Atomic decomposition and boundedness criterion of operators on multi-parameter hardy spaces of homogeneous type

The main purpose of this paper is to derive a new ( p, q)-atomic decomposition on the multi-parameter Hardy space Hp (X1 × X2 ) for 0 p0 p ≤ 1 for some p0 and all 1 q ∞, where X1 × X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both Lq (X1 × X2 ) (for 1 q ∞) and Hardy space Hp (X1 × X2 ) (for 0 p ≤ 1). As an application, we prove that an operator T1, which is bounded on Lq (X1 × X2 ) for some 1 q ∞, is bounded from Hp (X1 × X2 ) to Lp (X1 × X2 ) if and only if T is bounded uniformly on all (p, q)-product atoms in Lp (X1 × X2 ). The similar boundedness criterion from Hp (X1 × X2 ) to Hp (X1 × X2 ) is also obtained.     

Albanese and Picard 1-motives

We define, in a purely algebraic way, 1-motives Alb⁺(X), Alb⁻(X), Pic⁺(X), and Pic⁻(X) associated with any algebraic scheme X over an algebraically closed field of characteristic zero. For X over C of dimension n, the Hodge realizations are, respectively, H^{2n − 1} (X, Z(n))/(torsion), H₁ (X, Z)/(torsion), H¹ (X, Z(1)), and H_{2n − 1} (X, Z(1 n))/(torsion).     

A note on ball-covering property of Banach spaces

By a ball-covering B of a Banach space X, we mean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere SX of X; and X is said to have the ball-covering property (BCP) provided it admits a ball-covering by countably many balls. In this note we give a natural example showing that the ball-covering property of a Banach space is not inherited by its subspaces; and we present a sharp quantitative version of the recent Fonf and Zanco renorming result saying that if the dual X^∗ of X is w^∗ separable, then for every ε>0 there exist a (1+ε)-equivalent norm on X, and an R>0 such that in this new norm SX admits a ball-covering by countably many balls of radius R. Namely, we show that R=R(ε) can be taken arbitrarily close to (1+ε)/ε, and that for X=?₁[0,1] the corresponding R cannot be equal to 1/ε. This gives the sharp order of magnitude for R(ε) as ε→0.     

Integration by parts on the law of the reflecting Brownian motion

We prove an integration by parts formula on the law of the reflecting Brownian motion in the positive half line, where B is a standard Brownian motion. In other terms, we consider a perturbation of X of the form X^ε=X+εh with h smooth deterministic function and ε>0 and we differentiate the law of X^ε at ε=0. This infinitesimal perturbation changes drastically the set of zeros of X for any ε>0. As a consequence, the formula we obtain contains an infinite-dimensional generalized functional in the sense of Schwartz, defined in terms of Hida's renormalization of the squared derivative of B and in terms of the local time of X at 0. We also compute the divergence on the Wiener space of a class of vector fields not taking values in the Cameron-Martin space.     

Boundedness of para-product operators on spaces of homogeneous type

We obtain the boundedness of Calderón-Zygmund singular integral operators T of non-convolution type on Hardy spaces H ^p (X) for 1/(1 + ε) < p ? 1, where X is a space of homogeneous type in the sense of Coifman and Weiss (1971), and ε is the regularity exponent of the kernel of the singular integral operator T. Our approach relies on the discrete Littlewood-Paley-Stein theory and discrete Calderón’s identity. The crucial feature of our proof is to avoid atomic decomposition and molecular theory in contrast to what was used in the literature.     

Asymptotic behaviour of solutions near a turning point: The example of the Brusselator equation

The Brusselator equation is an example of a singularly perturbed differential equation with an additional parameter. It has two turning points: at x=0 and x=-1. We study some properties of so-called canard solutions, that remain bounded in a full neighbourhood of 0 and in the largest possible domain; the main goal is the complete asymptotic expansion of the difference between two values of the additional parameter corresponding to such solutions. For this purpose we need a study of behaviour of the solutions near a turning point; here we prove that, for a large class of equations, if 0 is a turning point of order p, any solution y not exponentially large has, in some sector centred at 0, an asymptotic behaviour (when ε→0) of the form ∑Y_n(x/ε^′)ε^′n, where ε^′p+1=ε, for x=ε^′X with X large enough, but independent of ε. In the Brusselator case, we moreover compute a Stokes constant for a particular nonlinear differential equation.     

A parabolic free boundary problem with double pinning

We study the Cauchy problem for the equation ∂_tu_ε−Δu_ε=−β_ε(u_ε) in (0,∞)×Rⁿ as $\text{ε→0}$, where the nonlinearity β_ε is assumed to converge to a measure concentrated at $\text{0}$. In this paper we allow for sign changes of β_ε and u_ε. The solutions are uniformly Lipschitz continuous in space and Hölder continuous in time. We show that each limit of u_ε is a solution of the free boundary problem ∂_tu−Δu=0 in {u>0}∩(0,∞)×Rⁿ,|∇u⁺|²−|∇u⁻|²=g on (∂{u>0}∪∂{u<0})∩((0,∞)×Rⁿ) in the sense of domain variations. Depending on the structure of the nonlinearity $\text{β}{}_{\mathrm{\epsilon }}\text{,}$ the function g in the condition on the free boundary need not be a constant.     

