Decomposition in blocks at the level of wavelet coefficients and T（1 ） theorem on Hardy space

This paper deals with the estabhshment of T(1) theorem on Hardy space H^l under condition of weak regldarity. An operator or a function is identified on the basis of their wavelet coefficients which axe regrouped on some blocks. The actions of each block operator ( pseudo-annular operator) on each block function(atom) axe exactly analyzed to establish T( 1 ) theorem on Hardy space.     

Property (ω) for operator matrices

In this note,property (ω)and property 1 (ω)are variants of Weyl’s theorem. By means of topological uniform descent, the sufficient and necessary conditions of a bounded linear operator defined on a Hilbert space that satisfies property 1 (ω) and property (ω)is studied. Moreover, property 1 (ω)and property (ω)of 2 2operator matrices are discussed as well.     

B-Valued Dyadic Derivative

The principles of the new maximal operator H＊ we defined are discussed. We prove that it is bounded from martingale Hardy-Lorentz L^Xp.q[0,1） to the Lorentz L^Xp.q[0,1） for 1/2〈 p〈∞, 0〈~ q ≤ ∞, where X is any Banach space. When the Banach space X has the RN property, the sequence dnHnf converges to f a.e. Meanwhile the convergence in L^Xp norm for 1≤p〈∞ is a consequence of that the family functions K （n∈N） is an approximate identity.     

THE CONTINUITY OF THE CARATHEODORY OPERATOR

Let X be a set of points on which a completely additive measure is given. Lot F(x.y) be a real-valued function which is defined for every point xeX and for every real number y and satisfies the Caratheodory conditions: F(x, y) is continuous in y for almost every x, and measurable in x for every y. It was proved by B. B. that the operator F defined by the equationis continuous in the space of the measurable functions on X. It was recently proved by M. A. that, if the given measure is continuous, the operator F, whenever it transforms the whole space If into Lq(1≤p,q<+∞) is continuous and baunded.     

ON THE THEORY OF NON-LINEAR OPERATOR EQUATIONS ON CONJUGATELY SIMILAR SPACES AND ITS SOME APPLICATIONS

Conjugately similar space is a special kind of Banach space and wasintroduced by Nakano.The theory of non-linear operator equations on conjugatelysimilar spaces was discussed by S.Yamamuro(sce[?]).The purpese of thispaper is to discuss further this theory and to give its some applications.This paper contains three sections.In§1,we introduce some defin tions usedlater and some elementary properties of the conjugately similar spaces.In§2,we prove the following theorems:Theorem1.Let R be a conjugately similar space and f(x)be a real,Fréchet-differentiable at any point of R,weakly lower semi-continuous functional.If there     

SOME PROPERTIES OF SUPER-DECOMPOSABLE OPERATORS

In this paper,the following main resvlts have been proved:1.Let T∈B(X) be a super-decomposable operator,then for any hyperinvariant subspace Y of T,the operator T/Y is super-decomposable.2.The operator T∈B(X) is super-decomposable if and only if T/Y is super-decomposable for any hyperinvariant subspace Y of T.3. If T∈B(X) has on open spectral resolvent,then T is superdecomposable.     

《一条摄动定理》一文的补遗

In this paper, a simple proof of the following perturbation theorem is given:Suppose that i) T is a self-adjoint operator in the Hilbert space with discrete spectrum σ(T)={λ_n|n=1,2,…}, λ_1≤λ_2≤… and λ_n=cn~P(1+0(1/n)),c>0, p>0.ii) Let 0≤v<1. Let P be a linear operator such that thedomain of P contains the domain of T~v and PT~(-v) is bounded.iii) p(1-v)>1.Then T+P has Property S and T+P and T have an asymptotically common decomposition. Furthermore, if v≤1/2, T+P has the strong Property S.     

On the boundary value problem for harmonic maps of the Poincaré disc

The boundary value problem for harmonic maps of the Poincare disc is discussed. The emphasis is on the non-smoothness of the given boundary values in the problem. Let T . be a subspace of the universal Teichmüller space, defined as a set of normalized quasisymmetric homeomorphisms h of the unit circle S onto itself where h admits a quasiconformal extension to the unit disc D with a complex dilatation μ satisfying where ρ(z)|dz|2 is the Poincare metric of D. Let B . be a Banach space consisting of holomorphic quadratic differentials φ in D with norms It is shown that for any given quasisymmetric homeomorphism h : S1→S1∈ T . , there is a unique quasiconformal harmonic map of D with respect to the Poincare metric whose boundary corresponding is h and the Hopf differential of such a harmonic map belongs to B .     

