Algebraic Sum of Unbounded Normal Operators and the Square Root Problem of Kato

µ − a + β . K•§ (0.12) – k˜‡
ë•©zµ [1] L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequality with weights, Compos. Math. 53 (1984) 259–275. [2] K. Chou, C. Chu, On the best constant for a weighted Sobolev–Hardy inequality, J. Lond. Math. Soc. (2) 48 (1993) 137–151. [3] F. Catrina, Z. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of external functions, Comm. Pure Appl. Math. 54 (2) (2001) 229–258. [4] X.J. Huang, X.P. Wu, C.L. Tang, Multiple positive solutions for semilinear elliptic equations with critical weighted Hardy-Sobolev exponents, Nonlinear Anal. 74 (2011) 2602–2611 [5] L. Ding, C.L. Tang, Existence and multiplicity of positive solutions for a class of semilinear elliptic equations involving Hardy term and Hardy-Sobolev critical exponents, J. Math. Anal. Appl. 339 (2008) 1073–1083. [6] L. Huang, X.P. Wu, C.L. Tang, Existence and multiplicity of solutions for semilinear elliptic equations with critical weighted Hardy-Sobolev exponents, Nonlinear Anal. 71 (2009) 1916–1924. [7] P. H. Rabinowitz. Minimax methods in critical point theory with applications to differential equations, Conference Board of the Mathematical Sciences, vol. 65 Providence, RI: American Mathematical Society, 1986. [8] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987. [9] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (4) (1983) 437–477. [10] D. Kang, G. Li, S. Peng, Positive solutions and critical dimensions for the elliptic problems involving the Caffarelli-Kohn-Nirenberg inequalities, J. Jilin. Univ. Sci. 46 (2008) 423–427.

Dirichlet空间上Toeplitz乘积的有界性于涛;陆俏飞【摘要】对于Dirichlet空间中的函数f和g,应用Berezin变换,研究Dirichlet空间上Toeplitz乘积TfT-g的有界性,分别给出TfT-g在Dirichlet空间上有界的充分条件和必要条件.【期刊名称】《浙江师范大学学报（自然科学版）》【年(卷),期】2014(037)001【总页数】6页(P20-25)【关键词】Dirichlet空间;Toeplitz算子;Berezin变换;Bergman空间【作者】于涛;陆俏飞【作者单位】浙江师范大学数理与信息工程学院,浙江金华321004;浙江师范大学数理与信息工程学院,浙江金华321004【正文语种】中文【中图分类】O177.20 引言本文中，令D是复平面C中的开单位圆盘，dA表示D上的正规化面积测度.把所有满足下式的D上函数u构成的空间称为Sobolev空间:其中,和表示弱导数意义下的偏导数.用 W1,2(D)表示Sobolev空间，它是一个具有如下内积的Hilbert空间:〉L2.其中,符号〈\5,\5〉L2表示Lebesque空间L2(D,dA)上的内积.称由Sobolev空间W1,2(D)中所有在原点处为0的解析函数构成的子空间为Dirichlet空间，记为D,它是一个具有如下内积的Hilbert空间:Dirichlet空间的再生核为即对任意的f∈D,f(z)=〈f,Kz〉.在单位圆盘的Hardy空间H2上，有界Toeplitz算子都是由有界符号函数诱导的.2个Toeplitz算子的乘积有界，并不意味着它们本身都是有界的.1994年,Sarason[1]提出猜想:对于f,g∈H2,H2上Toeplitz乘积有界的充要条件是(1)式(1)中,代表u在单位圆盘内的Poisson扩张.Sarason在文献[1]中提到,Treil已经证明了条件(1)是必要的.文献[2]给出了当符号函数 f和g是外函数时，Toeplitz算子乘积在Hardy空间H2上有界且可逆的充要条件，支持了Sarason的上述猜想.在一般情形下,文献[3]证明了如下比条件(1)强一些的条件是充分的：存在ε>0,使得(2)Sarason在文献[1]中同时提出了Bergman空间上的类似问题;文献[4]给出与Hardy 空间情形类似的回答，此时条件(1)和条件(2)中的Piosson扩张改为Bergman空间上的Berezin变换;文献[5-6]分别将该结果推广到单位圆盘和单位球上的加权Bergman空间;文献[7-9]分别将该结果推广到多圆盘和单位球上的Bergman空间.本文的目的是在Dirichlet空间上考虑Toeplitz乘积有界性的条件. Sobolev空间W1,∞(D)定义为其中,L∞(D)是D上所有本质有界可测函数构成的空间.空间W1,∞(D)的范数定义为令P为由W1,2(D)到D上的正交投影，则P是可表示为如下的积分算子:(3)给定u∈W1,∞(D)，定义D上以u为符号的Toeplitz算子Tu为Tu f=P(uf).(4)容易看出,对于每个u∈W1,∞(D),Toeplitz算子 Tu是有界的.Dirichlet空间上Toeplitz算子已被广泛研究.