Anisotropic Hardy-Lorentz spaces and their applications

Let p ∈(0, 1], q ∈(0, ∞] and A be a general expansive matrix on Rn. We introduce the anisotropic Hardy-Lorentz space H~(p,q)_A(R~n) associated with A via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic and the molecular decompositions, the radial and the non-tangential maximal functions, and the finite atomic decompositions. All these characterizations except the ∞-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on Rn.As applications, we first prove that Hp,q A(Rn) is an intermediate space between H~(p1,q1)_A(Rn) and H~(p2,q2)_A(R~n) with 0 p1 p p2 ∞ and q1, q, q2 ∈(0, ∞], and also between H~(p,q1)_A(Rn) and H~(p,q2)_A(R~n) with p ∈(0, ∞)and 0 q1 q q2 ∞ in the real method of interpolation. We then establish a criterion on the boundedness of sublinear operators from H~(p,q)_A(R~n) into a quasi-Banach space; moreover, we obtain the boundedness of δ-type Calder′on-Zygmund operators from H~(p,∞)_A(R~n) to the weak Lebesgue space L~(p,∞)(R~n)(or to H~p_A(R~n)) in the ln λcritical case, from H~(p,q)_A(R~n) to L~(p,q)(R~n)(or to H~(p,q)_A(R~n)) with δ∈(0,(lnλ)/(ln b)], p ∈(1/(1+,δ),1] and q ∈(0, ∞], as well as the boundedness of some Calderon-Zygmund operators from H~(p,q)_A(R~n) to L~(p,∞)(R~n), where b := | det A|,λ_:= min{|λ| : λ∈σ(A)} and σ(A) denotes the set of all eigenvalues of A.     

Marcinkiewicz积分交换子在Herz型空间中的弱型估计

用μΩ表示Marcinkiewicz积分,μΩ,b表示μΩ与函数b∈BMO(R~n)生成的交换子．本文证明了交换子μΩ,b是从Herz型Hardy空间H■_q~(n(1-(1/q)),p)(R~n)到弱Herz空间W■_q~(n(1-(1/q)),p)(R~n)有界的,其中0＜p≤1,1＜q＜∞．     

一类两点边值问题的正解个数

本文讨论边值问题y"+λ(yp+μy+yq)=0,y(-1)=y(1)=0,其中λ＞0是正参数,μ≥0.对(1-p)(1-q)＞0的情形得出了正解的存在唯一性.对(1-p)(1-q)＜0的情形,其主要结论是若p＞1＞q＞-(25+23p)/(23+25p),μ≥0,则存在λ*＞0,使得当0＜λ＜λ*时,此边值问题恰好存在两个正解,当λ=λ*时,存在唯一正解,当λ＞λ*时,不存在正解.     

Triebel-Lizorkin型空间在Ahlfors n-正则区域上的限制

设n≥2.对于任意的Ahlfors n-正则域??R~n,通过分数阶的Hajlasz-梯度,本文刻画了Triebel-Lizorkin型空间F_(p,q)~(α,τ)(R~n)在?上的迹空间F_(p,q)~(α,τ)(R~n)|?,其中参数α、τ、p和q满足α∈(0, 1), p∈(n/(n+α), ∞), q∈(n/(n+α), ∞], τ∈(0,1/p+(1-α)/n).(0.1)反之,对于任意的区域??R~n及满足(0.1)且τ≥1/p-α/n的参数α、τ、p和q,若迹空间F_(p,q)~(α,τ)(R~n)|?能通过Haj lasz梯度刻画,则?是Ahlfors n-正则域.     

由Hrmander向量场构成的抛物方程的W_*~(1,p)正则性

设X_1,…,X_q(qn)是有界区域ΩR~n上的一组光滑Hrmander向量场.考虑下面的散度型抛物方程:u_t+X_i~*(a_(ij)(x,t)X_ju)=X_i~*f_i,其中X_i~*是X_i的形式伴随,系数a_(ij)(x,t)(i,j=1,2,…,q)是定义在Ω_T=Ω×(0,T)∈R~(n+1)上满足一致椭圆性条件的有界可测函数.在系数属于VMO_(loc)∩L~∞函数空间的情况下,得到了抛物方程弱解的内W_*~(1,p)正则性.     

Estimates of Some Integral Operators with Bounded Variable Kernels on the Hardy and Weak Hardy Spaces over R~n

In this paper,we first introduce L~(σ_1)-(log L)~(σ_2) conditions satisfied by the variable kernelsΩ(x,z) for 0≤σ_1≤1 and σ_2≥0.Under these new smoothness conditions,we will prove the boundedness properties of singular integral operators T_Ω,fractional integrals T_(Ω,α) and parametric Marcinkiewicz integrals μ_Ω~ρ with variable kernels on the Hardy spaces H~p(R~n) and weak Hardy spaces WH~P(R~n).Moreover,by using the interpolation arguments,we can get some corresponding results for the above integral operators with variable kernels on Hardy-Lorentz spaces H~(p,q)(R~n) for all p q ∞.     

