Endp oint Estimates for Commutators of Fractional Integrals Asso ciated to Op erators with Heat Kernel Bounds

Let L be the infinitesimal generator of an analytic semigroup on L~2(R~n)with pointwise upper bounds on heat kernel,and denote by L~(-α/2)the fractional integrals of L.For a BMO function b(x),we show a weak type Llog L estimate of the commutators [b,L~(-α/2)](f)(x) = b(x)L~(-α/2)(f)(x)-L~(-α/2)(bf)(x).We give applications to large classes of differential operators such as the Schr¨odinger operators and second-order elliptic operators of divergence form.     

Local solvability of partial differential operators on the Heisenberg group

In this article the following class of partial differential operators is examined for local solvability: Let P(X, Y) be a homogeneous polynomial of degree n ≥ 2 in the non-commuting variables X and Y. Suppose that the complex polynomial P(iz, 1) has distinct roots and that P(z, 0) = zⁿ. The operators which we investigate are of the form P(X, Y) where X = δ_x and Y = δ_y + xδ_w for variables (x, y, w) ∈ ?³. We find that the operators P (X, Y) are locally solvable if and only if the kernels of the ordinary differential operators P(iδ_x, ± x)* contain no Schwartz-class functions other than the zero function. The proof of this theorem involves the construction of a parametrix along with invariance properties of Heisenberg group operators and the application of Sobolev-space inequalities by Hörmander as necessary conditions for local solvability.     

一类沿曲面的奇异积分算子

记Σ＿（ｎ－２）为ＩＲ￣（ｎ－１）中的单位球面。本文证明了当Ω为ＩＲ￣（ｎ－１）上的零次齐次函数，满足消失性条件，及时，沿任意曲面（ｔ，Г（｜ｔ｜））的主值奇异积分算子及其极大算子在Ｌ￣２（ＩＲ￣ｎ）上是有界的，此处ｂ为ＩＲ￣（ｎ－１）上的有界径向函数，ｘ∈ＩＲ￣（ｎ－１），ｘ＿ｎ∈ＩＲ，及     

On the Similarity of Some Differential Operators to Self-Adjoint Ones

The paper is devoted to the study of the similarity to self-adjoint operators of operators of the form , in the space with weight . As is well known, the answer to this problem in the case is positive; it was obtained by using delicate methods of the theory of Hilbert spaces with indefinite metric. The use of a general similarity criterion in combination with methods of perturbation theory for differential operators allows us to generalize this result to a much wider class of weight functions .     

参数边界条件下奇型Sturm-Liouville算子的半逆问题

Sturm-Liouville算子的半逆问题讨论由一组谱和半区间上势函数唯一确定整个区间上势函数q(x).本文利用Koyunbakan和Panakhov的方法和[13]的结论,讨论(0,π)上的奇型Sturm-Liouville问题满足-y″+[q(x)-1/4sin2x]y=λy,参数边界条件y(0,λ)=0或y′(0,λ)-hy(0,λ)=0和y′(π,λ)+(aλ+b)y(π,λ)=0,证明一组谱和(π/2,π)上的势函数q(x)唯一确定(0,π)上的势函数q(x).     

齐型空间上的Toeplitz型算子

设X是齐型空间.设T_(j,1)和T_(j,2)是具有非光滑核的奇异积分算子,或者是±II(I是恒等算子).令Toeplitz型算子T_b=■T_(j,1)M_T_(j,2),其中M_bf(x)=b(x)f(x).研究了当b∈BMO(X)时,T_b(f)在加权情况下的有界性,以及当b∈BMO(X)时,与经典Carderon-Zygmund算子相联的T_b(f)在Morrey空间上的有界性.     

Calderón-Zygmund算子在H_b~p空间上的有界性

众所周知,如果Calderón-Zygmund算子T满足T~*(1)=0,则算子T在H~p,n/(n+ε)相似文献   

On Semitransitive Collections of Operators

A collection F of operators on a vector space V is said to be semitransitive if for every pair of nonzero vectors x and y in V there exists a member T of F such that either Tx = y or Ty = x (or both). We study semitransitive algebras and semigroups of operators. One of the main results is that if the underlying field is algebraically closed, then every semitransitive algebra of operators on a space of dimension n contains a nilpotent element of index n. Among other results on semitransitive semigroups, we show that if the rank of nonzero members of such a semigroup acting on an n-dimensional space is a constant k, then k divides n.     

One-Dimensional Degenerate Diffusion Operators

The aim of this paper is to present some results about generation, sectoriality and gradient estimates both for the semigroup and for the resolvent of suitable realizations of the operators $$A^{\gamma, b}u(x) = {\gamma}xu^{\prime\prime} (x) + bu^{\prime} (x),$$ with constants γ > 0 and b ≥ 0, in the space C([0, ∞]).     

