Uniform quotient mappings of the plane with non-discrete point inverses

We construct uniform quotient mappings of the plane which have points with non-discrete inverse and good modulus of continuity. In particular, we find a monotone uniform quotient mappingf: ℝ² → ℝ² having a point with nontrivial inverse such that lim _r→0[Ω(r)/r ^γ]=0 for anyγ<1/2, where Ω is the modulus of continuity off. Supported by DFG grant Hi 584/2-1.     

Riesz transforms for a non-symmetric Ornstein-Uhlenbeck semigroup

Let (ℋ _t ) _t≥0 be the Ornstein-Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix −λ(I+R), where λ>0 and R is a skew-adjoint matrix and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on L ²(γ _∞). We investigate the weak type 1 estimate of the Riesz transforms for (ℋ _t ) _t≥0. We prove that if the matrix R generates a one-parameter group of periodic rotations then the first order Riesz transforms are of weak type 1 with respect to the invariant measure γ _∞. We also prove that the Riesz transforms of any order associated to a general Ornstein-Uhlenbeck semigroup are bounded on L ^p (γ _∞) if 1<p<∞. The authors have received support by the Italian MIUR-PRIN 2005 project “Harmonic Analysis” and by the EU IHP 2002-2006 project “HARP”.     

Spectral invariance for pseudodifferential operators on weighted function spaces

The spectrum of each symmetric ψ DO of the symbol class S⁰ _{1, γ}, 0≤γ<1, acting on B³ _p,q(w(x)) and F³ _p,q(w(x)), is independent of the choice ofs ∈ℝ, 0<p≤∞ (p<∞ in the F-case), 0<q≤∞ and the weight w(x)∈W.     

The Maximal Operator Associated to a Nonsymmetric Ornstein–Uhlenbeck Semigroup

Let (ℋ _t ) _t≥0 be the Ornstein–Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix λ(R−I), where λ>0 and R is a skew-adjoint matrix, and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein–Uhlenbeck semigroups which are normal on L ²(γ _∞). We prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ_* f(x)=sup  _t≥o |ℋ _t f(x)| is of weak type 1 with respect to the invariant measure γ _∞. We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L ^p (γ _∞) if and only if 1<p≤∞.      

Markov Jump Processes Approximating a Non-Symmetric Generalized Diffusion

Consider a non-symmetric generalized diffusion X(⋅) in ℝ ^d determined by the differential operator $A(\mbox{\boldmath{$A(\mbox{\boldmath{. In this paper the diffusion process is approximated by Markov jump processes X _n (⋅), in homogeneous and isotropic grids G _n ⊂ℝ ^d , which converge in distribution in the Skorokhod space D([0,∞),ℝ ^d ) to the diffusion X(⋅). The generators of X _n (⋅) are constructed explicitly. Due to the homogeneity and isotropy of grids, the proposed method for d≥3 can be applied to processes for which the diffusion tensor $\{a_{ij}(\mbox{\boldmath{$\{a_{ij}(\mbox{\boldmath{ fulfills an additional condition. The proposed construction offers a simple method for simulation of sample paths of non-symmetric generalized diffusion. Simulations are carried out in terms of jump processes X _n (⋅). For piece-wise constant functions a _ij on ℝ ^d and piece-wise continuous functions a _ij on ℝ² the construction and principal algorithm are described enabling an easy implementation into a computer code.     

A stochastic approximation algorithm with multiplicative step size modification

An algorithm of searching a zero of an unknown function ϕ: ℝ → ℝ is considered: x _t = x _t−1 − γ _t−1 y _t , t = 1, 2, ..., where y _t = ϕ(x _t−1) + ξ _t is the value of ϕ measured at x _t−1 and ξ _t is the measurement error. The step sizes γ _t > 0 are modified in the course of the algorithm according to the rule: γ _t = min{uγ _t−1, } if y _t−1 y _t > 0, and γ _t = dγ _t−1, otherwise, where 0 < d < 1 < u, > 0. That is, at each iteration γ _t is multiplied either by u or by d, provided that the resulting value does not exceed the predetermined value . The function ϕ may have one or several zeros; the random values ξ _t are independent and identically distributed, with zero mean and finite variance. Under some additional assumptions on ϕ, ξ _t , and , the conditions on u and d guaranteeing a.s. convergence of the sequence {x _t }, as well as a.s. divergence, are determined. In particular, if P(ξ ₁ > 0) = P (ξ ₁ < 0) = 1/2 and P(ξ ₁ = x) = 0 for any x ∈ ℝ, one has convergence for ud < 1 and divergence for ud > 1. Due to the multiplicative updating rule for γ _t , the sequence {x _t } converges rapidly: like a geometric progression (if convergence takes place), but the limit value may not coincide with, but instead, approximate one of the zeros of ϕ. By adjusting the parameters u and d, one can reach arbitrarily high precision of the approximation; higher accuracy is obtained at the expense of lower convergence rate.      

