OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON TRIEBEL SPACES

The authors establish operator-valued Fourier multiplier theorems on Triebel spaces on R^N, where the required smoothness of the multiplier functions depends on the dimension N and the indices of the Triebel spaces. This is used to give a sufficient condition of the maximal regularity in the sense of Triebel spaces for vector-valued Cauchy problems with Dirichlet boundary conditions.

Dyadic Bivariate Fourier Multipliers for Multi-Wavelets in L^2(R^2)

The single 2 dilation orthogonal wavelet multipliers in one dimensional case and single A-dilation(where A is any expansive matrix with integer entries and|det A|=2) wavelet multipliers in high dimensional case were completely characterized by the Wutam Consortium(1998) and Z. Y. Li, et al.(2010). But there exist no more results on orthogonal multivariate wavelet matrix multipliers corresponding integer expansive dilation matrix with the absolute value of determinant not 2 in L~2(R~2). In this paper, we choose 2I2=(~2~0)as the dilation matrix and consider the 2 I2-dilation orthogonal multivariate waveletΨ = {ψ, ψ, ψ},(which is called a dyadic bivariate wavelet) multipliers. We call the3 × 3 matrix-valued function A(s) = [ f(s)], where fi, jare measurable functions, a dyadic bivariate matrix Fourier wavelet multiplier if the inverse Fourier transform of A(s)( ψ(s), ψ(s), ψ(s)) ~T=( g(s), g(s), g(s))~ T is a dyadic bivariate wavelet whenever(ψ, ψ, ψ) is any dyadic bivariate wavelet. We give some conditions for dyadic matrix bivariate wavelet multipliers. The results extended that of Z. Y. Li and X. L.Shi(2011). As an application, we construct some useful dyadic bivariate wavelets by using dyadic Fourier matrix wavelet multipliers and use them to image denoising.

Operator-valued Fourier Multipliers on Periodic Triebel Spaces

We establish operator-valued Fourier multiplier theorems on periodic Triebel spaces, where the required smoothness of the multipliers depends on the indices of the Triebel spaces. This is used to give a characterization of the maximal regularity in the sense of Triebel spaces for Cauchy problems with periodic boundary conditions.

SOME MULTIPLIER THEOREMS FOR ANISOTROPIC HARDY SPACES——In Memory of Professor Yongsheng Sun

Let A be a symmetric expansive matrix and H^p（R^n） be the anisotropic Hardy space associated with A. For a function m in L∞（R^n）, an appropriately chosen function η in Cc^∞（R^n） and j ∈ Z define mj（ξ） = m（A^jξ）η（ξ）. The authors show that if 0 〈 p 〈 1 and mj belongs to the anisotropic nonhomogeneous Herz space K1^1/P^-1,p（R^n）, then m is a Fourier multiplier from H^p（R^n） to L^V（R^n）. For p = 1, a similar result is obtained if the space K1^0.1（R^n） is replaced by a slightly smaller space K（w）. Moreover, the authors show that if 0 〈 p 〈 1 and if the sequence {（mj）^v} belongs to a certain mixednorm space, depending on p, then m is also a Fourier multiplier from H^p（R^n） to L^v（R^n）.

BOUNDS FOR MULTILINEAR OPERATORS UNDER AN INTEGRAL TYPE CONDITION ON MORREY SPACES

In this paper,we study a boundedness property of the Adams type for multilinear fractional integral operators with the multilinear L^(r′,α)-Hörmander condition and their commutators with vector valued BMO functions on a Morrey space and a predual Morrey space.Moreover,we give an endpoint estimate for multilinear fractional integral operators.As an application,we obtain the boundedness of multilinear Fourier multipliers with limited Sobolev regularity on a Morrey space.

