带有分布导数的反应扩散方程在Rn中全局吸引子的存在性

该文考虑Rⁿ中一类带有加权项V(x)和分布导数的反应扩散方程解的长时间行为,证明了(L²(Rⁿ),L²(Rⁿ)∩L^p(Rⁿ))-全局吸引子的存在性;而且对任意的δ₁∈[0,+∞),该吸引子能够在L²∩L^p+δ₁-拓扑下吸引L²(Rⁿ)中的有界集.     

对称Sobolev函数到加权函数空间的嵌入

证明Sobolev空间W^(1,p)(R(1,p)(Rⁿ)上对称函数到某类加权Ln)上对称函数到某类加权L^p空间存在紧嵌入定理,进而,作为应用,证明在一定条件下,一类非线性项涉临界Sobolev指标的拟线性椭圆方程具有有限能量解的正解.     

THE WEIGHTED KATO SQUARE ROOT PROBLEM OF ELLIPTIC OPERATORS HAVING A BMO ANTI-SYMMETRIC PART

Let n≥ 2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in Rⁿ.In this article,we consider the weighted Kato square root problem for L.More precisely,we prove that the square root L^1/2 satisfies the weighted L^p estimates ■ for any p∈(1,∞) and ω∈ A_p(Rⁿ)(the class of Muckenhoupt weights),and that ■ for any p ∈(1,2+ε) and ■(the class o...     

Dynamical behaviors of non-autonomous fractional FitzHugh-Nagumo system driven by additive noise in unbounded domains

The regularity of random attractors is considered for the non-autonomous fractional stochastic FitzHugh-Nagumo system.We prove that the system has a pullback random attractor that is compact in Hs(Rⁿ)×L²(Rⁿ)and attracts all tempered random sets of L²(Rⁿ)×L²(Rⁿ)in the topology of Hs(Rⁿ)×L²(Rⁿ)with s∈(0,1).By the idea of positive and negative truncations,spectral decomposition in bounded domains,and tail estimates,we achieved the desired results.     

ISOMORPHISMS OF VARIABLE HARDY SPACES ASSOCIATED WITH SCHRÖDINGER OPERATORS

Let L:=-△+V be the Schrodinger operator on Rⁿwith n≥3,where V is a non-negative potential satisfying△^-1(V)∈L^∞(Rⁿ).Let w be an L-harmonic function,determined by V,satisfying that there exists a positive constantδsuch that,for any x∈Rn,0<δ≤w(x)≤1.Assume that p(·):Rⁿ→(0,1]is a variable exponent satisfying the globally log-H?lder continuous condition.In this article,the authors show that the mappings HL^p(·))(Rⁿ)■f■wf∈H^p(·)(Rⁿ)and HL^p(·)(Rⁿ)■f■(-△)^1/2L^-1/2(f)∈H^p(·)(Rⁿ)are isomorphisms between the variable Hardy spaces HL^p(·)(Rⁿ),associated with L,and the variable Hardy spaces H^p(·)(Rⁿ).     

欧氏空间中升序变换半群的格林关系和正则元

设L(Rⁿ)表示n维欧氏空间Rn)表示n维欧氏空间Rⁿ的所有线性变换构成的集合,||α||表示向量α的欧氏长度,由欧氏长度建立起向量间的序关系.令:O+(Rn的所有线性变换构成的集合,||α||表示向量α的欧氏长度,由欧氏长度建立起向量间的序关系.令:O+(Rⁿ)={f∈L(Rn)={f∈L(Rⁿ)|(?)α∈Rn)|(?)α∈Rⁿ,||f(α)||≥||α||},则O+(Rn,||f(α)||≥||α||},则O+(Rⁿ)是欧氏空间Rn)是欧氏空间Rⁿ的所有升序变换构成的集合,其在交换的合成运算下构成一个半群,讨论了O+(Rn的所有升序变换构成的集合,其在交换的合成运算下构成一个半群,讨论了O+(Rⁿ)的格林关系和正则元.     

与对数函数相关的Q型空间的调和延拓问题

本文研究了一类与对数函数相关的n维Q型空间——Q_log,λ^m(Rⁿ).首先给出Q_log,λ^m(Rⁿ)的定义和一些基本性质.进而利用Poisson积分和调和函数空间H_log,λ^m(R₊ⁿ⁺¹),得到了Q_log,λ^m(Rⁿ)的调和延拓,以及H_log,λ^m(R₊ⁿ⁺¹)的边值问题.     

各向异性变指标Hardy-Lorentz空间上的Fourier变换及其应用

设p(·)是Rⁿ上的一个满足全局log-H?lder连续性条件的可测函数,其本性上确界p₊和下确界p_-满足0 -≤p₊≤1.另设q∈(0, 1], A是一个伸缩矩阵, Hp_A⁽p(·)),^q(Rⁿ)表示通过径向主极大函数定义的各向异性变指标Hardy-Lorentz空间.本文利用Hp_A⁽p(·)),^q(Rⁿ)的原子分解,证明了该空间中的元素f的Fourier变换■在缓增分布意义下等于Rⁿ上的一个连续函数F.进一步地,本文得到了上述函数F的一个点态控制,即它被f的Hp_A⁽p(·)),^q(Rⁿ)范数和相关于A的转置矩阵的齐次拟模的乘积点态控制.作为应用,本文还获得了F在原点的高阶收敛性以及H...     

Lp Decay Rate for a Nonlinear Convection Diffusion Reaction Equation in R~n

This paper studies the asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in Rⁿ.Firstly,the global existence and uniqueness of classical solutions for small initial data are established.Then,we obtain the L^p,2≤p≤+∞decay rate of solutions.The approach is based on detailed analysis of the Green function of the linearized equation with the technique of long wave-short wave decomposition and the Fourier analysis.     

变指标弱Herz空间上的变量核分数次积分算子

应用原子分解理论与核函数Ω(x,z)的性质,证明了变量核分数次积分T_Ω,α是从变指标Herz-Hardy空间H■₍q(·))^α,p(Rⁿ)(HKK₍q(·))^α,p(Rⁿ))到变指标弱Herz空间W■₍q(·))^α,p(Rⁿ)(WK₍q(·))^α,p(Rⁿ))上的有界算子,从而拓宽了以往的相关研究结果.     

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).     

本刊英文版Vol.29(2013),No.7论文摘要(英文)

<正>Rough Marcinkiewicz Integrals with Mixed Homogeneity on Product Spaces Feng LIU Huo Xiong WU Abstract In this paper,the parabolic Marcinkiewicz integral operators on the product spaces R^m×Rm×Rⁿ(m,n≥2)are studied.The Ln(m,n≥2)are studied.The L^p-boundedness for such operators are established under rather weak size conditions of the kernels,which essentially improve or extend certain previous results.     

振荡超奇性Hilbert变换的Sobolev有界性

主要研究R~n上沿曲线Γ(t)=(t~(p_1),t~(p_2),…,t~(p_n))的振荡超奇性Hilbert变换H_(n,α,β)=∫_0~1 f(x-Γ(t))e~(it-β)t~(-1-α),在Sobolev空间上的有界性,其中0p_1P_2…P_n,αβ0.证明了对于0γ(nα)/((n+1))(p_1+α),当|1/p-1/2|(β-(n+1)[α-(β+p_1)γ])/(2β)时,H_(n,α,β)是从L_γ~2(R~n))到L~2(R~n)的有界算子.特别地,当β≥(α-γp_1)/(γ+1/(n+1))等时,H_(n,α,β)是从L_γ~2(R~n)到L~2(R~n)的有界算子·     

一类p-Kirchhoff方程基态解的存在性与唯一性

对于下面p-Kirchhoff型泛函■我们证明了约束在流形■上全局极小点或山路型临界点的存在性与唯一性,且这些临界点是某个Gagliardo-Nirenberg不等式的最优化子,特别当p∈(1,2]时,它们在不计平移意义下是唯一的.我们扩展了已有文献中p=2的情形的相关结果.     

The commutator of the Kato square root for second order elliptic operators on ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let L=-div(A▽) be a second order divergence form elliptic operator, and A be an accretive, n×n matrix with bounded measurable complex coefficients in Rⁿ. We obtain the L^p bounds for the commutator generated by the Kato square root √L and a Lipschitz function, which recovers a previous result of Calderón, by a different method. In this work, we develop a new theory for the commutators associated to elliptic operators with Lipschitz function. The theory of the commutator with Lipschitz function is distinguished from the analogous elliptic operator theory.     

向量值哈代-洛伦茨空间

对于1r∞与巴拿哈空间B=L~r(Ω,F,μ),我们研究了欧几里得空间R~n上B-值缓分布构成的哈代-洛伦茨空间H~(p,q)(R~n,B)及哈代-洛伦茨空间之间的内插,其中0p∞和0q≤∞,获得了H~(p,q)(R~n,B)的一系列等价的刻画及其原子分解.若Ω={1},则H~(p,q)(R~n,B)=H~(p,q)(R~n)是经典的情形;若Ω=Z是整数集且μ是Z上的计数测度并且r=2,0p∞及q=∞,则H~(p,q)(R~n,B)=H~(p,∞)(R~n,e~2)转化为Grafakos和He在文[Weak Hardy spaces,Preprint,2014]中讨论的情形.     

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.     

乘积型Morrey空间中乘积Hardy型算子的最佳界

本文得到了乘积Hardy型算子flm在乘积Morrey空间Lq,λ(Rn×…×Rn)和齐次中心Morrey空间Bq,λ(Rn×...× Rn)上的算子范数.基于旋转方法,我们推广了傅尊伟等人的结果(见[Houston J.Math.,2012,38(1):225-244]).     

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.     

高维Hausdorff算子在Hp(Rn)上的有界性

本文主要研究以下形式的Hausdorff算子H_Φf(x)=∫_RⁿΦ(u₁….,u_n)f(u₁x₁,…,u_nx_n)du₁…du_n,其中Φ是Rⁿ上的缓增分布.当n≥2,0Φ在H^p(Rⁿ)上有界当且仅当Φ≡0.进一步,当n≥2,n/n+1Φ有合适定义,那么H_Φ在H^p(Rⁿ)上有界当且仅当Φ是常数.这些结果都表明Hausdorff算子H_Φ在H^p(Rⁿ)上的有界性很复杂.此外,我们将H_Φ转化成卷积型算子,得到H_Φ在Lebesgue空间上有界的一些新的结果.