Explosion et normes Lp pour l'équation des ondes nonlinéaire cubiqueBlow-up and Lp norms for the cubic nonlinear wave equation

We find blow-up solutions of nonlinear wave equations with cubic nonlinearity, in any number of space dimensions, and study the asymptotic behavior of their L^p norms and “energy”. The L^p norm blows up if the blow-up surface has an interior non-degenerate minimum and p?n/2. For less smooth right-hand sides, and 0<ε<1, we give examples for which the L^p norm blows up if p?n/(1+ε); their Cauchy data are unbounded, but blow-up is not instantaneous. Applications to nonlinear optics are briefly outlined. To cite this article: G. Cabart, S. Kichenassamy, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 903–908.     

Torsion and Abelianization in Equivariant Cohomology

Let X be a topological space upon which a compact connected Lie group G acts. It is well known that the equivariant cohomology H ^* _G (X; Q) is isomorphic to the subalgebra of Weyl group invariants of the equivariant cohomology H ^* _T (X; Q), where T is a maximal torus of G. This relationship breaks down for coefficient rings k other than Q. Instead, we prove that under a mild condition on k the algebra H ^* _G (X; k) is isomorphic to the subalgebra of H ^* _T (X; k) annihilated by the divided difference operators.     

On convergence rates and the expectation of sums of random variables

Let X_i be iidrv's and S_n=X₁+X₂+…+X_n. When EX²₁<+∞, by the law of the iterated logarithm $\text{(S}{}_{\mathrm{n}}\text{?α}{}_{\mathrm{n}}\text{)}\text{(n log n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{→0 a.s.}$ for some constants α_n. Thus the r.v. $\text{Y=sup}{}_{\mathrm{n?1}}\text{[|S}{}_{\mathrm{n}}\text{?α}{}_{\mathrm{n}}\text{|?(δn log n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{]}{}^{\mathrm{+}}$ is a.s.finite when δ>0. We prove a rate of convergence theorem related to the classical results of Baum and Katz, and apply it to show, without the prior assumption EX²₁<+∞ that EY^h<+∞ if and only if $\text{E}\text{|X}{}_1\text{|}{}^{\mathrm{2+h}}\text{[log|X}{}_1\text{|]}{}^{-1}\text{<+∞}$ for 0<h<1 and δ> $\text{h}\text{E}\text{(X}{}_1\text{?}\text{E}\text{X}{}_1\text{)}{}^2$, whereas $\text{E}\text{Y}{}^{\mathrm{h}}\text{=+∞}$ whenever h>0 and $\text{0<δ<h}\text{E}\text{(X}{}_1\text{?}\text{E}\text{X}{}_1\text{)}{}^2$.     

Uniform approximation by some Hermite interpolating splines

In this paper some upper bound for the error ∥ s-f ∥_∞ is given, where f ε C¹[a,b], but s is a so-called Hermite spline interpolant (HSI) of degree 2q ?1 such that f(x_i) = s(x_i), f′(rmx_i) = s′(x_i), s^(j) (x_i) = 0 (i = 0, 1, …, n; j = 2, 3, …, q ?1; n > 0, q > 0) and the knots x_i are such that a = x₀ < x₁ < … < x_n = b. Necessary and sufficient conditions for the existence of convex HSI are given and upper error bound for approximation of the function fε C¹[a, b] by convex HSI is also given.     

On the consistency of some partition theorems for continuous colorings,and the structure of ℵ1-dense real order types

We present some techniques in c.c.c. forcing, and apply them to prove consistency results concerning the isomorphism and embeddability relations on the family of ?₁-dense sets of real numbers. In this direction we continue the work of Baumgartner [2] who proved the axiom BA stating that every two ?₁-dense subsets of $\text{R}$ are isomorphic, is consistent. We e.g. prove Con(BA+(2^?₀>?₂)). Let <K^H,<> be the set of order types of ?₁-dense homogeneous subsets of $\text{R}$ with the relation of embeddability. We prove that for every finite model <L, <->: Con(MA+ <K^H, <-> ? <L, <->) iff L is a distributive lattice. We prove that it is consistent that the Magidor-Malitz language is not countably compact. We deal with the consistency of certain topological partition theorems. E.g. We prove that MA is consistent with the axiom OCA which says: “If X is a second countable space of power ?₁, and {U₀,\h.;,U_n?1} is a cover of D(X)

XxX-}<x,x>¦x?X} consisting of symmetric open sets, then X can be partitioned into {X_i \brvbar; i ? ω} such that for every i ? ω there is l<n such that D(X_i)?U_l”. We also prove that MA+OCA [xrArr] 2 ?₀ = ?₂.     