ABSTRACT CAPACITY OF REGIONS AND COMPACT EMBEDDING WITH APPLICATIONS

The weighted Sobolev-Lions type spaces W pl,γ（Ω; E0, E） = W pl,γ（Ω; E） ∩ Lp,γ （Ω; E0） are studied, where E0, E are two Banach spaces and E0 is continuously and densely embedded on E. A new concept of capacity of region Ω ∈ Rn in W pl,γ（; E0, E） is introduced. Several conditions in terms of capacity of region Ω and interpolations of E0 and E are found such that ensure the continuity and compactness of embedding operators. In particular, the most regular class of interpolation spaces Eα between E0 and E, depending of α and l, are found such that mixed differential operators Dα are bounded and compact from W pl,γ（Ω; E0, E） to Eα-valued Lp,γ spaces. In applications, the maximal regularity for differential-operator equations with parameters are studied.

HLDER CONTINUOUS SOLUTIONS FOR SECOND ORDER INTEGRO-DIFFERENTIAL EQUATIONS IN BANACH SPACES

We study Hlder continuous solutions for the second order integro-differential equations with infinite delay （P1）： u′′（t）＋cu′（t）＋∫t-∞β（t-s）u′（s）ds＋∫t-∞γ（t-s）u（s）ds = Au（t）-∫t-∞δ（t-s）Au（s）ds ＋ f（t）on the line R, where 0 〈 α 〈 1, A is a closed operator in a complex Banach space X, c ∈ C is a constant, f ∈ Cα（R,X） and β,γ,δ∈L1（R＋）.Under suitable assumptions on the kernels β, γ and δ, we completely characterize the Cα- well-posedness of （P_1） by using operator-valued Cα-Fourier multipliers.

Some Remarks on the Restriction Theorems for the Maximal Operators on R^d

The aim of this paper is to give a simple proof of the restriction theorem for the maximal operators on the d-dimensional Euclidean space Ra, whose theorem was proved by Carro-Rodriguez in 2012. Moreover, we shall give some remarks of the restriction theorem for the linear and the multilinear operators by Carro-Rodriguez and Rodriguez, too.

The Quality Properties of Integral Type Problems for Wave Equations and Applications

In this paper, the integral problem for linear and nonlinear wave equations is studied. The equation involves abstract operator A in Hilbert space H. Here, assuming enough smoothness on the initial data and the operators, the existence, uniqueness, regularity properties of solutions are established. By choosing the space H and A, the regularity properties of solutions of a wide class of wave equations in the field of physics are obtained.

REGULARIZED SEMIGROUPS GENERATED BY PSEUDODIFFERENTIAL OPERATORS ON C~α

Let Op(f) be a pseudodifferential operator with symbol f∈ S m ρ,0 having constant coefficients. We prove that Op(f) generates a regularized semigroup or cosine function on C α (R n) (0<α<1) when the symbol f(ξ) and its derivatives satisfy certain growth conditions.

Embedding and Maximal Regular Differential Operators in Sobolev-Lions Spaces

This study focuses on vector-valued anisotropic Sobolev-Lions spaces associated with Banach spaces E0, E. Several conditions are found that ensure the continuity and compactness of embedding operators that are optimal regular in these spaces in terms of interpolations of spaces E0 and E. In particular, the most regular class of interpolation spaces Eα between E0, E depending on α and the order of space are found and the boundedness of differential operators D^α from this space to Eα-valued Lp,γ spaces is proved. These results are applied to partial differential-operator equations with parameters to obtain conditions that guarantee the maximal Lp,γ regularity and R-positivity uniformly with respect to these parameters.

Well-Posedness of Equations with Fractional Derivative

We study the well-posedness of the equations with fractional derivative D^αu（t） = Au（t） ＋ f（t）,0≤ t ≤ 2π, where A is a closed operator in a Banach space X, α 〉 0 and D^α is the fractional derivative in the sense of Weyl. Using known results on LP-multipliers, we give necessary and/or sufficient conditions for the LP-well-posedness of this problem. The conditions we give involve the resolvent of A and the Rademacher boundedness. Corresponding results on the well-posedness of this problem in periodic Besov spaces, periodic Triebel-Lizorkin spaces and periodic Hardy spaces are also obtained.