算子的拟相似与(Q)类算子

本文引进定义于L(H)上的集值函数β(S)和(Q)类算子(指本质谱含于其一切拟相似算子的本质谱的算子),用β(S)刻划 (Q)类算子的特征;证明(Q)类算子范围广泛,次可分解算子(包括次标量算子,M-亚正常算子,半亚正常算子等等)是其中的一部分;(Q)类算子在L(H)中稠密.     

Solutions of the Sylvester Equation in C*-Modular Operators

Ukrainian Mathematical Journal - We study the solvability of the Sylvester equation AX + YB = C and the operator equation AXD + FYB = C in the general setting of adjointable operators between...     

Diophantine方程（a^m-1）（6^n-1）=x^2的一点注记

设a,b是适合min（a,b）〉1,2｜a,2＋b以及v（6—1）是正奇数,其中v（b-1）表示整除b-1的2的最高次数．本文运用初等方法以及同余性质,研究了方程（a^m-1）（b^n-1）=x^2的可解性．对某些特殊素数P,证明了该方程无解．证明了如果存在适合P≡±E3（mod8）的奇素数P,可使a≡-1（modP）...     

Multiplication and Compact-friendly Operators

During the last few years the authors have studied extensively the invariant subspace problem of positive operators; see [6] for a survey of this investigation. In [4] the authors introduced the class of compact-friendly operators and proved for them a general theorem on the existence of invariant subspaces. It was then asked if every positive operator is compact-friendly. In this note, we present an example of a positive operator which is not compact-friendly but which, nevertheless, has a non-trivial closed invariant subspace.In the process of presenting this example, we also characterize the multiplication operators that commute with non-zero finite-rank operators. We show, among other things, that a multiplication operator M commutes with a non-zero finite-rank operator if and only the multiplier function is constant on some non-empty open set.     

Algebraic Integrability of Macdonald Operators and Representations of Quantum Groups

In this paper we construct examples of commutative rings of difference operators with matrix coefficients from representation theory of quantum groups, generalizing the results of our previous paper [ES] to the q-deformed case. A generalized Baker–Akhiezer function is realized as a matrix character of a Verma module and is a common eigenfunction for a commutative ring of difference operators. In particular, we obtain the following result in Macdonald theory: at integer values of the Macdonald parameter k, there exist difference operators commuting with Macdonald operators which are not polynomials of Macdonald operators. This result generalizes an analogous result of Chalyh and Veselov for the case q=1, to arbitrary q. As a by-product, we prove a generalized Weyl character formula for Macdonald polynomials (= Conjecture 8.2 from [FV]), the duality for the -function, and the existence of shift operators.     

Bochner-Riesz算子在加权Morrey空间上的一些估计

要本文将得到Bochner-Riesz算子T_R~((n-1)/2)在加权Morrey空间L~(p,k)(w)上的一些强型和弱型估计,1≤P<∞且0相似文献   

粗糙核奇异积分交换子在Triebel-Lizorkin空间的有界性

应用核的分解,讨论了粗糙核奇异积分算子 Tf（x）=p.v.∫R^nΩ（x-y）/｜x-y｜^nf（y）dy 和BMO（R^n）函数b生成的交换子[b,T]的有界性．证明了当Ω∈L（logL）^2（S^n-1）时,[b,T]是Triebel—Lizorhn空间Fp^α,q（R^n）上的有界算子．     

On a Class of Weakly Hyperbolic Operators

The paper considers Cauchy problem in the Gevre type multianisotropic spaces. Necessary and sufficient conditions for unique solvability of this problem are obtained and the properties of operators (polynomials) that are hyperbolic with a specified weight are investigated.     

Lipschitz Operators and the Solvability of Non-linear Operator Equations

Let U and V be Banach spaces, L and N be non-linear operators from U into V. L is said to be Lipschitz if LI(L) := sup{‖Lx - Ly‖. ‖x - y‖^-1 : x ≠ y} is finite. In this paper, we give some basic properties of Lipschitz operators and then discuss the unique solvability, exact solvability,approximate solvability of the operator equations Lx = y and Lx Nx = y. Under some conditions we prove the equivalence of these solvabilities. We also give an estimation for the relative-errors of the solutions of these two systems and an application of our method to a non-linear control system.     

Vekua Integral Operators on Riemann Surfaces

On an arbitrary (in general, noncompact) Riemann surface R, we study integral operators T and ∏ analogous to the operators introduced by Vekua in his theory of generalized analytic functions. By way of application, we obtain necessary and sufficient conditions for the solvability of the nonhomogeneous Cauchy--Riemann equation $\bar \partial f = F$ in the class of functions f exhibiting Λ₀-behavior in the vicinity of the ideal boundary of R.     

关于广义二次算子

设B(H)表示定义在希尔伯特空间H,上的所有有界线性算子的全体.如果A∈B(H)满足二次算子方程A2=αA βP,其中α,β∈C,P是一个非零的幂等算子且AP=PA=A,则称A为广义二次算子.记L(P)为关于幂等算子P的广义二次算子之集.我们用算子谱论的方法研究了L(P)的谱和群逆等相关性质,并推广了R. W. Farebrother和G. Trenkler的结论.