiable continuous function on ℝ² such that the range of its gradient mapping is disconnected. The work is a part of the research project MSM 0021620839 financed by MSMT and it was also partly supported by GAČR 201/06/0198 and GAČR 201/06/0018.     

Halves of a real Enriques surface

The real partE _ℝ of a real Enriques surfaceE admits a natural decomposition in two halves,E _ℝ=E _ℝ ⁽¹⁾ ∪E _ℝ ⁽²⁾ , each half being a union of components ofE _ℝ. We classify the triads (E _ℝ;E _ℝ ⁽¹⁾ ,E _ℝ ⁽²⁾ ) up to homeomorphism. Most results extend to surfaces of more general nature than Enriques surfaces. We use and study in details the properties of Kalinin's filtration in the homology of the fixed point set of an involution, which is a convenient tool not widely known in topology of real algebraic varieties.     

Boundedness of Marcinkiewicz integrals and their commutators in H1 (?n × ?m)

The authors establish the boundedness of Marcinkiewicz integrals from the Hardy space H ¹ (ℝ ⁿ × ℝ ^m ) to the Lebesgue space L ¹(ℝ ⁿ × ℝ ^m ) and their commutators with Lipschitz functions from the Hardy space H ¹ (ℝ ⁿ × ℝ ^m ) to the Lebesgue space L ^q (ℝ ⁿ × ℝ ^m ) for some q > 1.     

Discrete Calderón’s Identity,Atomic Decomposition and Boundedness Criterion of Operators on Multiparameter Hardy Spaces

In this paper we establish a discrete Calderón’s identity which converges in both L ^q (ℝ ^n+m ) (1<q<∞) and Hardy space H ^p (ℝ ⁿ ×ℝ ^m ) (0<p≤1). Based on this identity, we derive a new atomic decomposition into (p,q)-atoms (1<q<∞) on H ^p (ℝ ⁿ ×ℝ ^m ) for 0<p≤1. As an application, we prove that an operator T, which is bounded on L ^q (ℝ ^n+m ) for some 1<q<∞, is bounded from H ^p (ℝ ⁿ ×ℝ ^m ) to L ^p (ℝ ^n+m ) if and only if T is bounded uniformly on all (p,q)-product atoms in L ^p (ℝ ^n+m ). The similar result from H ^p (ℝ ⁿ ×ℝ ^m ) to H ^p (ℝ ⁿ ×ℝ ^m ) is also obtained.     

A characterization of the higher dimensional groups associated with continuous wavelets

A subgroup D of GL (n, ℝ) is said to be admissible if the semidirect product of D and ℝ ⁿ , considered as a subgroup of the affine group on ℝ ⁿ , admits wavelets ψ ∈ L²(ℝ ⁿ ) satisfying a generalization of the Calderón reproducing, formula. This article provides a nearly complete characterization of the admissible subgroups D. More precisely, if D is admissible, then the stability subgroup D_x for the transpose action of D on ℝ ⁿ must be compact for a. e. x. ∈ ℝ ⁿ ; moreover, if Δ is the modular function of D, there must exist an a ∈ D such that |det a| ≠ Δ(a). Conversely, if the last condition holds and for a. e. x ∈ ℝ ⁿ there exists an ε > 0 for which the ε-stabilizer D _x ^ε is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups.     

Dirichlet forms for diffusion in ℝ<Superscript>2</Superscript> and jumps on fractals: The regularity problem

We define a Dirichlet form ɛ describing diffusion in ℝ ^d and jumps in a fractal Γ ⊂ ℝ ^d . The jump measure J is defined as an image of a jump measure j of a process in a non-Archimedean metric space. As the result the jump intensity depends on the hierarchical structure of Γ rather than the geometric distance in ℝ ^d . For a class of fractals in ℝ² we find a condition on the measure j so that the Dirichlet form ɛ is regular. The condition is given in terms of Hausdorff dimension of Γ.     

Boundedness of Hausdorff operators on some product Hardy type spaces

We study Hausdorff operators on the product Besov space B01,1 (Rn × Rm) and on the local product Hardy space h1 (Rn × Rm).We establish some boundedness criteria for Hausdorff operators on these functio...     

On the spaces C k (ℝ) ∩ L p (ℝ)

We show that the Fréchet-Sobolev spaces C(ℝ) ∩ L ^p (ℝ) and C ^k (ℝ) ∩ L ^p (ℝ) are not isomorphic for p ≠ 2 and k ∈ ℕ. Research supported by the Italian MURST.     

