AREA INTEGRAL FUNCTIONS AND H∞ FUNCTIONAL CALCULUS FOR SECTORIAL OPERATORS ON HILBERT SPACES

Area integral functions are introduced for sectorial operators on Hilbert spaces. We establish the equivalence relationship between the square and area integral functions. This immediately extends McIntosh/Yagi＇s results on H∞ functional calculus of sectorial operators on Hilbert spaces to the case when the square functions are replaced by the area integral functions.

Weyl Type Theorem for Bounded Linear Operator and Its Functional Calculus

Let H be a complex infinite dimensional Hilbert space and B(H)be the algebra of all bounded linear operators on H.In this paper,we mainly study the operators that satisfy both a-Weyl's theorem and property(R).Also,the operators whose functional calculus satisfies the two properties are also explored.We give the features for the operator or its functional calculus for which both a-Weyl's theorem and property(R)hold.

Restriction Theorems on Métiver Groups Associated to Joint Functional Calculus

The authors get on Metivier groups the spectral resolution of a class of operators m（L, -Δ）, the joint functional calculus of the sub-Laplacian and Laplacian on the centre, and then give some restriction theorems together with their asymptotic estimates, asserting the mix-norm boundedness of the spectral projection operators Pμ^m for two classes of functions re（a, b） =（a^α＋b^β）^γ or （1＋a^α＋b^β）^γ,with α,β〉0,γ≠0.

LITTLEWOOD-PALEY THEOREM FOR SCHR DINGER OPERATORS

Let H be a Schroedinger operator on R^n. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.

AREA INTEGRAL FUNCTIONS FOR SECTORIAL OPERATORS ON L^p SPACES

Area integral functions are introduced for sectorial operators on L^p-spaces. We establish the equivalence between the square and area integral functions for sectorial operators on L^p spaces. This follows that the results of Cowling, Doust, McIntosh, Yagi, and Le Merdy on H^∞ functional calculus of seetorial operators on LP-spaces hold true when the square functions are replaced by the area integral functions.

THE EXPONENTIAL OF QUASI BLOCK-TOEPLITZ MATRICES

The matrix Wiener algebra,W_(N):=M_(N)(W)of order N>0,is the matrix algebra formed by N×N matrices whose entries belong to the classical Wiener algebraWof functions with absolutely convergent Fourier series.A block-Toeplitz matrix T(a)=[A_(i,j)]i,j≥0is a block semi-infinite matrix such that its blocks A_(i,j) are finite matrices of order N,A_(i,j)=A^(r,s) whenever i-j=r-s and its entries are the coefficients of the Fourier expansion of the generator a:T→M_(N)(C).Such a matrix can be regarded as a bounded linear operator acting on the direct sum of N copies of L^(2)(T).We show that exp(T(a))differes from T(exp(a))only in a compact operator with a known bound on its norm.In fact,we prove a slightly more general result:for every entire function f and for every compact operator E,there exists a compact operator F such that f(T(a)+E)=T(f(a))+F.We call these T(a)+E′s matrices,the quasi block-Toeplitz matrices,and we show that via a computation-friendly norm,they form a Banach algebra.Our results generalize and are motivated by some recent results of Dario Andrea Bini,Stefano Massei and Beatrice Meini.

TIME DECAY FOR SCHRDINGER EQUATION WITH ROUGH POTENTIALS

We obtain certain time decay and regularity estimates for 3D Schroedinger equation with a potential in the Kato class by using Besov spaces associated with Schroedinger operators.

Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility

We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics,a nonlocal formulation of continuum mechanics.We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation.A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero.We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible.We fnd that,for polynomial degrees greater than or equal to two,there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal difusion are asymptotically compatible.We verify these fndings through extensive numerical experiments.

INTERPOLATION THEOREMS FOR SELF-ADJOINT OPERATORS

We prove a complex and a real interpolation theorems on Besov spaces and Triebel-Lizorkin spaces associated with a selfadjoint operator L, without assuming the gradient estimate for its spectral kernel. The result applies to the cases where L is a uniformly elliptic operator or a Schrdinger operator with electro-magnetic potential.

