THE BOUNDEDNESS OF OPERATORS ON WEIGHTED MULTI-PARAMETER LOCAL HARDY SPACES

Though atomic decomposition is a very useful tool for studying the boundedness on Hardy spaces for some sublinear operators,untill now,the boundedness of operators on weighted Hardy spaces in a multi-parameter setting has been established only by almost orthogonality estimates.In this paper,we mainly establish the boundedness on weighted multi-parameter local Hardy spaces via atomic decomposition.

BOUNDEDNESS OF VECTOR-VALUED OPERATORS ON WEIGHTED HERZ SPACES

In this paper, the authors establish the weighted weak and strong type norm inequalities in the set of weighted Herz-type spaces for a vector-valued analogue o f the Hardy-Littlewood maximal operator; and using this,the authors obtain the weighted inequalies for a wide class of sublinear singular operators defined on Rn which include the Calderon-Zygmund operators as special cases. The fractional versions of these results are also given.

HYPONORMALITY OF BLOCK TOEPLITZ OPERATORS ON THE WEIGHTED BERGMAN SPACES

In this paper we consider the block Toeplitz operators T_Φ on the weighted Bergman space A_α~2(D,C^n) and we give a necessary and sufficient condition for the hyponormality of block Toeplitz operators with symbol in the class of functions Φ=F+G* with matrix-valued polynomial functions F and G with degree 2.

BOUNDEDNESS OF GENERALIZED FRACTIONAL INTEGRALS IN WEIGHTED HERZ-TYPE SPACES

In this paper, the authors investigate the boundedness of the generalized fractional integrals of Perez on the weighted Herz spaces, the weighted weak Herz spaces and the weighted Herz-type Haray spaces for general weights.

Direct and Reverse Carleson Conditions on Generalized Weighted Bergman-Orlicz Spaces

Let ■ be the open unit disk in the complex plane ■.For α>-1,let dA_α(z)=(1+α)1-|z|~2αd A(z) be the weighted Lebesgue measure on ■.For a positive function ω∈L^1(■,dA_α),the generalized weighted Bergman-Orlicz space A_ω~ψ(■,dA_α)isthe space of all analytic functions such that ||f||_ω~ψ=∫_■ψ(|f(z)|)ω(z)dA_α(z)<∞,where ψ is a strictly convex Orlicz function that satisfies other technical hypotheses.Let G be a measurable subset of ■,we say G satisfies the reverse Carleson condition for A_ω~ψ(■,dA_α) if there exists a positive constant C such that ∫_Gψ(|f(z)|)ω(z)dA_α(z)≥C∫_■ψ(|f(z)|)ω(z)dA_α(z),for all f∈A_ω~ψ(■,dA_α).Let μ be a positive Borel measure,we say μ satisfies the direct Carleson condition if there exists a positive constant M such that for all f∈A_ω~ψ(■,dA_α),∫_■ψ(|f(z)|)dμ(z)≤M∫_■ψ(|f(z)|)ω(z)dA_α(z).In this paper,we study the direct and reverse Carleson condition on the generalized weighted Bergman-Orlicz space A_ω~ψ(■,dA_α).We present conditions on the set G such that the reverse Carleson condition holds.Moreover,we give a sufficient conditionfor the finite positive Borel measureμto satisfy the direct carleson condition on the generalized weighted Bergman-Orlicz spaces.

Renormalized Solutions for Nonlinear Parabolic Systems with Three Unbounded Nonlinearities in Weighted Sobolev Spaces

We prove an existence result without assumptions on the growth of some nonlinear terms, and the existence of a renormalized solution. In this work, we study the existence of renormalized solutions for a class of nonlinear parabolic systems with three unbounded nonlinearities, in the form b1(x,u1) t div a(x, t, u1, Du1) +div Φ1(u1) + f1(x,u1,u2) = 0 in Q, b2(x,u2) t div a(x, t, u2, Du2) +div Φ2(u2) + f2(x,u1,u2) = 0 in Q, in the framework of weighted Sobolev spaces, where b(x,u) is unbounded function on u, the Carathe′odory function aisatisfying the coercivity condition, the general growth condition and only the large monotonicity, the function φiis assumed to be continuous on R and not belong to(L1 loc(Q))N.

WEIGHTED HARDY SPACES ON HOMOGENEOUS GROUPS

This paper is devoted to study the atom decomposition of the weighted Hardy space on a homogeneous group X. Some results are extensions of the objects studied in Euclidean harmonic analysis.

WEAK TYPE INEQUALITIES FOR FRACTIONAL MAXIMAL OPERATOR ON WEIGHTED ORLICZ SPACES

The fractional maximal operator on homogeneous s pace (X,d,u)is defined as In this paper ,the su fficient and necessary conditions for the to be of weak type and extra weak type will be given.

MEAN APPROXIMATION BY DILATATIONS IN BERGMAN SPACES ON THE UPPER HALF-PLANE

We study sufficient conditions on radial and non-radial weight functions on the upper half-plane that guarantee norm approximation of functions in weighted Bergman,weighted Dirichlet,and weighted Besov spaces on the upper half-plane by dilatations and eventually by analytic polynomials.

两类加权空间间的积分算子与离散算子的有界性及算子范数估计

Using the weight coefficient method, we first discuss semi-discrete Hilbert-type inequalities, and then discuss boundedness of integral and discrete operators and operator norm estimates based on Hilbert-type inequalities in weighted Lebesgue space and weighted normed sequence space.

ON F(p,q,s)SPACES

The family of spaces F(p,q,s)was introduced by the author in 1996.Since then,there has been great development in the theory of these spaces,due to the fact that these spaces include many classical function spaces,and have connections with many other areas of mathematics.In this survey we present some basic properties and recent results on F(p,q,s)spaces.

STRONG EQUIVALENCES OF APPROXIMATION NUMBERS AND TRACTABILITY OF WEIGHTED ANISOTROPIC SOBOLEV EMBEDDINGS

In this article,we study multivariate approximation defined over weighted anisotropic Sobolev spaces which depend on two sequences a={a j}j≥1 and b={b j}j≥1 of positive numbers.We obtain strong equivalences of the approximation numbers,and necessary and sufficient conditions on a,b to achieve various notions of tractability of the weighted anisotropic Sobolev embeddings.

Toeplitz Operator Related to Singular Integral with Non-Smooth Kernel on Weighted Morrey Space

Let T_1 be a singular integral with non-smooth kernel or ± I, let T_2 and T_4 be the linear operators and let T_3= ±I. Denote the Toeplitz type operator by T^b= T_1M^bI_αT_2+T_3I_αM^bT_4,where M^bf=bf, and I_α is the fractional integral operator. In this paper, we investigate the boundedness of the operator T^b on the weighted Morrey space when b belongs to the weighted BMO space.

Global Attractor for High-dimensional Spacially Discrete FitzHugh-Nagumo System in Weighted Space

In this paper,We study the global attractor and its properties on in nite lattice dynamical system FitzHugh-Nagumo in a weighted space lσ^2×lσ^2.We prove the existence and uniqueness of the solution to the lattice dynamical system FitzHugh-Nagumo in lσ^2×lσ^2.Then we get a bounded absorbing set,which suggests the existence of global attractors.Finally,we study the uniform boundedness and the upper semicontinuity of the global attractor.

On Approximation by Two Kinds Modified Durrmeyer Rational Interpolation Operators in Lω^M Spaces

This paper discusses the approximation problem of two kinds Durrmeyer rational interpolation operators in Orlicz spaces with weight functions,and gives a kind of Jackson type estimation of approximation order by means of continuous modulus, Hardy-Littlewood maximal function, convexity of N function and Jensen inequality.

Quasilinear Degenerated Elliptic Systems with Weighted in Divergence Form with Weak Monotonicity with General Data

We consider, for a bounded open domain Ω in Rn and a function u : Ω → Rm, the quasilinear elliptic system: (1). We generalize the system (QES)(f,g) in considering a right hand side depending on the jacobian matrix Du. Here, the star in (QES)(f,g) indicates that f may depend on Du. In the right hand side, v belongs to the dual space W-1,P’(Ω, ω*, Rm), , f and g satisfy some standard continuity and growth conditions. We prove existence of a regularity, growth and coercivity conditions for σ, but with only very mild monotonicity assumptions.

Maximal Operators of Multilinear Singular Integrals on Weighted Hardy Type Spaces

In this paper,the authors show that the maximal operators of the multilinear Calderón-Zygmund singular integrals are bounded from a product of weighted Hardy spaces into a weighted Lebesgue spaces,which essentially extend and improve the previous known results obtained by Grafakos and Kalton(2001)and Li,Xue and Yabuta(2011).The corresponding estimates on variable Hardy spaces are also established.

Weighted Estimates of Variable Kernel Fractional Integral and Its Commutators on Vanishing Generalized Morrey Spaces with Variable Exponent

In this paper,the authors obtain the boundedness of the fractional integral operators with variable kernels on the variable exponent generalized weighted Morrey spaces and the variable exponent vanishing generalized weighted Morrey spaces.And the corresponding commutators generated by BMO function are also considered.

ANOTHER CHARACTERIZATIONS OF MUCKENHOUPT A_p CLASS

This manuscript addresses Muckenhoupt A_p weight theory in connection to Morrey and BMO spaces. It is proved that ω belongs to Muckenhoupt A_p class,if and only if Hardy-Littlewood maximal function M is bounded from weighted Lebesgue spaces L^p(ω) to weighted Morrey spaces M_q^p(ω) for 1<q<p<∞. As a corollary,if M is(weak) bounded on M_q^p(ω),then ω∈A_p. The A_p condition also characterizes the boundedness of the Riesz transform R_j and convolution operators T_e on weighted Morrey spaces. Finally,we show that ω∈A_p if and only if ω∈ BMO^(p′)(ω) for 1≤p<∞ and 1/p+1/p′= 1.

THE EXISTENCE OF A NONTRIVIAL WEAK SOLUTION TO A DOUBLE CRITICAL PROBLEM INVOLVING A FRACTIONAL LAPLACIAN IN R^N WITH A HARDY TERM

In this paper,we consider the existence of nontrivial weak solutions to a double critical problem involving a fractional Laplacian with a Hardy term:(−Δ)s u−γu|x|2s=|u|2∗s(β)−2 u|x|β+[Iμ∗Fα(⋅,u)](x)fα(x,u),u∈H˙s(R n),(0.1)(1)where s∈(0,1),0≤α,β<2s<n,μ∈(0,n),γ<γH,Iμ(x)=|x|−μ,Fα(x,u)=|u(x)|2#μ(α)|x|δμ(α),fα(x,u)=|u(x)|2#μ(α)−2 u(x)|x|δμ(α),2#μ(α)=(1−μ2n)⋅2∗s(α),δμ(α)=(1−μ2n)α,2∗s(α)=2(n−α)n−2s andγH=4 sΓ2(n+2s4)Γ2(n−2s4).We show that problem(0.1)admits at least a weak solution under some conditions.To prove the main result,we develop some useful tools based on a weighted Morrey space.To be precise,we discover the embeddings H˙s(R n)↪L 2∗s(α)(R n,|y|−α)↪L p,n−2s2 p+pr(R n,|y|−pr),(0.2)(2)where s∈(0,1),0<α<2s<n,p∈[1,2∗s(α))and r=α2∗s(α).We also establish an improved Sobolev inequality,(∫R n|u(y)|2∗s(α)|y|αdy)12∗s(α)≤C||u||θH˙s(R n)||u||1−θL p,n−2s2 p+pr(R n,|y|−pr),∀u∈H˙s(R n),(0.3)(3)where s∈(0,1),0<α<2s<n,p∈[1,2∗s(α)),r=α2∗s(α),0<max{22∗s(α),2∗s−12∗s(α)}<θ<1,2∗s=2nn−2s and C=C(n,s,α)>0 is a constant.Inequality(0.3)is a more general form of Theorem 1 in Palatucci,Pisante[1].By using the mountain pass lemma along with(0.2)and(0.3),we obtain a nontrivial weak solution to problem(0.1)in a direct way.It is worth pointing out that(0.2)and 0.3)could be applied to simplify the proof of the existence results in[2]and[3].