Some new Triebel-Lizorkin spaces on spaces of homogeneous type and their frame characterizations

Let(X,p,μ)d,θ be a space of homogeneous type,(?) ∈(0,θ],|s|＜(?) andmax{d/(d＋(?)),d/(d＋s＋(?))}＜q≤∞.The author introduces the new Triebel-Lizorkin spaces (?)_∞q~s(X) and establishes the framecharacterizations of these spaces by first establishing a Plancherel-P(?)lya-type inequalityrelated to the norm of the spaces (?)_∞q~s(X).The frame characterizations of the Besovspace (?)_pq~s(X) with|s|＜(?),max{d/(d＋(?)),d/(d＋s＋(?))}＜p≤∞ and 0＜q≤∞and the Triebel-Lizorkin space (?)_pq~s(X)with|s|＜(?),max{d/(d＋(?)),d/(d＋s＋(?))}＜p＜∞ and max{d/(d＋(?)),d/(d＋s＋(?))}＜q≤∞ are also presented.Moreover,the au-thor introduces the new TriebeI-Lizorkin spaces b(?)_∞q~s(X) and H(?)_∞q~s(X) associated to agiven para-accretive function b.The relation between the space b(?)_∞q~s(X) and the spaceH(?)_∞q~s(X) is also presented.The author further proves that if s＝0 and q＝2,thenH(?)_∞q~s(X)＝(?)_∞q~s(X),which also gives a new characterization of the space BMO(X),since (?)_∞q~s(X)＝BMO(X).     

THE EXISTENCE OF RADIAL LIMITS OF ANALYTIC FUNCTIONS IN BANACH SPACES

Let X be a comPlex Banach space and let D be the open unit disc in the complex plane.We shall denote by H"(D, X) the Banach space consisting of all uniformly bounded X-vaued analytic functions defined on D equipped with the norm llflloo = suP lIf(z)Il. Az eDcomplex Banach space X is said to have the analytic Radon-NikOdym property if eachelemellt f E Hoo(D,X) has radial limits almost everywhere on the torus T = {e": 0 E[0, 2x]} (see [1]), this means that for almost all 0 C [0,27l, 9W…     

关于某类非线性方程解的存在区间问题

<正> §1.问题与结果考虑非线性方程(E) y″+f(x,y)=0.当 f 满足条件(H) f∈C{a≤x≤b;|y|<∞},且当 y(?)0时 y·f>0时,我们称方程(E)为(E)型方程.Atkinson [1]、Nehari [2]、Moroney [3]、Pimbley [4]等人曾经认为(或默认为),(E)型方程的任一解都能开拓到整个区间[a,b].我们曾在文[5]中指出,这个问题值得讨论.接着本文初稿和[6]构造了反例,阐明此结论不成立;同时研究了(E)型方程任一解都能开拓到 b 的充分条件.所得部份结果如下:     