例如:文献[10]研究了具有连续可导符号Toeplitz算子的Fredholm性质;文献[11]研究了当符号函数是调和函数时，Toeplitz算子的交换性和正规性等;文献[12]研究了W1,∞(D)符号Toeplitz算子的交换性、乘积和模有限秩交换等代数性质.对于u∈W1,2(D)，式(4)的右边对于f∈D ∩W1,∞(D)还是有意义的，此时Toeplitz算子Tu可以看作D上的稠定义算子.若再设v∈W1,2(D)，按照式(3)和式(4),对于f∈D ∩W1,∞(D),上述积分是可积的.因此,可将TvTu看作从D到H(D)(D上所有解析函数的空间)的稠定义算子.本文的主要结果是Toeplitz乘积在 D上有界的一个充分条件和一个必要条件.定理1 设f,g∈D，如果存在ε>0，使得那么在Dirichlet空间上有界.其中，表示 f的Berezin变换.定理2 设f,g ∈D，f≠0,如果在Dirichlet空间上有界，那么1 预备知识对于w∈D，定义分式线性变换φw 为则φw是单位圆盘上的一个自同构.事实上，这个映射是对合的，即=φw，且这个变换的实Jacobian行列式为因此有如下的变量变换公式：(5)设函数f∈L1(D)，则 f的Berezin变换定义为D上的一个函数(6)对于w∈D，定义W1,2(D)上的算子Uw为Uw f=f(w)- f ∘ φw.易见算子Uw 是D上的自伴酉算子.特别地，当φ∈W1,∞(D)解析时，有UwTφUw=Tφ ∘ φw+Uw(φ)⊗Kw;(7)当φ∈W1,∞(D)共轭解析时，有UwTφUw=Tφ ∘ φw.(8)关于Uw 的定义、式(4)及式(5)可参阅文献[13].对于f,g∈D,定义 D上一秩算子 f⊗g为(f⊗g)h=〈 h,g 〉f,h ∈D.如果T和S是有界线性算子，那么T( f⊗g)S*=(T f)⊗(Sg).对于w ∈D，令τw表示函数Uw(z)=w-φw，其中z表示D上的恒等映射.引理1[13] 在Dirichlet空间D上,是个一秩算子，且等于z⊗z.引理2[13] 设w∈D，在Dirichlet空间上,τw⊗引理3 设w∈D，则在Dirichlet空间D上，有τw⊗证明对于w∈D，应用引理1和引理2 及式(7)、式(8)可得τw⊗τw=Uw(z⊗z)⊗引理3证毕.引理4 设w∈D，则有证明令u∈D，v∈D，对于w∈D，有所以又因为所以可以得到类似可得因此,引理4证毕.引理5 令 f，g∈D，对于w∈D，有其中,‖\5‖H2表示Hardy空间 H2 的范数.证明令f∈D，w∈D,则由|τw(eiθ)|≥1-|w|，可以得到引理5的第1个不等式.令u∈D，v∈D，有事实上，所以可得因此,(9)式(9)中的最后一个等式通过式(5)和式(6)获得.引理6[13] 如果F∈D,G∈D，那么2 主要结果的证明令P0为Lp(D)上核为的积分算子,即当1<p<∞ 时,P0是 Lp-有界的[14].下面对Toeplitz算子作一些估计，这些估计将用于Toeplitz乘积有界性的充分性的证明.引理7 对给定的ε>0，令δ=(2+ε)/(1+ε)，则1)令g∈D，u∈D，对任意的w∈D，有2)令f∈D，v∈D，对任意的w∈D，有证明 1)若g∈D,u∈D，则对任意的w∈D，有因此,从而应用Hölder′s不等式，可得其中,第3个不等号使用了不等式类似地，由于所以2)令f∈D，v∈D，对任意的w∈D，有所以与1)的证明类似，有引理7证毕.定理1的证明要证明在Dirichlet空间上是有界的，只须证明存在常数C，满足对所有的u,v∈D,有成立即可.由内积公式(引理 6)得其中:由引理 7可得[P0(|u′|δ)(w)]1/δ[P0(|v′|δ)(w)]1/δdA(w)≤因为2/δ>1,从而P0是L2/δ-有界的，所以存在常数C>0,使得由Cauchy-Schwarz不等式可得因此,得到关于I1的如下估计:存在常数C>0,使得完全类似地，可以对I2和I3作出估计.通过对I1，I2和I3的估计，得到存在某个常数M>0，使得因此,Toeplitz算子乘积是有界的.定理1证毕.定理2的证明由引理3，有Tf(τw⊗又因为Tf(τw⊗⊗及引理 4 给出的估计可以得到⊗由引理4和引理5可以得到所以从而得到定理2的结论.定理2证毕.参考文献：[1]Sarason D.Products of Toeplitz operators[C]//Khavin V P,Nikol′ski N K.Linear and complex analysis problem book 3.New York:Springer-Verlag,1994:318-319.[2]Cruz-Uribe D.The invertibility of the product of unbounded Toeplitz operators[J].Integral Equations Operator Theory,1994,20(2):231-237. [3]Zheng Dechao.The distribution function inequality and products of Toeplitz operators and Hankel operators[J].J Funct Anal,1996,138(2):477-501.[4]Stroethoff K,Zheng Dechao.Product of Hankel and Toeplitz operator on the Bergman space[J].J Funct Anal,1999,169(1):289-313.[5]Stroethoff K,Zheng Dechao.Bounded Toeplitz products on weighted Bergman spaces[J].J Oper Theory,2008,59(2):277-308.[6]Stroethoff K,Zheng Dechao.Bounded Toeplitz products on Bergman spaces of the unit ball[J].J Math Anal Appl,2007,325(1):114-129.[7]Stroethoff K,Zheng Dechao.Bounded Toeplitz products on the Bergman space of the polydisk[J].J Math Anal Appl,2003,278(1):125-135.[8]Park J D.Bounded Toeplitz products on the Bergman space of the unit ball in Cn[J].Integral Equations Operator Theory,2006,54(4):571-584.[9]Lu Yufeng,Liu Chaomei.Toeplitz and Hankel products on Bergman spaces of the unit ball[J].Chin Ann Math Ser B,2009,30B(3):293-310. [10]Cao Guangfu.Fredholm properties of Toeplitz operators on Dirichlet space[J].Pacific J Math,1999,188(2):209-224.