用Bcklund变换确定神经传导反应扩散方程的显示精确行波解

文献[1]讨论了反应扩散方程的形如u(x_1,t)=q(x-ct)的行波解.令ξ=x-ct,给出该方程的BackIund变换为q_x=p(q),q_t=-cp(q).显然,p=p(q)∈C~1[0,1]∩C~2(0,1)应满足p((dp)/(dq) c)=-f(q).若c=0,则p=±(-2∫_0~(q(ξ))f(τ)dτ)~(1/2);若c≠0,则必须从方程(dp)/(dq)=-c-f(q)/p,p(0)=p(1)=0,p(q)>0,q∈(0,1)出发寻求传播较快的行波.如果p和f分别为次数m和n的多项式,那么n=2m-1.在m=1和2情形下求得的传播速度与生物物理学家用实验     

C~n中单位球上一般Hardy型空间的几种等价刻画

设p 0, S≥0, q+n≥0, q+s≥0,本文探讨了C~n中单位球B上一般Hardy型空间H~(p,q,s)(B)的几种等价刻画.同时,本文还给出了单位球内双变点球面积分所有情形的双向估计.     

非交换加权Lorentz空间的对偶空间

设μ是一个半有限von Neumann代数.对于0P∞,0q≤∞,定义了非交换加权Lorentz空间Λ_ω~(p,q)(μ)及其associate空间Λ_ω~(p,q)(μ)',给出了空间Λ_ω~(p,q)(μ)'和Λ_ω~(p,q)(μ)'的一些基本性质.应用这些性质,还给出了非交换加权Lorentz空间Λ_ω~p(μ),0P∞的对偶空间.     

具有有界位势的Lagrange系统的周期解

本文考虑Lagrange系统 d/dt ?/?(q,q)-?/?q(q,q)=0 (?)这儿?(q,ξ)=1/2 wum form i,j=1 to n a_(ij)(q)eξξ-V(q),q∈R~n,ξ∈R~n 。假设位势函数V有界,并且lim V(q)/|q|~2 =δ>0,V(q)=V(-q),则系统(?)具有一个以适当大的正数T为周期的非平凡解。     

非齐型空间上分数型Marcinkiewicz积分算子的加权估计

令（X，d，μ）为满足所谓上倍双倍条件和几何双倍条件的度量测度空间.设M_β，ρ，q为（X，d，μ）上的分数型Marcinkiewicz积分算子.在本文中，作者证明了若β ∈[0，∞），ρ ∈（0，∞），q ∈（1，∞）且M_β，ρ，q在L²（μ）上有界，则M_β，ρ，q是从加权Lebesgue空间L^p（w）到加权弱Lebesgue空间L^p，∞（w）上有界和从加权Morrey空间L^p，κ，η（ω）到加权弱Morrey空间WL^p，κ，η（ω）上有界.     

On Characterization of Poisson Integrals of Schrödinger Operators with Morrey Traces

Let L be a Schr?dinger operator of the form L =-Δ + V acting on L~2(R~n) where the nonnegative potential V belongs to the reverse H?lder class B_q for some q ≥ n. In this article we will show that a function f ∈ L~(2,λ)(R~n), 0 λ n, is the trace of the solution of L_u =-u_(tt) + L_u =0, u(x, 0) = f(x), where u satisfies a Carleson type condition sup x_B,r_Br_B~(-λ)∫_0~(rB)∫_(B(x_B,r_B))t|u(x,t)|~2dxdt≤C∞.Its proof heavily relies on investigate the intrinsic relationship between the classical Morrey spaces and the new Campanato spaces L_L~(2,λ)(R~n) associated to the operator L, i.e.L_L~(2,λ)(R~n)=L~(2,λ)(R~n).Conversely, this Carleson type condition characterizes all the L-harmonic functions whose traces belong to the space L~(2,λ)(R~n) for all 0 λ n.     

Jacobian-determinant equation in the critical Besov spaces

Let Ω be a bounded domain in R~n with smooth boundary. Here we consider the following Jacobian-determinant equation det u(x)=f(x),x∈Ω;u(x)=x,x∈?Ω where f is a function on Ω with min_Ω f = δ 0 and Ωf(x)dx = |Ω|. We prove that if f ∈B_(p1)~(np)(Ω) for some p∈(n,∞), then there exists a solution u ∈ B_(p1)~(np+1)(Ω)C~1(Ω) to this equation. On the other hand, we give a simple example such that u ∈ C_0~1(R~2, R~2) while detu does not lie in B_(p1)~(2p)(R~2) for any p∞.     