ON PGLAR PRODUCT OPERATOR

Let H be a Hilbert space, and let A be a linear bounded operator on H. For \(\lambda \in \rho (A)\), the \({U_\lambda } = {(A - \lambda )^{ * - 1}}(A - \lambda )\) is called polar.Produot operator. In this paper, we discuss the properties of \({U_\lambda }\) and the relation between \({U_\lambda }\) and A. We obtain tbe following results. Definition. Let B be a linear bounded operator on H, suppose \(0 \in \rho (B)\). For every \(x,y \in H\), we definite \([x,y] = (Bx,y)(H,B)\)(or (H, [·,·]) is called a non- degenerate bilinear space (it is obvious that if B=B*,then (H,B)is a space with an indefinite metric; and that if B>0, then (H,B) is a Hilbert Space. If an operator U(A) satisfies \[[Ux,Uy] = [x,y]([Ax,y] = [x,Ay]),x,y \in H\] then the operator U(A) is called a wvitary (self adjoint) on (H,B). Theorem I . Suppose A is a linear bounded operator on H, (1) If \(0 \in \rho (A)\), then \(U = {A^{ * - 1}}A\) is a unitary operator on (H,A) or (H, A*), and \(\sigma (U) = \frac{1}{{\sigma (U)}}\). (2) If there is a complex number \(\alpha \), such that \({\mathop{\rm Im}\nolimits} A \ge \alpha > 0\) then a)\(0 \in \rho (A)\), and the operator \(U = {A^{ * - 1}}A\) is a unitary on Hilbert space \((H,{\mathop{\rm Im}\nolimits} A) and 1 \in \rho (U)\);b) there exist two Hilbert spaces \((H,{v_1}),(H,{v_2})\), such that A, A* are all the unitary operator from (H,v1) onto (H,v2), and there are two spectral measures \(\{ E_\lambda ^i,\lambda \in [\alpha ,\chi ] \subset (0,2\pi )\} ,i = 1,2,\), such that \(AE_\Delta ^1H \subset E_\Delta ^2H,{A^ * }E_\Delta ^1H \subset E_\Delta ^2H\) for any \(\Delta = (\lambda ,u] \subset (0,2\pi ]\). (3) If \(0 \in \rho (A) \cap \rho ({\mathop{\rm Im}\nolimits} A)\) then the operator \(U = {A^{ * - 1}}A\) is a unitary on \((H,{\mathop{\rm Im}\nolimits} A)\). with an indefinite met He, and \(1 \in \rho (U)\). (4) For any complex number \(\lambda = r{e^{i\theta }},\left| \lambda \right| > \left\| A \right\|\), then \({U_\lambda }\) must be a unitary operator on the Hilbert space \(\left( {H,{\mathop{\rm Im}\nolimits} (\frac{1}{{i\lambda }}A + iI)} \right),and - {e^{i2\theta }} \in \rho ({U_\lambda })\) Theorem 2. (1) A is a normal operator iff there exists a complex number \(\lambda \), \(\lambda \in \rho (A)\), such that \(\frac{{\partial {U_\lambda }}}{{\partial \lambda }}{U_\lambda } = {U_\lambda }\frac{{\partial {U_\lambda }}}{{\partial \lambda }}\),where \(\frac{\partial }{{\partial \lambda }}\) is the directional derivative. (2) If there exists a complex number \(\alpha ,{\mathop{\rm Im}\nolimits} A \ge \alpha > 0\) then A is a normal operator iff \(U = {A^{ * - 1}}A\) is also. (3) If A is a hyperiwrmal or a subnormal, then for every \(\lambda \in \rho (A),\sigma ({U_\lambda })\) lies on the circle. 3, 4期 关于极?积算子炉一U 499 Theorem 3? Let A be a linear bounded operator on Hm Suppose 0￡p(J.)? (1) If U=А*~гА is a unitary operator in a certain non-degenerate bilinear space (H} 1 J |Л|>1 入 J Ш=1 solmble iff (l)cr(ü) =-=i=-, (2) the operotor Ui= f XdE\ is unitarity equivalent to cr(C7) J |л|>1 the operator ül_1 == ШЕ1, 、 J 1Л|>1 If the conditions (1),(2) are satified, then we have (3) the subspace JJi?^2 of H reduces any solution of the equation A*"1 A — U, and 1 AI (Ягея*)1 = ^Uo where Ъ is any in vertible self-adjoint on (i?i?J?2) L, and bv Uo (Uo = Jiai i ЫЕ1 )}⑷41Я1фя, = , Ла = A1ü2, if the operator V is to realize TJt and Ut^unitary equivalence^ then Ai = VSf where S is any invertible on Нг and SvUi. " Corollary. Swpp)se operator U is a normal on a certain Hilbert space(Hy v) (^the U is similar to a certain normal operator on IT), if U satisfies the conditions (1)、(2) of Theorem^ on ?S,v) ? Then the general form of the solution of А*~гА = U is A = vA'y where A' is same as A in Theorem^. Theorem 5. Let U be a linear bounded operator on H. Suppose O?p(J.)and p(ü) is a simply connected region, then the equation А*~гЛ = U is a solvable iff there exists a certain space {H,v) with a indefinite metric, such that U is a unitary operator on (Hf 丨）. If is a unitary operator on ?H,v),then there exists a particular solution of I X+$ А*~гА = U: Ar = 2e v [ (?7 —X)_1+X], where eie ? p (JJ), and the general form of the solution is Аж АУ, wliere V is any in vertible self-adjoint on (H, -u)and VvU,     

Bers型空间和复合算子

For α∈（0,∞),let Hα^∞（or Hα^∞,0)denote the collection of all functions f which are analytic on the unit disc D and satisfy |f(z)|(1-|z|^2)^α=O(1)(or|f(z)|(1-|z|^2)^α=o(1) as |z|→1).Hα^∞,0)is called a Bers-type space (or a little Bers-type space).In this paper,we give some basic properties of Hα^∞，Cψ，the composition operator associated with a symbol function ψ which is an analytic self map of D,is difined by Cψf=f o ψ,We characterize the boundedness,and compactness of Cψ which sends one Bers-type space to another function space.     