Multi-parameter singular radon transforms I: The <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> theory

The purpose of this paper is to study the L ² boundedness of operators of the form f ↦ ψ(x) ∫ f (γ _t (x))K(t)dt, where γ _t (x) is a C ^∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝ ^N × ℝ ⁿ , satisfying γ ₀(x) ≡ x, ψ is a C ^∞ cut-off function supported on a small neighborhood of 0 ∈ ℝ ⁿ , and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝ ^N . The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L ². The case when K is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderón- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of L ^p boundedness, while the third paper deals with the special case when γ is real analytic.     

Weak Morrey spaces and strong solutions to the Navier-Stokes equations

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp*,λ(Rn) on the basis of Lorentz space Lp,∞ = Lp*(Rn)(in particular, Mp*,0(Rn) = Lp,∞, if p ＞ 1), and study some fundamental properties of them; Second,bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato's method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp*,λ(Rn) (1＜p≤n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp*,n-p(Rn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato's results.     

Removable singularities and non-uniqueness of solutions of a singular diffusion equation

 We prove that the solution u of the equation u _t =Δlog u, u>0, in (Ω\{x ₀})×(0,T), Ω⊂ℝ², has removable singularities at {x ₀}×(0,T) if and only if for any 0<α<1, 0<a<b<T, there exist constants ρ₀, C ₁, C ₂>0, such that C ₁ |x−x ₀|^α≤u(x,t)≤C ₂|x−x ₀|^−α holds for all 0<|x−x ₀|≤ρ₀ and a≤t≤b. As a consequence we obtain a sufficient condition for removable singularities at {∞}×(0,T) for solutions of the above equation in ℝ²×(0,T) and we prove the existence of infinitely many finite mass solutions for the equation in ℝ²×(0,T) when 0≤u ₀∉L ¹ (ℝ²) is radially symmetric and u ₀L _loc ¹(ℝ²). Received: 16 December 2001 / Revised version: 20 May 2002 / Published online: 10 February 2003 Mathematics Subject Classification (1991): 35B40, 35B25, 35K55, 35K65     

Pseudodifferential Operators on Local Hardy Spaces

In this work, we study the continuity of pseudodifferential operators on local Hardy spaces h ^p (ℝ ⁿ ) and generalize the results due to Goldberg and Taylor by showing that operators with symbols in S _1,δ ⁰(ℝ ⁿ ), 0≤δ<1, and in some subclasses of S _1,1⁰(ℝ ⁿ ) are bounded on h ^p (ℝ ⁿ ) (0<p≤1). As an application, we study the local solvability of the planar vector field L=∂ _t +ib(x,t)∂ _x , b(x,t)≥0, in spaces of mixed norm involving Hardy spaces. Work supported in part by CNPq, FINEP, and FAPESP.     

Diophantinc approximation by linear forms on manifolds

The following Khintchine-type theorem is proved for manifoldsM embedded in ℝ ^k which satisfy some mild curvature conditions. The inequality |q·x| <Ψ(|q|) whereΨ(r) → 0 asr → ∞ has finitely or infinitely many solutionsqεℤ ^k for almost all (in induced measure) points x onM according as the sum Σ _r ^{= 1/∞} Ψ(r)r ^k−2 converges or diverges (the divergent case requires a slightly stronger curvature condition than the convergent case). Also, the Hausdorff dimension is obtained for the set (of induced measure 0) of point inM satisfying the inequality infinitely often whenψ(r) =r ^−t . τ >k − 1.     

