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

Compositions of isometric immersions¶in higher codimension

Given a submanifold M ⁿ of Euclidean space ℝ ^{n + p} with codimension p≤6, under generic conditions on its second fundamental form, we show that any other isometric immersion of M ⁿ into ℝ ^{n + p + q} , 0≤q≤n− 2p−1 and 2q≤n+ 1 if q≥ 5, must be locally a composition of isometric immersions. This generalizes several previous results on rigidity and compositions of submanifolds. We also provide conditions under which our result is global. 14 March 2001     

BV has the bounded approximation property

We prove that the space BV (ℝ ^{n) of functions with bounded variation on} ℝ ^{n has the bounded approximation property}.     

An Ito formula for domain-valued processes driven by stochastic flows

 We consider a natural class of stochastic processes taking values in the space of smoothly bounded domains in ℝ ⁿ with compact closure. These processes are generated by stochastic flows on ℝ ⁿ which are obtained as the solutions of stochastic differential equations on ℝ ⁿ . We establish an Ito formula for smooth domain functionals, applied to processes in this class. Received: 2 March 2001 / Revised version: 10 January 2002 / Published online: 22 August 2002     

Operators commuting with a discrete subgroup of translations

We study the structure of operators from the Schwartz space S(ℝ ⁿ ) into the tempered distributions S′(ℝ ⁿ ) that commute with a discrete subgroup of translations. The formalism leads to simple derivations of recent results about the frame operator of shift-invariant systems, Gabor, and wavelet frames.     

Germes de d\'etermination infinie dans des classes lisses

We give sufficient conditions of infinite determinacy (in the sense of Mather) with respect to right equivalence, in subrings of C ^∞ (ℝ ⁿ ,0) defined by estimates on successive derivatives, such as rings of Gevrey germs. In this quantitative setting, a defect (hidden in the classical C ^∞ case) appears between the regularity of equivalent germs and the regularity of local diffeomorphisms of (ℝⁿ,0) giving the equivalence. Our conditions yield precise estimates of this defect, related to some suitable Łojasiewicz exponents of critical loci. We also show that the result is sharp in general. Received: Received: 27 May 1998     

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.     

Global convergence of a class of non-interior point algorithms using Chen-Harker-Kanzow-Smale functions for nonlinear complementarity problems

We propose a class of non-interior point algorithms for solving the complementarity problems(CP): Find a nonnegative pair (x,y)∈ℝ ²ⁿ satisfying y=f(x) and x _i y _i =0 for every i∈{1,2,...,n}, where f is a continuous mapping from ℝ ⁿ to ℝ ⁿ . The algorithms are based on the Chen-Harker-Kanzow-Smale smoothing functions for the CP, and have the following features; (a) it traces a trajectory in ℝ ³ⁿ which consists of solutions of a family of systems of equations with a parameter, (b) it can be started from an arbitrary (not necessarily positive) point in ℝ ²ⁿ in contrast to most of interior-point methods, and (c) its global convergence is ensured for a class of problems including (not strongly) monotone complementarity problems having a feasible interior point. To construct the algorithms, we give a homotopy and show the existence of a trajectory leading to a solution under a relatively mild condition, and propose a class of algorithms involving suitable neighborhoods of the trajectory. We also give a sufficient condition on the neighborhoods for global convergence and two examples satisfying it. Received April 9, 1997 / Revised version received September 2, 1998? Published online May 28, 1999     

Asymptotic behavior of solutions to a wave equation with a non linear dissipative term in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this note we investigate the asymptotic behavior of solutions to the wave equation:u"-Δu+g(u')=0 in ℝⁿxℝ₊.     

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.     

Complex<Emphasis Type="BoldItalic">n</Emphasis>-dimensional manifolds with a real<Emphasis Type="BoldItalic">n</Emphasis><Superscript>2</Superscript>-dimensional automorphism group

The main theorem of this article is a characterization of non compact simply connected complete Kobayashi hyperbolic complex manifold of dimension n≽ 2 with real n ²-dimensional holomorphic automorphism group. Together with the earlier work [11, 12] and [13] of Isaev and Krantz, this yields a complete classification of the simply-connected, complete Kobayashi hyperbolic manifolds with dim_ℝ Aut (M) ≽ (dim_ℂ M)².     

On the theory of mappings with finite area distortion

It is shown that mappings in ℝⁿ with finite distortion of area in all dimensions 1 ≤ k ≤ n − 1 satisfy certain modulus inequalities in terms of inner and outer dilatations of the mappings; in particular, generalizations of the well-known Poletskii inequality for quasiregular mappings are proved. The theory developed is applicable, for example, to the class of finitely bi-Lipschitz mappings, which is a natural generalization of the bi-Lipschitz mappings, as well as isometries and quasi-isometries in ℝⁿ.     

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.     

On the maximal operator associated with certain rotational invariant measures

The aim of this work is to investigate the integrability properties of the maximal operator Mu,associated with a non-doubling measure μ defined on Rn. We start by establishing for a wide class of radial and increasing measures μ that Mu is bounded on all the spaces Lu^p（R^n）,P〉1.Also,we show that there is a radial and increasing measure p for which Mμ does not map Lμ^p（R^n） into weak Lμ^p（R^n）,1≤p〈∞.     

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.     

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.     

Brownian bridge on hyperbolic spaces and on homogeneous trees

Let B be the Brownian motion on a noncompact non Euclidean rank one symmetric space H. A typical examples is an hyperbolic space H _n , n > 2. For ν > 0, the Brownian bridge B ^(ν) of length ν on H is the process B _t , 0 ≤t≤ν, conditioned by B ₀ = B _ν = o, where o is an origin in H. It is proved that the process converges weakly to the Brownian excursion when ν→ + ∞ (the Brownian excursion is the radial part of the Brownian Bridge on ℝ³). The same result holds for the simple random walk on an homogeneous tree. Received: 4 December 1998 / Revised version: 22 January 1999     

On classification of orthogonal multiplications à la do carmo-wallach

The space of range-equivalence classes of full orthogonal multiplications F: ℝ ⁿ ×ℝ ⁿ →ℝ ^p , n≤p ≤n ², is shown to be a compact convex body lying in so(n)⊗so(n). Furthermore, the dimension of the space of equivalence classes is determined to be (n ²(n−1)²)/4−n(n−1).     

The Cauchy problem for nonhomogeneous Navier-Stokes equations inL q([0,T);L p)

In this paper the Cauchy problem for a class of nonhomogeneous Navier-Stokes equations in the infinite cylinderS _T =ℝⁿ x [0,T) is considered. We construct a unique local solution inL ^q([0,T);L ^p (ℝ ⁿ )) for a class of nonhomogeneous Navier-Stokes equations provided that initial data are inL ^r (ℝ ⁿ ), wherer>1 is an exponent determined by the structure of nonlinear terms andp,q are such that 2/q=n(1/r−1/p). Meanwhile under suitable conditions we also obtain thatu(t)≠L ^q([0,∞];L ^p (ℝ ⁿ )) provided that initial data are sufficiently small. This work is supported by the National Natural Sciences Foundation of China and the Foundation of LNM Laboratory of Institute of Mechanics of the Chinese Academy of Sciences.