FIXED POINT THEOREMS FOR MULTI-VALUED MAPPINGS IN LOCALLY CONVEX SPACES

<正> Throughout the present paper it is assumed that E is a real,locally convex,Hausdorfftopological vector space,and X(?)E a nonempty compact convex subset.A multi-valuedmapping F:X→2~E is said to be upper(lower) semicontinuous if for any open(closed) setU(?)E the set{x∈X|F(x)(?)U)is open (closed) in X;or,equivalently,for any closed(open) set V(?)E the set{X∈X|F(x)∩ V(?)(?)}is closed (open) in X.Moreover,F     

Holomorphic Vector Bundle on Hopf Manifolds with Abelian Fundamental Groups

Let X be a Hopf manifolds with an Abelian fundamental group.E is a holomorphic vectorbundle of rank r with trivial pull-back to W=C~n-{0}.We prove the existence of a non-vanishingsection of L(?)E for some line bundle on X and study the vector bundles filtration structure of E.These generalize the results of D.Mall about structure theorem of such a vector bundle E.     

On approximation of smooth functions from null spaces of optimal linear differential operators with constant coefficients

For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln(ψ) = n ∑ k=0 akψ(k), where the constant coefficients ak ∈ R may be adapted to f . We prove that for each f ∈ C(n)(I), there is a selection of coefficients {a1, ,an} and a corresponding linear combination Sn( f ,t) = n ∑ k=1 bkeλkt of functions ψk(t) = eλkt in the nullity of L which satisfies the following Jackson’s type inequality: f (m) Sn(m )( f ,t) ∞≤ |an|2n|Im|1/1q/ep|λ|λn|n|I||nm1 Ln( f ) p, where |λn| = mka x|λk|, 0 ≤ m ≤ n 1, p,q ≥ 1, and 1p + q1 = 1. For the particular operator Mn(f) = f + 1/(2n) f(2n) the rate of approximation by the eigenvalues of Mn for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed.     

The Lebesgue Constants and Gibbs Phenomenon for Non-negative (f , dn) Summability Method

The ( f,d_n) -summability method is defined as follows^[1,4]: Let f be a nonconstant function, analytic in |z | < R for R > l, and let {d_n} be a sequence of complex numbers,such that for all n,$d_n \ne -f(1)$.Suppose that the elements of the metrix A = (a_nk) are given by the relations $a_00=1,a_0k=0(k \geq 1)$ $[\prod\limits_{j = 1}^n {\frac{{f(z) + {d_j}}}{{f(1) + {d_j}}} = \sum\limits_{k = 0}^\infty {{a_{nk}}{z^k}} } \]$ A sequence {S_n} is said to be ( f, d_n), —summable to s, if \sigma_n = \sum\limits_{k=0}^\infty \arrow s as n \arrow \infty. The ( f, d_n) —summability method is said to be non-negative if for all n, d_n> 0 and the Maclaurin coefficients of f are real and non-negative. The Lebesgue constants for the ( f,d_n)-method are defined by $L_n(A)=2/\pi \int_0^\pi /2 {\frac{|\sum\limits_{k=0}^\infty {a_nk sin(2k+1)t|}{sint}dt}$ In this parer we prove the following two theorems.     

直径不超过4的树的割宽

§ 1　IntroductionThe cutwidth problem for graphs,as well as a class of optimal labeling and embed-ding problems,have significant applications in VLSI designs,network communicationsand other areas (see [2 ] ) .We shall follow the graph-theoretic terminology and notation of [1 ] .Let G=(V,E)be a simple graph with vertex set V,| V| =n,and edge set E.A labeling of G is a bijec-tion f:V→ { 1 ,2 ,...,n} ,which can by regarded as an embedding of G into a path Pn.Fora given labeling f of G,th…     

A NECESSARY AND SUFFICIENT CONDITION OF EXISTENCE OF GLOBAL SOLUTIONS FOR SOME NONLINEAR HYPERBOLIC EQUATIONS

ＡＮＥＣＥＳＳＡＲＹＡＮＤＳＵＦＦＩＣＩＥＮＴＣＯＮＤＩＴＩＯＮＯＦＥＸＩＳＴＥＮＣＥＯＦＧＬＯＢＡＬＳＯＬＵＴＩＯＮＳＦＯＲＳＯＭＥＮＯＮＬＩＮＥＡＲＨＹＰＥＲＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＺＨＡＮＧＱＵＡＮＤＥ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅ...     

Maximal inequalities for a continuous semimartingale

Let X =( X t ) t S 0 be a continuous semimartingale given by d X t = f ( t ) w ( X t )d d M ¢ t + f ( t ) σ ( X t )d M t , X 0 =0, where M =( M t , F t ) t S 0 is a continuous local martingale starting at zero with quadratic variation d M ¢ and f ( t ) is a positive, bounded continuous function on [0, X ), and w , σ both are continuous on R and σ ( x )>0 if x p 0. Denote X 𝜏 * =sup 0 h t h 𝜏 | X t | and J t = Z 0 t f ( s ) } ( X s )d d M ¢ s ( t S 0) for a nonnegative continuous function } . If w ( x ) h 0 ( x S 0) and K 1 | x | n σ 2 ( x ) h | w ( x )| h K 2 | x | n σ 2 ( x ) ( x ] R , n >0) with two fixed constants K 2 S K 1 >0, then under suitable conditions for } we show that the maximal inequalities c p , n log 1 n +1 (1+ J 𝜏 ) p h Á X 𝜏 * Á p h C p , n log 1 n +1 (1+ J 𝜏 ) p (0< p < n +1) hold for all stopping times 𝜏 .     

Convex Holes Produce Lower Bounds for Coefficients

Let D denote the open unit disk and $ f:D \to \bar {{\bf C}}$ be meromorphic and injective in D . Especially, we consider such f which have an expansion $$ f(z) = z + \sum \limits_{n=2}^{\infty }a_n(\;f\,)z^n $$ in a neighbourhood of the origin and map D onto a domain whose complement with respect to $\bar {{\bf C}}$ is convex. Let the set of these functions be denoted by Co . We fix | f m 1 ( X )| for f ] Co and determine the inner and outer radius of the ring domain which is the domain of variability of a 2 ( f ) for such f . Further, it is shown that f ] Co implies that $$ \phi (z) = z+2 {f'(z) \over f''(z)}$$ is holomorphic in D and maps D into itself. This implication in turn implies the inequalities | a n ( f )| S 1 for f ] Co and n = 2,3,4. In addition, we show that | a n ( f )| S 1/2 for f ] Co and all n S 2 .     

Depencence of solutions of the nonlinear Schrödinger equation on dissipation parameters

We consider an initial-boundary-value problem for the nonlinear Schrödinger equation in the complexvalued functionE=E(x,z): (1) $\partial _z E + i\Delta E + i\alpha \left| E \right|^p E + \beta \left| E \right|^q E = 0, q > p \geqslant 0, \beta > 0,$ (2) $\left. E \right|_{z = 0} = E_0 \in H^2 (\Omega ) \cap H_0^1 (\Omega ), \left. E \right|_{\partial \Omega } = 0, \Omega \subset R^2 , \partial \Omega \in C^2 .$ We investigate the behavior of the solution of problem (1)–(2) as β→0 and its closeness to the solution of the degenerate equation (β=0). Given the consistency conditionq(β)=p+εln(1/β), 00, we establish boundedness of the norm $\left\| E \right\|_{C([0,z_0 ]):H_0^1 (\Omega ))} + \left\| {\partial _z E} \right\|_{C([0,z_0 ]);L^2 (\Omega ))} $ for every finitez ₀>0 as β→0. For α≤0 and a fixedq, we prove uniform (in β) boundness of solutions of problem (1)–(2) on some interval [0,Z] and their convergence as β→0 to the solution of the degenerate problem (β=0) in the normC([0,Z];L ² (Ω)).     

Banach空间非扩张映像不动点强收敛定理

设E是具弱序列连续对偶映像自反Banach空间, C是E中闭凸集, T:C→ C是具非空不动点集F(T)的非扩张映像.给定u∈ C,对任意初值x0∈ C,实数列{αn}n∞=0,{βn}∞n=0∈ (0,1),满足如下条件:(i)sum from n=α to ∞α_n=∞, α_n→0;(ii)β_n∈[0,α) for some α∈(0,1);(iii)sun for n=α to ∞|α_(n-1) α_n|<∞,sum from n=α|β_(n-1)-β_n|<∞设{x_n}_(n_1)~∞是由下式定义的迭代序列:{y_n=β_nx_n (1-β_n)Tx_n x_(n 1)=α_nu (1-α_n)y_n Then {x_n}_(n=1)~∞则{x_n}_(n=1)~∞强收敛于T的某不动点.     

A criterion of vanishing of the spectral density based on homoclinic sums

Let (X, d) be a compact metric space, let T: X→X be a homeomorphism satisfying a certain suitable hyperbolicity assumption, and let μ be a Gibbs measure on X relative to T. Let λ be a complex number |λ|=1, and let f:X → ? be a Hölder continuous function. It is proved that $\sum\limits_{k \in \mathbb{Z}} {\lambda ^{ - k} } \left( {\int\limits_X {f(T^k x)\bar f(x)\mu (dx) - \left| {\int\limits_X {f(x)\mu (dx)} } \right|^2 } } \right) = 0$ if and only if ∑λ^?k(f(T^ky) ? f(T^kx)) = 0 for all x, y ε X such that $d(T^k x,T^k y)\xrightarrow[{|k| \to \infty }]{}0$ . Bibliography: 11 titles.     

图 P2×Cn的均匀邻强边色数

对图G(V,E),一正常边染色f若满足(1)对(V)uv∈E(G),f[u]≠f[v],其中f[u]={f(uv)|uv∈E};(2)对任意i≠j,有||E|-|Ej||≤1,其中Ei={e| e∈E(G)且f(e)=i}.则称f为G(V,E)的一k-均匀邻强边染色,简称k-EASC,并且称Xcas(G)=min{k|存在G(V,E)的一k-EASC为G(V,E)的均匀邻强边色数.本文得到了图P2×Cn的均匀邻强边色数.     

A Remark on a Problem of Rolf Nevanlinna

Let T ( f ) and N ( r,c ) denote the usual Nevanlinna characteristic and the counting function for the c -points of a meromorphic function f , respectively. Using a result of Miles and Shea ( Quart. J. Math. Oxford , 24 (2), (1973), 377-383) and two simple estimates for trigonometric functions, we show in connection with a 1929 problem of Nevanlinna for meromorphic functions f of finite order 1 < u < X $$ \limsup\limits_{r\rightarrow \infty } { N(r, 0)+N(r, \infty ) \over T(r, \,f)}\ge {2\sqrt 2 \over \pi} {|\sin \pi \lambda | \over D(\lambda )}\ge (0.9)\, {{|\sin \pi \lambda | \over {D(\lambda )}, }} $$ with D ( u ) = q +|sin ~ u | for $ q\le \lambda \le q + \fraca {1}{2} $ and D ( u ) = q + 1 for $ q + {\fraca {1} {2}} \le \lambda \lt q + 1 $ , where $ q = \lfloor \lambda \rfloor $ .     

奇异半线性发展方程的局部Cauchy问题

本文在Ｂａｎａｃｈ空间Ｅ中讨论如下问题ｄｕｄｔ＋１ｔσＡｕ＝Ｊ（ｕ），０＜ｔＴ，ｌｉｍｔ→０＋ｕ（ｔ）＝０，其中ｕ：（０，Ｔ］Ｅ，Ａ是与ｔ无关的线性算子．（－Ａ）是Ｅ上Ｃ０半群｛Ｔ（ｔ）｝ｔ０的无穷小生成元，常数σ１，Ｊ是一个非线性映射ＥＪ→Ｅ．它满足局部Ｌｉｐｓｃｈｉｔｚ条件．我们证明了当其Ｌｉｐｓｃｈｉｔｚ常数ｌ（ｒ）满足一定条件时．问题（Ｓ）有局部解，且在某函类中解唯一．设Ｊ（ｕ）＝｜ｕ｜γ－１ｕ＋ｆ（ｘ）（γ＞１），Ｅ＝Ｌｐ，ＥＪ＝Ｌｐγ时得到了与Ｗｅｉｓｌｅｒ［２］在非奇异情形类似的结果．     

ADJACENT COEFFIC ENTS OF UNIVALENT FUNCTIONS

Let the function f(z)=z sum from n=2 to ∞ a_nz~n ∈ S. It is obtained that-2.793<|a_n 1|-|a_n|<3.26,which is an improvement of the result in [1] or [2].