[11]Lee Y J.Algebraic propertise of Toeplitz operators on the Dirichlet space[J].J Math Anal Appl,2007,329(2):1361-1329.[12]Yu Tao.Toeplitz operators on the Dirichlet space[J].Integral Equations Operator Theory，2010,67(2):163-170.[13]Yu Tao.Operators on the orthogonal complement of the Dirichlet space[J].J Math Anal Appl,2009,357(1):300-306.[14]Axler S.Bergman spaces and their operators[C]//Conway J B,Morrel BB.Surveys of some recent results in operator theory.NewYork:Wiley,1988:1-50.。

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A.M.Cormack 共同获得诺贝尔医学奖 G.N.Hounsfield 寄语：如果你考试没有通过，不用太担心，只要你感到你的确理解了所学的 课程；将自己常用的推理方法充分使用后，通过对身边发生的事物基本要素的掌握，你就会 对你所能达到的理解能力和所掌握的知识感到吃惊。
基本原理：不损伤物体本身结构的情况下，发射各种可通过物体的讯号（各种射线，波， 粒子，电磁场等），然后通过对从体外接收到的信号。利用数学方法和计算机进行加工和处理， 获得物体内部结构的信息，形成物体内部结构的三维透视图像，也称为图像重建或图像恢复。
dτ
其中 χD (x) 为示性函数，D 为 R1 中的有届区域。
正问题：已知 z(t) ，利用上式求未来任意时刻的温度分布 u(x, t)
反问题：已知 u(0, t) = g(t) ，求 z(t) ，即第一类 Volterra 积分方程
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∫t H (t −τ )z(τ )dτ = g(t), t > 0 0
1.1 反问题的若干例子
背景：1923，Hadamard,线性偏微分方程的 Cauchy 问题时开始研究反问题的不适定性。 20 世纪 40 年代，Tikhonov，提出了变分正则化方法，《Solutions of ill-posed problems》,(Tikhonov,1977,中译本《不适定问题的解法》（王秉忱，1979，地质出版社）)， Landweber 和 Fridman，迭代正则化方法。Morozov 和 Groetsch 把不适定问题的正则化放在抽 象泛函空间进行完整描述。国内：冯康等。
∫ h φ( y) dy = f (h), h > 0
0 h− y
来求φ( y) 。
Abel 应用：地震学，利用地震波的传播时间来确定地壳运动的速度。等离子物理，用光谱法 测量和计算温度，电子密度，粒子密度等。

factor - Factor a Multivariate PolynomialCalling Sequencesfactor(a)factor(a,K)Parametersa - an expressionK - a field extension over which to factorDescription∙ The function factor computes the factorization of a multivariate polynomial with integer, rational, (complex) numeric, or algebraic number coefficients.∙ The factor function does NOT factor integers. Nor does it factor integer coefficients in a polynomial. Use the ifactor function to factor integers.∙ If the second argument K is not given, the polynomial is factored over the field implied by the coefficients. For example, if the coefficients are all integers then factor computes all irreducible factors with integer coefficients. Thus factor does not necessarily factor into linear factors. Note that any integer content (see first example below) is not factored. ∙ If the input, a, is a rational expression, then it is first ``normalized'' (see normal ) and the numerator and denominator of the resulting expression are then factored. This provides a ``fully-factored form'' which can be used to simplify an expression in the same way the normal function is used. However, it is more expensive to compute.∙ If the input, a, is a list, set, equation, range, series, relation, or function, then factor is applied recursively to the components of a.∙ If the second argument K is the keyword real or complex, a floating point factorization is performed over the reals and complexes respectively. At present this is only implemented for univariate polynomials.