Kernel theorems in the setting of mixed nonquasi-analytic classes

Let Ω₁⊂Rr and Ω₂⊂Rs be nonempty and open. We introduce the Beurling-Roumieu spaces D(ω₁,ω₂}(Ω₁×Ω₂), D(M,M^′}(Ω₁×Ω₂) and obtain tensor product representations of them. This leads for instance to kernel theorems of the following type: every continuous linear map from the Beurling space D(ω₁)(Ω₁) (respectively D(M)(Ω₁)) into the strong dual of the Roumieu space D{ω₂}(Ω₂) (respectively D{M^′}(Ω₂)) can be represented by a continuous linear functional on D(ω₁,ω₂}(Ω₁×Ω₂) (respectively D(M,M^′}(Ω₁×Ω₂)).     

Tensor product characterizations of mixed intersections of non quasianalytic classes and kernel theorems

Mixed intersections of non quasi‐analytic classes have been studied in [12]. Here we obtain tensor product representations of these spaces that lead to kernel theorems as well as to tensor product representations of intersections of non quasi‐analytic classes on product of open or of compact sets (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ultradifferentiable extension theorems: A survey

We survey ultradifferentiable extension theorems, i.e., quantitative versions of Whitney’s classical extension theorem, with special emphasis on the existence of continuous linear extension operators. The focus is on Denjoy–Carleman classes for which we develop the theory from scratch and discuss important related concepts such as (non-)quasianalyticity. It allows us to give an efficient and, to a fair extent, elementary introduction to Braun–Meise–Taylor classes based on their representation as intersections and unions of Denjoy–Carleman classes.     

Borel extractions of converging sequences in compact sets of Borel functions

It is well known by a classical result of Bourgain–Fremlin–Talagrand that if K is a pointwise compact set of Borel functions on a Polish space then given any cluster point f of a sequence (f_n)_nω in K one can extract a subsequence (f_nk)_kω converging to f. In the present work we prove that this extraction can be achieved in a “Borel way.” This will prove in particular that the notion of analytic subspace of a separable Rosenthal compacta is absolute and does not depend on the particular choice of a dense sequence.     

拟亚纯映射的最大型Borel方向

作者用几何方法研究了更广泛的K-拟亚纯映射;定义了K-拟亚纯映射的最大型Borel方向;证明了有限正级K-拟亚纯映射最大型Borel方向的存在性;并导出了最大型Borel方向的一个充要条件.     

Generic results in classes of ultradifferentiable functions

Let E be a Denjoy–Carleman class of ultradifferentiable functions of Beurling type on the real line that strictly contains another class F of Roumieu type. We show that the set S of functions in E that are nowhere in the class F   is large in the topological sense (it is residual), in the measure theoretic sense (it is prevalent), and that S∪{0}$S\cup \{0\}$ contains an infinite dimensional linear subspace (it is lineable). Consequences for the Gevrey classes are given. Similar results are also obtained for classes of ultradifferentiable functions defined imposing conditions on the Fourier–Laplace transform of the function.     

Whitney-type theorems on extension of functions on Carnot groups

We generalize the classical Whitney theorem which describes the restrictions of functions of various smoothness to closed sets of a Carnot group. The main results of the article are announced in [1].     

圆内亚纯函数与其微分多项式的公共Borel点

该文把A.P.Singh关于一类齐次微分多项式级的结果推广到更一般的微分多项式。并证明了：如果Q（f)≠0是圆内有限正级亚纯函数f的身长分多项式,则f^(k0Q(f)的Borel点必是f的Borel点,其中K0满足0≤K0≤min{K:f^(k)出现在Q（f)中}。     

Stability Zones in Whitney's Extension Problem for Ultradifferentiable Functions

We establish a connection between the growth rate of weight functions generating nonquasianalytic classes of ultradifferentiable functions of Beurling and Roumieu type and the validity of an analog of Whitney's extension theorem for these classes.     

单位圆内亚纯函数的Borel半径的分布

对于单位圆周上的任一非空闭集E,任意正数λ及任意实数μ（0≤μ≤λ）;构造了一个单位国内的λ级亚纯函数,以{（L（θ）|θ∈E｝为Borel半径集,并且｛L（θ）|θ|E｝都不是Julia半径,同时N（r,f）的级为θ.     

On the Range of Two Convolution Operators

Let _(ω)(ℝ) denote the non–quasianalytic class of Beurling type on ℝ. For μ, ν ∈ ′_(ω)(ℝ) we give necessary conditions for the inclusion T_ν( _(ω)(ℝ)) ⊂ T_μ( _(ω)(ℝ)), thus extending previous work of Malgrange and Ehrenpreis .     

Picard type theorems concerning certain small functions

Let f(z) be a meromorphic function in the complex plane, whose zeros have multiplicity at least k + 1(k ≥ 2). If sin z is a small function with respect to f(z), then f~(k)(z)-P(z) sin z has infinitely many zeros in the complex plane, where P(z) is a nonzero polynomial of deg(P(z)) ≠ 1.     

Application of the Borel transform to the study of the spectrum of integral equations whose kernels are entire functions of exponential type

Using the Borel transform, we study the spectrum of a class of non-compact integral operators whose kernels are of exponential type and square integrable on the real line. Our method also enables us to obtain an interesting characterization of a well-known integral equation involving the Bessel function

     

代数体函数与其系数函数的Borel方向间的关系

研究了代数体函数w(z)的Borel方向和确定该代数体函数的复方程A_k(z)w~k+_(Ak-1)(z)w~_(k-1)+…+A_0(z)=0的系数函数A_k(z),A_(k-1)(z),…,A_0(z)的Borel方向之间的关系.     

Carrier and nerve theorems in the extension theory

We show that a regular cover of a general topological space provides structure similar to a triangulation. In this general setting we define analogues of simplicial maps and prove their existence and uniqueness up to homotopy. As an application we give simple proofs of sharpened versions of nerve theorems of K. Borsuk and A. Weil, which state that the nerve of a regular cover is homotopy equivalent to the underlying space.

Next we prove a nerve theorem for a class of spaces with uniformly bounded extension dimension. In particular we prove that the canonical map from a separable metric -dimensional space into the nerve of its weakly regular open cover induces isomorphisms on homotopy groups of dimensions less than .

     

Some alternative principles and mixed minimax theorems involving two functions

Some alternative principles and quantitative-topological minimax theorems involving two functions are given. Consequently, Geraghty–Lin's minimax theorem and Terkelsen's minimax theorem are generalized. This revised version was published online in June 2006 with corrections to the Cover Date.     

关于Schr(o)der函数Borel方向的一个应用

本文应用Borel方向和充满圆的关系得到了方程F(z)=R(f(z))的一个充分必要条件,并给出它关于Schr(o)der方程f(sz)=R(f(z))的-个应用,这里s是一常数且|s|＞,R(w)是次数大于2的有理函数.     

单位圆内K-拟亚纯映射的Borel半径

§ 1.　IntroductionForthevaluedistributionofentriefunctionsandmeromorphicfunctions,manyimportantresultshavebeenobtained [1 ].Recently ,ProfessorSUNDao_chunandProfessorYANGLo[2 ]introducedtheK_quasimermorphicmappingsintheplane,whichismorewiderthantheclassofentireormermorphicfunctions ,andstudiedtheirvaluedistribution .Inthispaper,weobtainthesequenceoffillingdisksofK_quasimermorphicmappingsintheunitdisk ,andprovetheexistenceoftheBorelradiusofK_quasiconformalmappingsintheunitdisk .Inthefoll…