Kakeya极大算子及其分数次情形的正则性

研究中心Kakeya(Nikodym)极大算子K_N(N2)及其分数次情形K_(α,N)(0αd)的正则性.特别地,建立了中心分数次Kakeya极大算子K_(α,N)是从W~(1,p)(R~d)到W~(1,q)(R~d)上的有界连续算子,其中1p∞,q=dp/(d-αp)和0≤αd/p.还证明了中心Kakeya极大算子K_N是分数次Sobolev空间W~(s,p)(R~d),非齐次Triebel-Lizorkin空间F_s~(p,q)(R~d)以及非齐次Besov空间B_s~(p,q)(R~d)上的有界连续算子,其中0s1,1p,q∞.此外,也考虑分数次Kakeya极大函数的弱导数的两种点态估计以及其离散情形的正则性.     

Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy–Sobolev–Maz'ya term:-Δu- λu/|y|2=|u|pt-1u/|y|t+ μf(x), x ∈Ω,where Ω is a bounded domain in RN(N ≥ 2), 0 ∈Ω, x =(y, z) ∈ Rk× RN-kand pt =N +2-2t N-2(0 ≤ t ≤2). For f(x) ∈ C1(Ω)\{0}, we show that there exists a constant μ* 0 such that the problem possessesat least two positive solutions if μ∈(0, μ*) and at least one positive solution if μ = μ*. Furthermore,there are no positive solutions if μ∈(μ*, +∞).     

Molecular Characterization of Anisotropic Musielak–Orlicz Hardy Spaces and Their Applications

Let A be an expansive dilation on R~n and φ:R~n× [0,∞)→[0,∞) an anisotropic Musielak–Orlicz function.Let H_A~φ(R~n) be the anisotropic Hardy space of Musielak–Orlicz type defined via the grand maximal function.In this article,the authors establish its molecular characterization via the atomic characterization of H_A~φ(R~n).The molecules introduced in this article have the vanishing moments up to order s and the range of s in the isotropic case(namely,A:=2I_(n×n)) coincides with the range of well-known classical molecules and,moreover,even for the isotropic Hardy space H~p(R~n)with p∈(0,1](in this case,A:=2I_(n×n),φ(x,t) :=t~p for all x∈R~n and t∈[0,∞)),this molecular characterization is also new.As an application,the authors obtain the boundedness of anisotropic Calderón–Zygmund operators from H_A~φ(R~n) to L~φ(R~n) or from H_A~φ(R~n) to itself.     

一类新型BMO空间

引进了弱型有界平均震荡函数空间WBMO_q,1q∞,它是类似于弱型勒贝格空间L~(q,∞)所对应的BMO空间.证明了‖·‖*(BMO范数)与‖·‖_(WBMO_q)之间的等价特征刻画.作为应用,对于p∈(1,∞)和1/q=1/p-α/n,交换子[b,I_α]是从Lp到L~(q,∞)的有界算子,当且仅当局部可积函数b属于BMO空间,其中I_α表示分数次积分算子.另外,还引进以及学习了弱型的中心有界平均震荡空间W_q.     

双线性Fourier乘子交换子的有界性与紧性

假定T_σ是关于乘子σ的双线性Fourier乘子算子,其中σ满足如下Sobolev正则条件:对某个s∈(n,2n],有sup_(κ∈Z)‖σ_k‖W~s(R~(2m))∞.对于p_1,p_2,p∈(1,∞)且满足1/p=1/p_1+1/p_2和ω=(ω_1,ω_2)∈A_(p/t)(R~(2n)),建立了T_σ及其与函数b=(b_1,b_2)∈(BMO(R~n))~2生成的交换子T_(σ,b)由L~(p_1,λ)(ω_1)×L~(p_2,λ)(ω_2)到L~(p,λ)(v_w)的有界性;同时,在b_1,b_2∈CMO(R~n)(C_c~∞(R~n)在BMO拓扑下的闭包)的条件下,证明交换子T_(σ,b)是L~(p_1,λ)(ω_1)×L~(p_2,λ)(ω_2)到L~(p,λ)(v_w)的紧算子.为了得到主要结果,我们先后建立了几个双(次)线性极大函数在加多权Morrey空间上的有界性以及该空间中准紧集的判定.     

带多项式相位的高维振荡积分算子的有界性

考虑如下的振荡积分算子:T_(m,k,n)f(x):=∫_(R~n)e~(i(x_1~2+…+x_n~2))~m(y_1~2+…+y_n~2)~kf(y)dy,其中函数f为定义在R~n上的Schwartz函数,并且满足m,k0.本文给出算子T_(m,k,n).从L~p(R~n)(1≤p∞)到L~q(R~n)有界的一个充分必要条件.此外,我们还证明了算子T_(m,k,n)把L~1(R~n)映到C_0(R~n).