∙ If the second argument K is a single RootOf, a list or set of RootOfs, a single radical, or a list or set of radicals, then the expression is factored over the algebraic number field defined by K.Examples> factor(6*x^2+18*x-24);6() + x 4() - x 1> factor(6);6> ifactor(6); () 2() 3> factor((x^3-y^3)/(x^4-y^4)); + + x 2x y y 2() + y x ()+ x 2y 2 > factor(x^3+5);+ x 35> factor(x^3+5, 5^(1/3)); () - + x 2x 5()/135()/23() + x 5()/13 > factor(x^3+5, {5^(1/3),(-3)^(1/2)}); 14() - - 2x 5()/13-35()/13() - + 2x 5()/13-35()/13() + x 5()/13> factor(x^3+5.0);() + x 1.709975947() - + x 2 1.709975947x 2.924017740> factor(x^3+5,complex);()+x 1.709975947()-x +.8549879733 1.480882610I()-x -.8549879733 1.480882610I> factor(y^4-2,sqrt(2));()-y22()+y22> alias(alpha = RootOf(x^2-2)):factor(y^4-2,alpha);()-y2α()+y2α> factor(x^3+y^3);()+x y()- +x2x y y2> factor(x^3+y^3,(-3)^(1/2));1 4()- -2x y y-3()- +2x y y-3()+x ycombine - combine terms into a single termCalling Sequencecombine(f)combine(f, n)combine(f, n, opt1, opt2, ...)Parametersf - any expression, or set or list of expressionsn - a name or a list or set of namesopt1, opt2, ... - (optional) names of optionsDescription∙The combine function applies transformations which combine terms in sums, products, and powers into a single term. This function is applied recursively to the components of lists, sets, and relations; that is, f and n may be lists (or sets) of expressions and names, respectively.∙For many functions, the transformations applied by combine are the inverse of the transformations that are applied by expand. For example, consider the well-knownidentitysin(a+b) = sin(a)*cos(b) + cos(a)*sin(b)∙The expand function applies the identity from left to right whereas the combine function does the reverse.∙Subexpressions involving Int, Sum, and Limit are combined into one expression where possible using linearity; that is, c1*f(a,range) + c2*f(b,range) ==> f(c1*a+c2*b,range). ∙ A specific set of transformations is obtained by specifying a second (optional) argument n (a name) which is one or a list of the following:@@abs arctanconjugate exp icombineln piecewise polylogpower product Psiradical range signumtrigFor additional information and examples about the transformations applied by each of these, see combine[n].∙The combine function maps over functions. Therefore combine(f(a)) is the same as map(combine,f(a)).Examples> combine(Int(x,x=a..b)-Int(x^2,x=a..b));d ⎛⎠⎜⎜a b- x x 2x> combine(Limit(x,x=a)*Limit(x^2,x=a)+c); lim → x a + x 3c> combine(4*sin(x)^3,trig);- + ()sin 3x 3()sin x > combine(exp(x)^2*exp(y),exp);e ()+ 2x y> combine(exp(sin(a)*cos(b))*exp(cos(a)*sin(b)),[trig,exp]); e ()sin + a b> assume(y>0,z>0);combine(2*ln(y)-ln(z),ln);⎛⎝ ⎫⎭⎪⎪ln y~2z~> combine((x^a)^2,power);x ()2a> combine(Psi(-x)+Psi(x),Psi); + + 2()ψx π()cot πx 1x> combine([2*sin(x)*cos(x),2*cos(x)^2-1],trig);[],()sin 2x ()cos 2x> combine(exp(sin(a)*cos(b))*exp(cos(a)*sin(b)),[trig,exp]); e ()sin + a bsimplify - Apply simplification rules to an expressionCalling Sequencesimplify(expr)simplify(expr, n1, n2, ...)simplify(expr, assume=prop)simplify(expr, symbolic)Parametersexpr - any expressionn1, n2,... - (optional) names or sets or listsprop - any property (optional)Description∙ The simplify function is used to apply simplification rules to an expression. If only oneargument is present, then simplify will search the expression for function calls, squareroots, radicals, and powers. Next it will invoke the appropriate simplification procedures, which include: BesselI , BesselJ , BesselK , BesselY , D , Ei , GAMMA , RootOf , LambertW , dilog , exp , ln , sqrt , polylog , pochhammer , trig (for trig functions), hypergeom (for hypergeometrics), radical (occurrence of fractional powers), power (occurrence of powers), exp , ln ),and `@` (for operators).∙ In the case of two or more arguments where the additional arguments are names, simplify will only invoke the simplification procedures specified by the additional arguments. ∙ Further information on the simplification procedures supported is available in the help pages simplify[name] where name is one of: Ei , GAMMA , RootOf , @, hypergeom , ln , polar , power , radical , sqrt , trig∙ The symbolic option indicates that formal symbolic manipulation of expressions isallowed without regard to the analytical issue of branches for multi-valued functions. For example, the expression sqrt(x^2) will simplify to x under the symbolic option whereas without this option the simplified result must take into account the different possible values of the (complex) sign of x.∙ Note that when the symbolic option is specified, any branch of a multi-valued function can be chosen during the simplification process. The result of such an operation is in general not valid over the whole complex plane, and in fact it can lead to incorrect results in the common situation where the expressions are understood to represent analytical functions.∙ A user can make his or her own simplifications known to the simplify function bydefining a Maple procedure. If the procedure `simplify/f` is defined then the function call simplify(a,f) will invoke `simplify/f`(a).∙ The case of two or more arguments where the additional arguments are sets or lists is used for simplification with respect to side relations. See the subtopic simplify[siderels] for details.∙ Whenever the last argument is assume=prop , all the indeterminates in expr are assumed to have the property prop to compute the simplified expression.Examples> simplify(4^(1/2)+3);5> simplify((x^a)^b+4^(1/2), power); + ()x a b 2> simplify(exp(a+ln(b*exp(c))));b e ()+ a c> simplify(sin(x)^2+cos(x)^2, trig);1> e := cos(x)^5 + sin(x)^4 + 2*cos(x)^2 - 2*sin(x)^2 - cos(2*x): simplify(e);+ ()cos x 5()cos x 4> f := -1/3*x^5*y + x^4*y^2 + 1/3*x*y^3 + 1:simplify(f, {x^3 = x*y, y^2 = x+1});+ + - - + 1y 5y 42y 3y 2y> g:=sqrt(x^2);:= g x 2> simplify(g);()csgn x x> simplify(g,assume=real);x> simplify(g,assume=positive);x> simplify(g,symbolic);xseries - generalized series expansionCalling Sequenceseries(expr, eqn)series(expr, eqn, n)Parametersexpr - an expressioneqn - an equation (such as x = a) or name (such as x)n - (optional) a non-negative integerDescription∙ The series function computes a truncated series expansion of expr, with respect to the variable x, about the point a, up to order n. If a is infinity then an asymptotic expansion is given.