L P boundedness of area function associated with operators on product spaces

Let L be the infinitesimal generator of an analytic semigroup on L2 (Rn) with suitable upper bounds on its heat kernels. Assume that L has a bounded holomorphic functional calculus on L2(Rn). In this paper,we define the Littlewood- Paley g function associated with L on Rn × Rn, denoted by GL(f)(x1, x2), and decomposition, we prove that ‖SL(f)‖p ≈‖GL(f)‖p ≈‖f‖p for 1 ＜ p ＜∞.     

Von Neumann-Jordan Constants of Absolute Normalized Norms on C^n

In this note, we give some estimations of the Von Neumann-Jordan constant C _{N J} (∥·∥_ψ) of Banach space (ℂ ⁿ , ∥·∥_ψ), where ∥·∥_ψ is the absolute normalized norm on ℂ ⁿ given by function ψ. In the case where ψ and φ are comparable, n=2 and C _{N J} (∥·∥_ψ)=1, we obtain a formula of computing C _{N J} (∥·∥_ψ). Our results generalize some results due to Saito and others. Received May 11, 2002, Accepted November 20, 2002 This work is partly supported by NNSF of China (No. 19771056)     

On the global existence and uniqueness of solutions to Prandtl’s system

In this paper, we consider the Prandtl system for the non-stationary boundary layer in the vicinity of a point where the outer flow has zero velocity. It is assumed that U（t, x, y） = x^mU1（t, x）, where 0 〈 x 〈 L and m 〉 1. We establish the global existence of the weak solution to this problem. Moreover the uniqueness of the weak solution is proved.     

Inequalities for convex bodies and polar reciprocal lattices inR n

LetL be a lattice and letU be ano-symmetric convex body inR ⁿ . The Minkowski functional ∥ ∥ _U ofU, the polar bodyU ⁰, the dual latticeL ^*, the covering radius μ(L, U), and the successive minima λ _i (L,U)i=1,...,n, are defined in the usual way. Let ℒ _n be the family of all lattices inR ⁿ . Given a pairU,V of convex bodies, we define and kh(U, V) is defined as the smallest positive numbers for which, given arbitraryL∈ℒ _n andu∈R ⁿ /(L+U), somev∈L ^* with ∥v∥ _V ≤sd(uv, ℤ) can be found. Upper bounds for jh(U, U ⁰), j=k, l, m, belong to the so-called transference theorems in the geometry of numbers. The technique of Gaussian-like measures on lattices, developed in an earlier paper [4] for euclidean balls, is applied to obtain upper bounds for jh(U, V) in the case whenU, V aren-dimensional ellipsoids, rectangular parallelepipeds, or unit balls inl _p ⁿ , 1≤p≤∞. The gaps between the upper bounds obtained and the known lower bounds are, roughly speaking, of order at most logn asn→∞. It is also proved that ifU is symmetric through each of the coordinate hyperplanes, then jh(U, U ⁰) are less thanCn logn for some numerical constantC.     

Asymptotic stability and instability criteria for nonautonomous systems

The classical criterion of asymptotic stability of the zero solution of equations x′ = f(t, x) is that there exists a function V (t, x), a(∥x∥) ≤ V (t, x) ≤ b(∥x∥) for some a, b ∈ K such that [(V)\dot] \dot{V} (t, x) ≤ −c(∥x∥) for some c ∈ K. In this paper, we prove that if V^{(m + 1)} \mathop {V}\limits^{(m + {1})} (t, x) is bounded on some set [tk − T, tk + T] × BH(tk → +∞ as k → ∞), then the condition that [(V)\dot] \dot{V} (t, x) ≤ −c(∥x∥) can be weakened and replaced by that [(V)\dot] \dot{V} (t, x) ≤ 0 and − (−[(V)\dot] \dot{V} (tk, x)| + − [(V)\ddot] \ddot{V} (tk, x)| + ⋯ + − V^(m) \mathop {V}\limits^{(m)} (tk, x)|) ≤ −c′(∥x∥) for some c′ ∈ K. Moreover, the author also presents a corresponding instability criterion. [1–10]     

Strong commutativity-preserving generalized derivations on semiprime rings

Let R be a ring with a subset S. A mapping of R into itself is called strong commutativitypreserving （scp） on S, if [f（x）, f（y）] = [x, y] for all x, y ∈ S. The main purpose of this paper is to describe the structure of the generalized derivations which are scp on some ideals and right ideals of a prime ring, respectively. The semiprime case is also considered.     