THE CONTINUITY OF THE YPbUCOH OPERATOR

1. Let X and Y be two sets of points on each of which a completely additive measure is given. Let K(x,y,u) be a real-valued function which is defined for every pair of points (x,y) ∈ (X,Y) and for every real number u, such that, for almost every point x∈X it satisfies the Caratheodory condition with respect to (y,u): K(x,y,u) is measurable in y for every u and continuous in u for almost every y. For every measurable function f(y), the functionKxf(y) = K(x,y,f(y))is measurable in y for almost every x. If this funetion is integrable with respect to y for almost every x, the value of the integral yields a function Kf:defined for almost every a∈ X. We call the functional operator K the operator generated by the function K(x,y,u).     

Chaos in the sense of Li-Yorke and the order of the inverse limit space

Let I= [0,1] and ω0 be the first limit ordinal number. Assume that f: I→I is continuous , piece-wise monotone and the set of periods of f is { 2': i∈ {0} ∪ N}. It is known that the order of (I, f) is ω0 or ω0 + 1. It is shown that the order of the inverse limit space (I, f) is ω0 (resp. ω0 + 1) if and only if f is not (resp. is) chaotic in the sense of Li-Yorke.     

Multipliers on weighted Hardy spaces over locally compact Vilenkin groups

A weighted Hpω(G) multiplier theorem on the multiplier operator T associated with a function m∈ L∞ (Γ) is shown and the atomic decomposition of functions fin Hp*(G) is obtained, where G is a Vilenkin group, r its dual, 0 < p≤1 and ω is a weight on G which is more general than that proposed by C. W. Onneweer et al.     

On p-ω-hyponormal operators

In what follows,H means a complex Hilbert space.Call an invertible operator T log-hyponormal if log (T^*T)≥log(TT^*). Let T=U|T|be the polar decomposition of T.Aluthge in [1] defined the operatorT=|T|^1/2U|T|^1/2 which called the Aluthge transformation of T.An operator T is said to be ω-hyponormal if     

Integral Theorems Based on a New Gradient Operator Derived from Biomembranes (Part II): Applications

Introduction Yin et al.[1, 2] described a gradient operator ? derived from biomembranes with “the second gradient operator” defined on a curved surface. Yin[2] then used the second gradient operator to develop a set of integral theorems named “the second category of integral theorems” on curved surfaces, including the second divergence theorem, the second gradient theorem, the second curl theorem, and the second circulation theorem: d d ?2 dA C A? A = ?K∫∫ ∫ ∫∫i v si Li v A iv (1) …     

A REMARK ON CERTAIN CONVOLUTION OPERATOR

A certain operator D~(a+p-1) defined by convolutions (or Hadamard products) is introduced. The object of this paper is to give an application of the convolution operator D~(a+p-1) to the differential inequalities.     

RIEMANN PROBLEM FOR GENERALIZED HYPERANALYTIC FUNCTIONS ON SET OF COUNTABLE SMOOTH CLOSED CURVES

A continuously differntiable solution of hyercomplex elliptic equationDW+AW+BW=0, z=x+iy (1)is called generalized hyperanalytic function, here D is Douglis differentiat operator. Suppose L consists ofcountable soomth Jordan closed curves L_k k=1, 2, …, the finite domain surrounded by L_k is denotedG_k~+, All G_k~+ don't intersect one another, the positive directions of L_k are determined as usual, and {L_k}converges at a finite z_0. Set L= L∪(Z_0) , G~+ = G_k~+, G~-=C\G~+.This paper deals with the Riemann problem ; find a piecewise generalized hyperanalytic function w(z)in the whole plane C, satisfying the bounbary condition on L     

锥巴拿赫空间上(α,β)-导子的稳定性

The Hyers-Ulam stability of ( α,β ) - derivations from a unital ring R to a R -bimodulewhich is a cone Banach space with the cone norm ||·||P , whereP is a normal cone in a real Banach spaceE is investigated associated with the following functional equation f (( a + b) c ) = f ( a ) α ( c ) + β ( a ) f ( c ) + f (b) α ( c ) + β (b) f ( c ) using the fixed point method and the direct method respectively.     