Properties of the Flows Generated by Stochastic Equations with Reflection

We consider the properties of a random set ϕ _t (ℝ ₊ ^d ), where ϕ _t (x) is a solution of a stochastic differential equation in ℝ ₊ ^d with normal reflection from the boundary that starts from a point x. We characterize inner and boundary points of the set ϕ _t (ℝ ₊ ^d ) and prove that the Hausdorff dimension of the boundary ∂ϕ _t (ℝ ₊ ^d ) does not exceed d − 1. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1069 – 1078, August, 2005.     

Asymptotiques d’un systeme dynamique conservatif associe a une equation elliptique non lineaire

LetM be a compact riemannian manifold,h an odd function such thath(r)/r is non-decreasing with limit 0 at 0. Letf(r)=h(r)-γr and assume there exist non-negative constantsA andB and a realp>1 such thatf(r)>Ar ^P-B. We prove that any non-negative solutionu ofu _tt+Δ_gu=f(u) onM x ℝ⁺ satisfying Dirichlet or Neumann boundary conditions on ϖM converges to a (stationary) solution of Δ _g Φ=f(Φ) onM with exponential decay of ‖u-Φ‖C ²(M). For solutions with non-constant sign, we prove an homogenisation result for sufficiently small λ; further, we show that for every λ the map (u(0,·),u _t(0,·))→(u(t,·), u _t(t,·)) defines a dynamical system onW ^1/2(M)⊂C(M)×L ²(M) which possesses a compact maximal attractor.      

Construction of Optimal Cubature Formulas Related to?Computer Tomography

We study the problem of the optimization of approximate integration on the class of functions defined on the parallelepiped Π _d =[0,a ₁]×⋅⋅⋅×[0,a _d ], a ₁,…,a _d >0, having a given majorant for the modulus of continuity (relative to the l ₁-metric in ℝ ^d ). An optimal cubature formula, which uses as information integrals of f along intersections of Π _d with n arbitrary (d−1)-dimensional hyperplanes in ℝ ^d (d>1) is obtained. We also find an asymptotically optimal sequence of cubature formulas, whose information functionals are integrals of f along intersections of Π _d with shifts of (d−2)-dimensional coordinate subspaces of ℝ ^d (d>2).     

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.     

Singular spherical maximal operators on a class of two step nilpotent lie groups

LetH ⁿ ≅ℝ²ⁿ ⋉ℝ be the Heisenberg group and letμ _t be the normalized surface measure for the sphere of radiust in ℝ²ⁿ . Consider the maximal function defined byM f=sup _t>0|f*μ _t |. We prove forn≥2 thatM defines an operator bounded onL ^p (H ⁿ ) provided thatp>2n/(2n−1). This improves an earlier result by Nevo and Thangavelu, and the range forL ^p boundedness is optimal. We also extend the result to a more general class of surfaces and to groups satisfying a nondegeneracy condition; these include the groups of Heisenberg type. The second author was supported in part by the National Science Foundation.     

Non-Strebel points and variability sets

This paper studies the subset of the non-Strebel points in the universal Teichmuller space T. Let Z0 ∈ △be a fixed point. Then we prove that for every non-Strebel point h, there is a holomorphic curve γ : [0, 1]→ T with h as its initial point satisfying the following conditions.(1) The curve γ is on a sphere centered at the base-point of T, i.e. dT(id, γ(t)) = dT(id, h), (t ∈ [0, 1]).(2) For every t ∈ (0,1], the variability set Vγ(t)[Z0] of γ(t) has non-empty interior, i.e. Vγ(t) [Z0] ≠ .     

Inertial manifold for a hyperbolic equation with dissipation

Sufficient conditions for the existence of an inertial manifold are found for the equation u _tt + 2γu _t − Δu = f(u, u _t ), u = u(x, t), x ∈ Ω ⋐ ℝ ^N , u| _∂Ω = 0, t > 0 under the assumption that the function f satisfies the Lipschitz condition.     

Reciprocal polynomials with all zeros on the unit circle

Let f(x)=a _d x ^d +a _d−1 x ^d−1+⋅⋅⋅+a ₀∈ℝ[x] be a reciprocal polynomial of degree d. We prove that if the coefficient vector (a _d ,a _d−1,…,a ₀) or (a _d−1,a _d−2,…,a ₁) is close enough, in the l ¹-distance, to the constant vector (b,b,…,b)∈ℝ ^d+1 or ℝ ^d−1, then all of its zeros have moduli 1.