Excess of a Lattice

 A pair (x,y) of elements in a lattice is mismatching if x is join-irreducible, y is meet-irreducible, and xy. The excess of a lattice L is defined by ex(L):=|L|− min{|V _x |+|I _y |:(x,y) is mismatching}, where V _x (I _y ) is the principal filter (ideal) generated by x(y), respectively. In the first half of this paper, it is shown that a lattice has excess zero (one) if and only if it is isomorphic to a Boolean lattice (one of a chain of length two, a diamond, and a pentagon), respectively. In the second half, we show that for each relatively complemented lattice which is not Boolean, its size is at most five times its excess. Moreover we determine extremal configurations. Received: September 2, 1999 Final version received: July 25, 2000 Acknowledgments. The author thanks the referee for constructive comments. Present address: Bureau of Human Resources, National Personnel Authority, 1-2-3 Kasumigaseki, Chiyoda-ku, Tokyo 100-8913, Japan. e-mail: abetet@jinji.go.jp     

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.     

On the integrability of hyperbolic systems of riccati-type equations

We consider equations of the form U_xy = U * U_x, where U(x, y) is a function taking values in an arbitrary finite-dimensional algebra T over the field ℂ. We show that every such equation can be naturally associated with two characteristic Lie algebras, L_x and L_y. We define the notion of a ℤ-graded Lie algebraB corresponding to a given equation. We prove that for every equation under consideration, the corresponding algebraB can be taken as a direct sum of the vector spaces L_x and L_y if we define the commutators of the elements from L_x and L_y by means of the zero-curvature relations. Assuming that the algebra T has no left ideals, we classify the equations of the specified type associated with the finite-dimensional characteristic Lie algebras L_x and L_y. All of these equations are Darboux-integrable. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 2, pp. 261–275, November, 1997.     

Approximation Methods for Solving the Cauchy Problem

In this paper we give some new results concerning solvability of the 1-dimensional differential equation y′ = f(x, y) with initial conditions. We study the basic theorem due to Picard. First we prove that the existence and uniqueness result remains true if f is a Lipschitz function with respect to the first argument. In the second part we give a contractive method for the proof of Picard theorem. These considerations allow us to develop two new methods for finding an approximation sequence for the solution. Finally, some applications are given.     

Algebraic properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space

In this paper we investigate some algebra properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space in the Sobolev space. We completely characterize commuting dual Toeplitz operators with harmonic symbols, and show that a dual Toeplitz operator commutes with a nonconstant analytic dual Toeplitz operator if and only if its symbol is analytic. We also obtain the sufficient and necessary conditions on the harmonic symbols for SφSφψ= Sφψ.     

Some properties of Riesz potential associated with SchrSdinger operator

Let L = −Δ + V be the Schr?dinger operator on ℝ ^d , d ≥ 3, where Δ is the Laplacian on ℝ ^d and V ≢ 0 is a nonnegative function satisfying the reverse H?lder inequality. In this article, the author investigates some properties of the Riesz potential I _α ^L associated with L on the Campanato-type spaces Λ _L ^β and the Hardy-type spaces H _L ^p .     

On the range of the derivative of Gâteaux-smooth functions on separable banach spaces

We prove that there exists a Lipschitz function froml ¹ into ℝ² which is Gateaux-differentiable at every point and such that for everyx, y εl ¹, the norm off′(x) −f′(y) is bigger than 1. On the other hand, for every Lipschitz and Gateaux-differentiable function from an arbitrary Banach spaceX into ℝ and for everyε > 0, there always exist two pointsx, y εX such that ‖f′(x) −f′(y)‖ is less thanε. We also construct, in every infinite dimensional separable Banach space, a real valued functionf onX, which is Gateaux-differentiable at every point, has bounded non-empty support, and with the properties thatf′ is norm to weak* continuous andf′(X) has an isolated pointa, and that necessarilya ε 0. This work has been initiated while the second-named author was visiting the University of Bordeaux. The second-named author is supported by grant AV 1019003, A1 019 205, GA CR 201 01 1198.     