Note on Gradient Estimate of Heat Kernel for Schrodinger Operators

Let be a Schr?dinger operator on . We show that gradient estimates for the heat kernel of with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators. The latter decay property has been known to play an important role in the Littlewood-Paley theory for and Sobolev spaces. We are able to establish the result by modifying Hebisch and the author’s recent proofs. We give a counterexample in one dimension to show that there exists in the Schwartz class such that the long time gradient heat kernel estimate fails.

Lucas Symbolic Formulae and Generating Functions for Chebyshev Polynomials

This work shows that each kind of Chebyshev polynomials may be calculated from a symbolic formula similar to the Lucas formula for Bernoulli polynomials. It exposes also a new approach for obtaining generating functions of them by operator calculus built from the derivative and the positional operators.

THE CALCULUS OF GENERATING FUNCTIONS AND THE FORMAL ENERGY FOR HAMILTONIAN ALGORITHMS

In [2-4], symplectic schemes of arbitrary order are constructed by generating functions. However the construction of generating functions is dependent on the chosen coordinates. One would like to know that under what circumstance the construction of generating functions will be independent of the coordinates. The generating functions are deeply associated with the conservation laws, so it is important to study their properties and computations. This paper will begin with the study of Darboux transformation, then in section 2, a normalization Darboux transformation will be defined naturally. Every symplectic scheme which is constructed from Darboux transformation and compatible with the Hamiltonian equation will satisfy this normalization condition. In section 3, we will study transformation properties of generator maps and generating functions. Section 4 will be devoted to the study of the relationship between the invariance of generating functions and the generator maps. In section 5, formal symplectic erengy of symplectic schemes are presented.

Applications of Orlicz-Hardy spaces associated with operators satisfying Poisson estimates

Let L be a linear operator in L^2（R^n） and generate an analytic semigroup {e^-tL}t≥0 with kernel satisfying an upper bound estimate of Poisson type, whose decay is measured by θ（L） ∈ （0, ∞）. Let 4） be a positive, continuous and strictly increasing function on （0, ∞）, which is of strictly critical lower type pФ （n/（n ＋ θ（L））, 1]. Denote by HФ, L（R^n） the Orlicz-Hardy space introduced in Jiang, Yang and Zhou＇s paper in 2009. If Ф is additionally of upper type 1 and subadditive, the authors then show that the Littlewood-Paley g-function gL maps HФ, L（R^n） continuously into LФ（R^n） and, moreover, the authors characterize HФ, L（R^n） in terms of the Littlewood-Paley gλ^＊-function with λ ∈ （n（2/pФ ＋ 1）, ∞）. If Ф is further slightly strengthened to be concave, the authors show that a generalized Riesz transform associated with L is bounded from HФ, L（R^n） to the Orlicz space L^Ф（R^n） or the Orlicz-Hardy space HФ （R^n）; moreover, the authors establish a new subtle molecular characterization of HФ, L （R^n） associated with L and, as applications, the authors then show that the corresponding fractional integral L^-γ for certain γ∈ E （0,∞） maps HФ, L（R^n） continuously into HФ, L（R^n）, where Ф satisfies the same properties as Ф and is determined by Ф and λ and also that L has a bounded holomorphic functional calculus in HФ, L（R^n）. All these results are new even when Ф（t） = t^p for all t ∈ （0, ∞） and p ∈ （n/（n ＋ θ（L））, 1].

Elliptic Equations with Degenerate Coercivity： Gradient Regularity

In this paper, we prove higher integrability results for the gradient of the solutions of some elliptic equations with degenerate coercivity whose prototype is$ - {\rm div}\left( {a\left( {x,u} \right)Du} \right) = f$ in $D^' \left( \Omega \right),\,\,f \in L^r \left( \Omega \right),\,\,r > 1$where for example, a(x,u)=(1+|u|)^m/ with / ] (0,1). We study the same problem for minima of functionals closely related to the previous equation.

Limit behaviour of the minimal solution of a BSDE with singular terminal condition in the non Markovian setting

We use the functional Ito calculus to prove that the solution of a BSDE with singular terminal condition verifies at the terminal time:lim inf_(t→T)Y(t)=ξ=Y(T).Hence,we extend known results for a non-Markovian terminal condition.