Research of optimization to solve nonlinear equation based on granular computing

In this article, a real number is defined as a granulation and the real space is transformed into real granu-lar space[1]. In the entironment, solution of nonlinear equation is denoted by granulation in real granular space. Hence,the research of whole optimization to solve nonlinear equation based on granular computing is proposed[2]. In classicalcase, we solve usually accurate solution of problems. If can't get accurate solution, also finding out an approximate solutionto close to accurate solution. But in real space, approximate solution to close to accurate solution is very vague concept. Inreal granular space, all of the approximate solutions to close to accurate solution are constructed a set, it is a granulation inreal granular space. Hence, this granulation is an accurate solution to solve problem in some sense, such, we avoid to sayvaguely "approximate solution to close to accurate solution". We introduce the concept of granulation in one dimension real space. Any positive real number a together with movinginfinite small distance ε will be constructed an interval [a-ε,a ε], we call it as granulation in real granular space, denotedby ε(a) or [a]. We will discuss related properties and operations[3] of the granulations. Let one dimension real space be R, where each real number a will be generated a granulation, hence we get a granularspace R* based on real space R. Obviously, R∈R*. Infinite small number in real space R is only O, and there are three in-finite small granulations in real number granular space R* : [0], [ε] and [-ε]. As the graph in Fig. 1 shows. In Fig. 1,[-ε] is a negative infinite small granulation,[ε] is a positive infinite small granulation,[0] is a infinite small granulation.[a] is a granulation of real number a generating, it could be denoted by interval [a-ε,a ε] in real space [3-5].Letf(x)=0 be a nonliner equation,its graph in interval[-3,10]id showed in Fig.2.Where -3≤x≤10 Relation ρ(f‖,ε)is defied is follows:(x1,x2)∈ p(f‖,ε)iff |f(x1)- f(x2)|<εWhere ε is any given small real number.We have five appoximate solution sets on the nonliner equation f(x)=0 by ρ(f‖,ε)∧|f(x)|[a,b]max,to denote by granulations[xi1 xi2/2],[xi3 xi4/2],[xi5 xi6/2],[xi7 xi8/2]and[xi9 xi10/2]respectively,where |f(x)|[a,b]max denotes local maximum on x ∈[a,b].This is whole optimum on nonliear equation in interval [-3,10].We will get best opmension solution on nonliner equation via computing f(x)to use the five solutions dented by grandlation in one dimension real granlar space[2,5].     

μ Synthesis Method for Robust Control of Uncertain Nonlinear Systems

μ synthesis method for robust control of uncertain nonlinear systems is propored, which is based on feedback linearization. First, nonlinear systems are linearized as controllable linear systems by I/O linearization,such that uncertain nonlinear systems are expressed as the linear fractional transformations (LFTs) on the generalized linearized plants and uncertainty.Then,linear robust controllers are obtained for the LFTs usingμsynthesis method based on H∞ optimization.Finally,the nonlinear robust controllers are constructed by combining the linear robust controllers and the nonlinear feedback.An example is given to illustrate the design.     

The boundedness of multilinear operators of strongly singular integral operators on Hardy spaces

We obtain the boundedness of multilinear operator (BMO) of strongly singular integral operator (~T)A on Lp spaces based on the relation between commutators and multilinear operators. It is found that (~T)A is an (H1, L1) type operator, while TA is not.     

NEWTON IMBEDDING METHOD FOR NONLINEAR ELLIPTIC SYSTEM OF FIRST ORDER

1 Newton imbedding method for nonlinear elliptic complex equation of first orderWe discuss the following nonlinear elliptic complex equation of first order on D;where D is N+1 connected bounded domain. Without loss of generality, D is supposed to be N+ 1 con-nected circular domain in |z|< 1 and its boundary contour is = _j, _j= {|z-z_j|=r_j},j=1,…,N, _0= _N+1= {|z|=1},z=0∈D.First, we introduce two conditions satisfying (1. 1).Condition C; For any function W(z) continuous on D and any complex numbers V_1,V_2, the followinginequalities hold