Homogenization of a nonstationary model equation of electrodynamics

In L ₂(?₃;?₃), we consider a self-adjoint operator ? _ε , ε > 0, generated by the differential expression curl η(x/ε)^?1 curl??ν(x/ε) div. Here the matrix function η(x) with real entries and the real function ν(x) are periodic with respect to some lattice, are positive definite, and are bounded. We study the behavior of the operators cos(τ? _ε ^1/2 ) and ? _ε ^?1/2 sin(τ? _ε ^1/2 ) for τ ∈ ? and small ε. It is shown that these operators converge to cos(τ(?⁰)^1/2) and (?₀)^?1/2 sin(τ(?₀)^1/2), respectively, in the norm of the operators acting from the Sobolev space H ^s (with a suitable s) to ?₂. Here ?⁰ is an effective operator with constant coefficients. Error estimates are obtained and the sharpness of the result with respect to the type of operator norm is studied. The results are used for homogenizing the Cauchy problem for the model hyperbolic equation ? _τ ² v _ε = ?? _ε v _ε , div v _ε = 0, appearing in electrodynamics. We study the application to a nonstationary Maxwell system for the case in which the magnetic permeability is equal to 1 and the dielectric permittivity is given by the matrix η(x/ε).     

Strong Convergence Theorems for Two Parameter Vilenkin-Fourier Series

As main result we prove that certain means of the partial sums of two-parameter Vilenkin-Fourier series are uniformly bounded operators from H ^P to L ^p (0 < p ≦ 1). The Hardy space H ^p (0 < p ≦ 1) will be defined by means of a diagonal maximal function. As a consequence we obtain a so-called strong convergence theorem for the Vilenkin-Fourier partial sums. Some dual inequalities are also verified for BMO spaces.     

Convexity of the bounds induced by Markov's inequality

X is a nonnegative random variable such that EX^t < ∞ for 0≤ t < λ ≤ ∞. The (l??) quantile of the distribution of X is bounded above by [?^?1 EX^t]^1?t. We show that there exist positive ?₁ ≥ ?₂ such that for all 0 <?≤?₁ the function g(t) = [?^-1EX^t]^1?t is log-convex in [0, c] and such that for all 0 < ? ≤ ?₂ the function log g(t) is nonincreasing in [0, c].     

An interior point potential reduction algorithm for the linear complementarity problem

The linear complementarity problem (LCP) can be viewed as the problem of minimizingx ^T y subject toy=Mx+q andx, y?0. We are interested in finding a point withx ^T y <ε for a givenε > 0. The algorithm proceeds by iteratively reducing the potential function $$f(x,y) = \rho \ln x^T y - \Sigma \ln x_j y_j ,$$ where, for example,ρ=2n. The direction of movement in the original space can be viewed as follows. First, apply alinear scaling transformation to make the coordinates of the current point all equal to 1. Take a gradient step in the transformed space using the gradient of the transformed potential function, where the step size is either predetermined by the algorithm or decided by line search to minimize the value of the potential. Finally, map the point back to the original space. A bound on the worst-case performance of the algorithm depends on the parameterλ ^*=λ^*(M, ε), which is defined as the minimum of the smallest eigenvalue of a matrix of the form $$(I + Y^{ - 1} MX)(I + M^T Y^{ - 2} MX)^{ - 1} (I + XM^T Y^{ - 1} )$$ whereX andY vary over the nonnegative diagonal matrices such thate ^T XYe ?ε andX _jj Y _jj?n ². IfM is a P-matrix,λ ^* is positive and the algorithm solves the problem in polynomial time in terms of the input size, |log ε|, and 1/λ ^*. It is also shown that whenM is positive semi-definite, the choice ofρ = 2n+ \(\sqrt {2n} \) yields a polynomial-time algorithm. This covers the convex quadratic minimization problem.     

SVEP and compact perturbations

Let H be a complex separable infinite dimensional Hilbert space. In this paper, we prove that an operator T acting on H is a norm limit of those operators with single-valued extension property (SVEP for short) if and only if T^?, the adjoint of T, is quasitriangular. Moreover, if T^? is quasitriangular, then, given an ε>0, there exists a compact operator K on H with ‖K‖<ε such that T+K has SVEP. Also, we investigate the stability of SVEP under (small) compact perturbations. We characterize those operators for which SVEP is stable under (small) compact perturbations.     

A Variational Approach to Discontinuous Problems with Critical Sobolev Exponents

We employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form − Δu = λh(x)H(u − a)u^q + u^2* − 1 in R^N, where λ, a > 0 are parameters, h(x) is both nonnegative and integrable on R^N, H is the Heaviside function, 2* is the critical Sobolev exponent, and 0 ≤ q < 2* − 1. We obtain existence, multiplicity and regularity of solutions by distinguishing the cases 0 ≤ q ≤ 1 and 1 < q < 2* − 1.     

One characterization of symmetry

In this note we present one characterization of symmetry of probability distributions in Euclidean spaces which is formulated as follows. Let X and Y be independent and identically distributed random elements in a separable Euclidean space E. If Ee^h|X|<∞, h>0, then the distribution of X is symmetric if and only if E|(X−Y,t)|^p=E|(X+Y,t)|^p for some 0<p<2 and for any t∈E. The criterion is not correct when at least one of the conditions 0<p<2 or Ee^h|X|<∞ breaks.