Analytic functions over a field of power series

 We extend the notion of absolute convergence for real series in several variables to a notion of convergence for series in a power series field ℝ((t ^Γ)) with coefficients in ℝ. Subsequently, we define a natural notion of analytic function at a point of ℝ((t ^Γ))^m. Then, given a real function f analytic on a open box I of ℝ ^m , we extend f to a function f ^★ which is analytic on a subset of ℝ((t ^Γ)) ^m containing I. We prove that the functions f ^★ share with real analytic functions certain basic properties: they are , they have usual Taylor development, they satisfy the inverse function theorem and the implicit function theorem. Received: 5 October 2000 / Revised version: 19 June 2001 / Published online: 12 July 2002     

Modified Mini finite element for the Stokes problem in ℝ2 or ℝ3

We analyze a modified version of the Mini finite element (or the Mini* finite element) for the Stokes problem in ℝ² or ℝ³. The cross‐grid element of order one in ℝ³ is also analyzed. The stability is verified with the aid of the macroelement technique introduced by Stenberg. Each of these methods converges with first order in h as the Mini element does. Numerical tests are given for the Mini* element in comparison with the Mini element when Ω is a unit square on ℝ². This revised version was published online in June 2006 with corrections to the Cover Date.     

Éclatement des coefficients des séries entières et deux théorèmes de Gabrielov

Given a non trivial power series in ℝ ^m × ℝ ^k , it is in general not possible to choose a good direction in ℝ ^k in order to apply Weierstrass Preparation Theorem. Now, one can make it possible by blowing-up coefficients in ℝ ^m . This enables e. g. to prove in some natural way Gabrielov’s complement theorem, as well as Gabrielov’s fiber components theorem in subanalytic geometry.      

Laguerre minimal surfaces in ℝ3

Laguerre geometry of surfaces in R^3 is given in the book of Blaschke, and has been studied by Musso and Nicolodi, Palmer, Li and Wang and other authors. In this paper we study Laguerre minimal surface in 3-dimensional Euclidean space R^3. We show that any Laguerre minimal surface in R^3 can be constructed by using at most two holomorphic functions. We show also that any Laguerre minimal surface in R^3 with constant Laguerre curvature is Laguerre equivalent to a surface with vanishing mean curvature in the 3-dimensional degenerate space R0^3.     

Automorphismes des cônes convexes

One studies the subgroups of GL(m,ℝ) which preserve a properly convex cone of ℝ ^m and whose action on ℝ ^m is irreducible. In particular, one describes the Zariski closure of these subgroups. As an application, one describes the Zariski closure G of the subgroups of GL(m,ℝ) all of whose elements have nothing but positive eigenvalues. For instance, one can get the group G=GL(m,ℝ) if and only if m≠≡2 modulo4.

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Automorphismes des c?nes convexes

Résumé. On étudie les sous-groupes de GL(m,ℝ) qui préservent un c?ne convexe saillant de ℝ ^m et dont l’action sur ℝ ^m est irréductible. En particulier, on décrit les adhérences de Zariski possibles pour ces sous-groupes. Comme application, on décrit les adhérences de Zariski G possibles pour les sous-groupes de GL(m,ℝ) dont tous les éléments ont toutes leurs valeurs propres positives. Par exemple, le groupe G=GL(m,ℝ) convient si et seulement si m≠≡2 modulo4.

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Oblatum 22-I-1999 & 10-XI-1999?Published online: 21 February 2000     

The structure of area-minimizing double bubbles

We show that the least area required to enclose two volumes in ℝⁿ orS ⁿ forn ≥ 3 is a strictly concave function of the two volumes. We deduce that minimal double bubbles in ℝⁿ have no empty chambers, and we show that the enclosed regions are connected in some cases. We give consequences for the structure of minimal double bubbles in ℝⁿ. We also prove a general symmetry theorem for minimal enclosures ofm volumes in ℝⁿ, based on an idea due to Brian White. Supported in part by NSF DMS-9409166.     

Classification of topologically conjugate affine mappings

We consider affine mappings from ℝ ⁿ into ℝ ⁿ , n ≥ 1. We prove a theorem on the topological conjugacy of an affine mapping that has at least one fixed point to the corresponding linear mapping. We give a classification, up to topological conjugacy, for affine mappings from R into R and also for affine mappings from ℝ ⁿ into ℝ ⁿ , n > 1, having at least one fixed point and the nonperiodic linear part. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 134–139, January, 2009.     

Littlewood–Paley theorem on spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)</Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We point out that if the Hardy–Littlewood maximal operator is bounded on the space L ^p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L ^p (ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L ^p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L ^p(t) (ℝ ⁿ ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ ⁿ , if and only if p(t) = const. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008.