∙ If eqn evaluates to a name x then the equation x = 0 is assumed.∙ If the third argument n is present then it specifies the ``truncation order'' of the series calculations. This does not mean the truncation order of the actual series. See Order for more information about this. If n is not present, the ``truncation order'' is determined by the global variable Order. The user may assign any non-negative integer to Order. The default value of Order is 6. See Order for more information.∙ If the series is not exact then an ``order term'' (for example O(x^6) ) is the last term in the series.∙ It is possible to invoke series on user-defined functions. For example, if the procedure `series/f` is defined then the function call series(f(x,y),x) will invoke `series/f`(x,y,x) to compute the series. Note that this user-defined function `series/f` must return a series data structure, not just a polynomial (see type[series]).∙ If series is applied to an unevaluated integral then the series expansion of the integral will be computed (if possible).∙ The result of the series function is a generalized series expansion. This could be a Taylor series or a Laurent series or a more general series. Formally, the coefficients in a ``generalized series'' are such thatk1*(x-a)^eps < |coeff[i]| < k2/(x-a)^epsfor some constants k1 and k2, for any eps>0 as x approaches a. In other words, thecoefficients may depend on x but their growth must be less than the polynomial in x. O(1) represents such a coefficient, rather than an arbitrary constant.∙ Usually, the result of the series function is represented in the form of Maple's series data structure. For an explanation of the data structure, see the help page for type[series]. However, the result of the series function will be represented in ordinary sum-of-products form, rather than in Maple's series data structure, if it is a generalized series requiring fractional exponents.Examples> series(x/(1-x-x^2), x=0);+ + + + + x x 22x 33x 45x 5()O x 6> convert(%,polynom);+ + + + x x 22x 33x 45x 5> series(x+1/x, x=1, 3 ); + + 2() - x 12()O () - x 13> series(exp(x), x=0, 8 ); + + + + + + + + 1x 12x 216x 3124x 41120x 51720x 615040x 7()O x 8> series(exp(x)/x, x=0, 8 );+ + + + + + + + x -1112x 16x 2124x 31120x 41720x 515040x 6()O x 7> series(GAMMA(x), x=0, 2 );- + + x -1γ⎛⎝ ⎫⎭⎪⎪ + 112π212γ2x ()O x 2> series(x^3/(x^4+4*x-5),x=infinity);- + + 1x 4x 45x 5⎛⎝ ⎫⎭⎪⎪O 1x 7> int(exp(x^3), x ); -13()-1()/23⎛⎝ ⎫⎭⎪⎪⎪⎪⎪⎪⎪ - 23x ()-1()/13π3⎛⎝ ⎫⎭⎪⎪Γ23()-x 3()/13x ()-1()/13⎛⎝ ⎫⎭⎪⎪Γ,13-x 3()-x 3()/13 > series(%, x=0);+ + x 14x 4()O x 7> series(x^x, x=0, 3);+ + + 1()ln x x 12()ln x 2x 2()O x 3> s := series(sqrt(sin(x)), x=0, 4);:= s - + x 112x ()/52()O x ()/92> type(s, series);false> whattype(s);+。