Criteria for the single-valued metric generalized inverses of multi-valued linear operators in banach spaces

Let X, Y be Banach spaces and M be a linear subspace in X × Y = {{x, y}|x ∈ X, y ∈ Y }. We may view M as a multi-valued linear operator from X to Y by taking M (x) = {y|{x, y} ∈ M }. In this paper, we give several criteria for a single-valued operator from Y to X to be the metric generalized inverse of the multi-valued linear operator M . The principal tool in this paper is also the generalized orthogonal decomposition theorem in Banach spaces.     

Solvability of Elliptic Boundary Value Problems without Standard Landesman-Lazer Condition

Abstract In this paper we prove a very general result concerning solvability of the resonant problem: Δu + λ_κ u + g(x, u) = h (x); u = 0, x ∈∂Ω, which immediately gives three generalized Landesman-Lazer conditions. The most interesting application of the general result is concerned with the problem when λ_κ = λ₁, in which case we prove solvability results for it under conditions which are not the standard Landesman-Lazer condition or only partly enjoy it. Furthermore, we propose a new sign condition and give a comprehensive extension of a main result of Figueiredo and Ni.     

On the Duality Mapping Sets in Orlicz Sequence Spaces

The criteria for the weak compactness of duality mapping sets J(x) = {f∈X* : <f, x> = ∥f∥² = ∥x∥²} in Orlicz sequence spaces endowed either with the Luxemburg norm or with the Orlicz norm are obtained. Supported by the National Natural Science Foundation of China, Grants 19901007 and 19871020     

The Estimates for Sharp Maximal Functions of Multilinear Strongly Singular Integral Operators

In this paper, the author obtains that the multilinear operators of strongly singular integral operators and their dual operators are bounded from some L^p（R^n） to L^p（R^n） when the m-th order derivatives of A belong to L^p（R^n） for r large enough. By this result, the author gets the estimates for the Sharp maximal functions of the multilinear operators with the m-th order derivatives of A being Lipschitz functions. It follows that the multilinear operators are （L^p, L^p）-type operators for 1 〈 p 〈 ∞.     

On embedding trees into uniformly convex Banach spaces

We investigate the minimum value ofD =D(n) such that anyn-point tree metric space (T, ρ) can beD-embedded into a given Banach space (X, ∥·∥); that is, there exists a mappingf :T →X with 1/D ρ(x,y) ≤ ∥f(x) −f(y)∥ ≤ρ(x,y) for anyx,y εT. Bourgain showed thatD(n) grows to infinity for any superreflexiveX (and this characterized super-reflexivity), and forX =ℓ _p, 1 <p < ∞, he proved a quantitative lower bound of const·(log logn)^min(1/2,1/p). We give another, completely elementary proof of this lower bound, and we prove that it is tight (up to the value of the constant). In particular, we show that anyn-point tree metric space can beD-embedded into a Euclidean space, with no restriction on the dimension, withD =O(√log logn). This paper contains results from my thesis [Mat89] from 1989. Since the subject of bi-Lipschitz embeddings is becoming increasingly popular, in 1997 I finally decided to publish this English version. Supported by Czech Republic Grant GAČR 0194 and by Charles University grants No. 193, 194.     

The Kakeya maximal operator with a special base

LetM _ℛ f be the Kakeya maximal function in d-dimensional Euclidean space with some base ℛ, consisting of cylinders of eccentricity N. The inequality ∥M _ℛ f∥ _d ⩽c(logN)^ε∥ is shown for a base ℛ satisfying a direction condition, where ε and c are constants depending only on d. to the memory of Professor Ruilin Long The author is partly supported by the Grants-in-Aid for Scientific Research, The Ministry of Education, Science and Culture, Japan.     

The riquier problem for a nonlinear equation unresolved with respect to the Lévy iterated laplacian

We present a method of solving for the nonlinear equationf(U(x),Δ _L ² U(x)) = Δ _L U(x) (Δ _L is an infinite-dimensional Laplacian) unresolved with respect to an iterated infinite-dimensional Laplacian and for the Riquier problem for this equation. Ukrainian NII MOD, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 423–427, March, 1999.