LEIBNIZ′FORMULA OF GENERALIZED DIFFERENCE WITH RESPECT TO A CLASS OF DIFFERENTIAL OPERATORS AND RECURRENCE FORMULA OF THEIR GREEN′S FUNCTION

In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.

APPLICATION OF TWO OPERATORS TRANSFORM FROM CLASS A OPERATOR TO THE CLASS OF HYPONORMAL OPERATOR

In this article, we give an operator transform T （＊） from class A operator to the class of hyponormal operators. It is different from the operator transform T defined by M. Ch and T. Yamazaki. Then, we show that σ（T ） = σ（ T （＊）） and σa（T ）／{0} = σa（ T （＊））／{0}, in case T belongs to class A. Next, we obtain some relations between T and T （9）.

Gaussian Integral Formula for Quaternions Based on Class Operator's Formula of Rotation Group

Abstract There exists rare integration formulas over quaternions due to the noncommutativity of quarternions multplication. Based on the class operator＇s formula of rotation group we derive a Gaussian integral formula for quaternions, which is similar in form to the integration for coherent state＇s completeness relation.

Boundeness of a Class of Maximal Operators on H^p Spaces on Compact Lie Groups

In this paper we discuss the weak type(IP,I)boundedness of a class of maximal operators T and themaximal strong,mean boundedness of a family of the operators {T on the atomic IP spaces on compaet Lie groups.Also,we obtain the correspoding convergent rosults.

APPROXIMATION BY A CLASS OF AIODIFIED BERNSTEIN-DURRMEYER OPERATORS

We modify Bernstein-Durrmeyer operators by means of digonal matarix which overeome a difficulty in extending a Berens-Lorentz result to the Bernstein-Durrmeyer operators for second order of smoothness. The direct and converse theorems for these operators in L_p are also presented by Ditzian-Totik, modulus of smoothness.

On the Spectrum of a Class of Differential Operators and Embedding Theorems

The author considers the embedding problem of weighted Sobolev spaces Hnp in weighted Ls spaces Ls,r,and some sufficient conditions and necessary conditions are given, when weight functions satisfy certain conditions.The author uses the results obtained to the qualitative analysis of the spectrum of 2n-order weighted differential operator,and gives some sufficient conditions and necessary conditions to ensure that the spectrum is discrete.

The calculation for C-G coefficients

The eigenfunction method put forward by Chen Jin-quan is illustrated. We apply this theory to the space group D1_ 6h. The selection rules of this space are worked out in the points of higher symmetry A,K,H in the first Brillion Zone. The C-G coefficients are calculated for K.HA.

The C-G coefficients for the space group D_6~1h

In this paper, the eigenfunction method established by Chen Jin-quan is used to compute the C-G coefficients in regard to the coupling between symmetry points and lines in the first Brillouin zone of the structure D6h^1, space group. Therewith, the wave vector selection rule and the C-G series, are also given as the middle result of computing the C-G coefficients.

On a Class of Second Order Partial Differential Operators of Non-principal Type

§1.IntroductionThis paper deals with linear partial differential operators with real principalsymbol.Let P(x,D)be such an operator of mth order with C~∞ coefficients definedin an open subset Ω of R^n and p_m(x,ξ)be its principal symbol.According to thedefinition given by Duistermaat and Hmander(see[1]),P(x,D)is called ofprincipal type at x^0 ∈Ω if for any ξ∈R^n\0 satisfying p_m(x^0,ξ)=0,x=x^0 is not theprojection in Ω of the bicharacteristic strip of P(x,D)through(x^0,ξ).Under thiscondition,they proved that there exists a neighborhood U of x^0,U,such that forany real number s,

On the Compactness of Grunsky Differential Operators

Grunsky operators play an important role in classical geometric function theory and in the study of Teichmüller spaces.The Grunsky map is known to be holomorphic on the universal Teichmüller space.In this paper the authors deal with the compactness of a Grunsky differential operator.They will give upper and lower estimates of the essential norm of a Grunsky differential operator and discuss when a Grunsky differential operator is a p-Schatten class operator.

Carleson measures, BMO spaces and balayages associated to Schrdinger operators

Let L be a Schrdinger operator of the form L =-? + V acting on L^2(R^n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R^n) denote the BMO space associated to the Schrdinger operator L on R^n. In this article, we show that for every f ∈ BMO_L(R^n) with compact support, then there exist g ∈ L~∞(R^n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R^n), where S_(μ,P)=∫(R_+^(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-^(t(L)^(1/2))}t>0 on L^2(R^n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R^n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators.