A note on the multilinear singular integral operators

Lp(Rn) boundedness is considered for the multilinear singular integral operator defined by TAf(x) = ∫Rn Ω(x - y)/|x - y|n 1 (A(x) - A(y) - (△)A(y)(x - y))f(y)dy,where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one. A has derivatives of order one in BMO(Rn). We give a smoothness condition which is fairly weaker than that Ω∈ Lipα(Sn-1) (0 ＜α≤ 1) and implies the Lp(Rn) (1 ＜ p ＜ oo) boundedness for the operator TA. Some endpoint estimates are also established.     

An endpoint estimate for maximal multilinear singular integral operators

A weak type endpoint estimate for the maximal multilinear singular integral operator T*Af(x)=supε＞0|(f)(x-y)＞ε (Ω(x-y)/(|x-y|(n 1)))(A(x)-A(y)-▽A(y)(x-y))f(y)dy| is established, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, and A has derivatives of order one in BMO(Rn). A regularity condition on Ω which implies an LlogL type estimate of T*A is given.     

A BMO estimate for the multilinear singular integral operator

The behavior on the space L∞((R)n) for the multilinear singular integral operator defined by TAf(x)=∫Rn Ω(x-y)/|x-y|n 1(A(x)-A(y)-(△)A(y)(x-y))f(y)dy is considered, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, A has derivatives of order one in BMO((R)n). It is proved that if Ω satisfies some minimum size condition and an L1-Dini type regularity condition, then for f ∈ L∞((R)n), TAf is either infinite almost everywhere or finite almost everywhere, and in the latter case, TAf ∈ BMO((R)n).     

L^2（R^n） Boundedness for a Class of Multilinear Singular Integral Operators

The L^2(R^n) boundedness for the multilinear singular integral operators defined by TAf(x)=∫R^nΩ（x-y）/|x-y|^n 1(A(x)-A(y)-△↓A(y)(x-y))f(y)dy is considered,where Ω is homogeneous of degree zero,integrable on the unit sphere and has vanishing moment of order one,A has derivatives of order one in BMO(R^n） boundedness for the multilinear operator TA is given.     

A LIPSCHITZ ESTIMATE FOR MULTILINEAR OSCILLATORY SINGULAR INTEGRALS WITH ROUGH KERNELS

In this paper, for the multilinear oscillatory singular integral operators TA1,A2,..,Ar defined by (x) where P(x,y) is a nontrivial and real-valued polynomial defined on Rn × Rn, Ω(x) is homogeneous of degree zero on Rn, As(x) has derivatives of order ms in ∧βs (0 ＜βs ＜1),Rms 1 (As; x, y) denotes the (ms 1)-st remainder of the Taylor series of As at x expended about y(s=1,2,…,r),M=∑rs=1ms, the author proves that if 0<β=∑rs=1βs<1, and Ω∈ Lq(Sn-1) for some q ＞ 1/(1 - β), then for any p ∈ (1, ∞), and some appropriate 0 ＜ β ＜ 1, TA1,A2,…,Ar is bounded on Lp(Rn).     

Liouville type theorems for Schrdinger systems

We study positive solutions to the following higher order Schr¨odinger system with Dirichlet boundary conditions on a half space:(-△)α2 u(x)=uβ1(x)vγ1(x),in Rn+,(-)α2 v(x)=uβ2(x)vγ2(x),in Rn+,u=uxn==α2-1uxnα2-1=0,onRn+,v=vxn==α2-1vxnα2-1=0,onRn+,(0.1)whereαis any even number between 0 and n.This PDE system is closely related to the integral system u(x)=Rn+G(x,y)uβ1(y)vγ1(y)dy,v(x)=Rn+G(x,y)uβ2(y)vγ2(y)dy,(0.2)where G is the corresponding Green’s function on the half space.More precisely,we show that every solution to(0.2)satisfies(0.1),and we believe that the converse is also true.We establish a Liouville type theorem—the non-existence of positive solutions to(0.2)under a very weak condition that u and v are only locally integrable.Some new ideas are involved in the proof,which can be applied to a system of more equations.     

Liouville type theorems for a system of integral equations on upper half space

In this paper,we consider the following system of integral equations on upper half space {u(x) = ∫Rn + (1/|x-y|n-α-1/|-y|n-α) λ1up1(y) + μ1vp2(y) + β1up3(y)vp4(y) dy;v(x) = ∫Rn + (1/|x-y|n-α-1/|-y|n-α)(λ2uq1(y) + μ2vq2(y) + β2uq3(y)vq4(y) dy,where Rn + = {x =(x1,x2,...,xn) ∈ Rn|xn 0}, =(x1,x2,...,xn-1,-xn) is the reflection of the point x about the hyperplane xn= 0,0 α n,λi,μi,βi≥ 0(i = 1,2) are constants,pi≥ 0 and qi≥ 0(i = 1,2,3,4).We prove the nonexistence of positive solutions to the above system with critical and subcritical exponents via moving sphere method.     

Global Topological Linearization with Unbounded Nonlinear Term

§ 1.Statementof Theorem　 Consider the systemx′=Ax + f (x) ,y′=By +φ(x) +ψ(y) ,(1 )where x∈ Rn1,y∈ Rn2 ,f,φ andψ are locally Lipschitzian.If x is in Rn we denote itsEnclidend norm by| x| and if A is an n×n matrix we denote its operator norm by| A| .Let Reλ(A) be the real partof eigenvalues of A.　 Suppose that Reλ(A) <0 and Reλ(B) >0 .Without loss of generality,we may assamethatd| x| 2dtx′=Ax≤ -α| x| 2 , (2 )| e- Bt|≤ k . e-βt　 (t≥ 0 ) , (3 )whereα,β and k are all…     

THE ALEKSANDROV PROBLEM FOR UNIT DISTANCE PRESERVING MAPPING

1 IntroductionLet X and Y be two real metric spaces. A mappillg f: X ~ Y is called an isometryj ifd(f(x), f(y)) = d(x, y) for all x, y E X.A mapping f: X - Y satisfies the distance one preserving property (DOPP) if f for allx, y E X with d(x, y) = 1 it follows that d(f(x), f(y)) = 1.A mapping f: X ~ Y satisfies the strong distance one preserving property (SDOPP) ifffor all x, y E X with d(x, y) = 1 it follows that d(f(x), f(y)) = 1 and conversely.Problem(P) Let f: X - Y be a mappin…     

Symmetry and nonexistence of positive solutions to an integral system with weighted functions

Consider the system of integral equations with weighted functions in Rn,{u(x) =∫Rn|x-y|α-nQ(y)v(y)qdy1,v(x)=∫Rn|x-y|α-nK(y)u(y)pdy,where 0 < α < n,1/(p+1) + 1/(q+1)≥(n-α)/n1,α/(n-α) < p1q < ∞1,Q(x) and K(x) satisfy some suitable conditions.It is shown that every positive regular solution(u(x)1,v(x)) is symmetric about some plane by developing the moving plane method in an integral form.Moreover,regularity of the solution is studied.Finally,the nonexistence of positive solutions to the system in the case 0 <...     

位势型算子的加权不等式

Let Ф be a non-negative locally integrable function on R^n and satisfy some weak growth conditions, define the potential type operator TФ by TФf（x）=∫R^n Ф（x-y）f（y）dy. The aim of this paper is to give several strong type and weak type weighted norm inequalities for the potential type operator TФ.     

Evans functions and bifurcations of standing wave solutions in delayed synaptically coupled neuronal networks

Consider the following nonlinear singularly perturbed system of integral differential equations &amp;\frac{\partial u}{\partial t}+f(u)+w\\ =&amp;(\alpha-au)\int^{\infty}_0\xi(c)\left[\int_{\mathbb R}K(x-y) H\left(u\left(y,t-\frac1c|x-y|\right)-\theta\right){\rm d}y\right]{\rm d}c\\ &amp;+(\beta-bu)\int^{\infty}_0\eta(\tau)\left[\int_{\mathbb R}W(x-y)H\big(u(y,t-\tau)-\Theta\big){\rm d}y\right]{\rm d}\tau,\\ &amp;\frac{\partial w}{\partial t}=\varepsilon[g(u)-w], and the scalar integral differential equation &amp;\frac{\partial u}{\partial t}+f(u)\\ =&amp;(\alpha-a u)\int^{\infty}_0\xi(c)\left[\int_{\mathbb R}K(x-y) H\left(u\left(y,t-\frac1c|x-y|\right)-\theta\right){\rm d}y\right]{\rm d}c\\ &amp;+(\beta-bu)\int^{\infty}_0\eta(\tau)\left[\int_{\mathbb R}W(x-y)H\big(u(y,t-\tau)-\Theta\big){\rm d}y\right]{\rm d}\tau. There exist standing wave solutions to the nonlinear system. Similarly, there exist standing wave solutions to the scalar equation. The author constructs Evans functions to establish stability of the standing wave solutions of the scalar equation and to establish bifurcations of the standing wave solutions of the nonlinear system.     

Convergence of Solutions of General Dispersive Equations Along Curve

In this paper, the authors give the local L~2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R~+→R satisfies some suitable conditions and φ((-?)~(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L~2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper.     

ON A MULTILINEAR OSCILLATORY SINGULAR INTEGRAL OPERATOR （I）

ＯＮＡＭＵＬＴＩＬＩＮＥＡＲＯＳＣＩＬＬＡＴＯＲＹＳＩＮＧＵＬＡＲＩＮＴＥＧＲＡＬＯＰＥＲＡＴＯＲ（Ｉ）ＣＨＥＮＷＥＮＧＵＨＵＧＵＯＥＮＬＵＳＨＡＮＺＨＥＮＭａｎｕｓｃｒｉｐｔｒｅｃｅｉｖｅｄＯｃｔｏｂｅｒ１８，１９９４．ＲｅｖｉｓｅｄＤｅｃｅ...     

双线性分数次积分算子交换子在Morrey空间上有界的充分必要条件

In this paper,we obtain that b∈ BMO(R~n) if and only if the commutator[b,I_α]is bounded from the Morrey spaces L~(p_1,λ_1)(R~n)×L~(p_2,λ_2)(R~n) to L~(q,λ)(R~n),for some appropriate indices p,q,λ,μ.Also we show that b ∈ Lip_β(R~n) if and only if the commutator[b,I_α]is bounded from the Morrey spaces L~(p_1,λ_1)(R~n)×L~(p_2,λ_2)(R~n) to L~(q,λ)(R~n),for some appropriate indices p,q,λ,μ.     

Weighted Compact Commutator of Bilinear Fourier Multiplier Operator

Let T_σ be the bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that sup κ∈Z∥σ_κ∥W~s(R~(2n)) ∞ for some s ∈ (n, 2n]. In this paper, it is proved that the commutator generated by T_σ and CMO(R~n) functions is a compact operator from L~(p1)(R~n, w_1) × L~(p2)(R~n, w_2) to L~p(R~n, ν_w) for appropriate indices p_1, p_2, p ∈ (1, ∞) with1 p=1/ p_1 +1/ p_2 and weights w_1, w_2 such that w = (w_1, w_2) ∈ A_(p/t)(R~(2n)).     

沿超曲面的强奇异积分算子在调幅空间上的有界性

设■该文主要讨论了上述奇异积分算子在广义的调幅空间上的有界性,其中粗糙核Ω∈L~1(S~(n-2))h(y)为有界的径向函数,而γ(y)是满足一定条件的超曲面.     

Rough bilinear fractional integrals with variable kernels

We study the rough bilinear fractional integral

Existence of semiclassical states for a coupled Schrödinger system with potentials and nonlocal nonlinearities

In this paper, we first study a Schrödinger system with nonlocal coupling nonlinearities of Hartree type $$\left\{\begin{array}{ll} -\varepsilon^{2}\Delta u +V_1(x)u = \left ( \int \limits_{\mathbb{R}^{3}} \frac{u^{2}}{|x-y|}{\rm d}y \right)u\,+\, {\beta} \left ( \int \limits_{\mathbb{R}^{3}} \frac{v^{2}}{|x-y|}{\rm d} y \right)u,\\ -\varepsilon^{2} \Delta v +V_2(x)v = \left(\int \limits_{\mathbb{R}^{3}} \frac{v^{2}}{|x-y|}{\rm d}y \right)v \,+ \, {\beta} \left ( \int \limits_{\mathbb{R}^{3}} \frac{u^{2}}{|x-y|}{\rm d}y \right)v. \end{array}\right.$$ Using variational methods, we prove the existence of purely vector ground state solutions for the Schrödinger system if the parameter ${\varepsilon}$ is small enough. Secondly, we also establish some existence results for the coupled Schrödinger system with critical exponents.     

Heat Kernel Upper Estimates for Symmetric Jump Processes with Small Jumps of High Intensity

We consider the following non-local operator