Estimate of fourier transforms with respect to the system of generalized eigenfunctions of the Schrödinger operator with Stummel-type potential

Suppose thatА is a nonnegative self-adjoint extension to { } of the formal differential operator−Δu+q(x)u with potentialq(x) satisfying the condition {

Boundedness of Marcinkiewicz integral related to multilinear commutators with variable kernels on Hardy spaces

Let→b=(b1,b2,…,bm),bi∈∧βi(Rn),1≤I≤m,βi＞0,m∑I=1βi=β,0＜β＜1,μΩ→b(f)(x)=(∫∞0|F→b,t(f)(x)|2dt/t3)1/2,F→b,t(f)(x)=∫|x-y|≤t Ω(x,x-y)/|x-y|n-1 mΠi=1[bi(x)-bi(y)dy.We consider the boundedness of μΩ,→b on Hardy type space Hp→b(Rn).     

A note on regularity for a class of quasilinear elliptic equations

The following regularity of weak solutions of a class of elliptic equations of the form are investigated.     

Averaging technique for the oscillation of second order damped elliptic equations

By using the averaging technique, we establish some oscillation theorems for the second order damped elliptic differential equation N↑∑↓i,j=1 Di[AIY（x）Djy]＋N↑∑↓i=1 bi（x）Diy＋c（x）f（y）=0 which extend and improve some known results in the literature.     

Bifurcation problems for a class of degenerate quasilinear elliptic equations

In this paper we consider the bifurcation problem -div A（x, u）=λa（x）｜u｜^p-2u＋f（x,u,λ） in Ω with p 〉 1.Under some proper assumptions on A（x,ξ）,a（x） and f（x,u,λ）,we show that the existence of an unbounded branch of positive solutions bifurcating Irom the principal eigenvalue of the problem --div A（x, u）=λa（x）｜u｜^p-2u.     

Oscillation Theorems for Nonlinear Second Order Elliptic Equations

Some oscillation theorems are given for the nonlinear second order elliptic equationsum from i,j=1 to N D_i[a_(ij)(x)Ψ(y)||▽y||~(p-2)D_(jy)] c(x)f(y)=0.The results are extensions of modified Riccati techniques and include recent results of Usami.     

On the riemann zeta-function and the divisor problem II

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If E ^*(t)=E(t)-2πΔ^*(t/2π) with , then we obtain

On the decay rate of (p, A)-lacunary series

A power series with radius of convergence equal 1 is called a (p,A)-lacunary one if n_k ≥ Ak^p, A > 0, 1 < p < ∞. It is proved that if 1 < p < 2 and f(x) is a (p,A)-lacunary series that satisfies the condition

Oscillation of second order semilinear elliptic equations with damping

We obtain some new Kamenev-type oscillation theorems for the second order semilinear elliptic differential equation with damping N ∑i,j=1Di[aij（x）Djy]＋N∑i=1bi（x）Diy＋c（x）f（y）=0 under quite general assumptions. These results are extensions of the recent results of Sun [Sun, Y. G.： New Kamenev-type oscillation criteria of second order nonlinear differential equations with damping. J. Math. Anal. Appl., 291, 341-351 （2004）] in a natural way. In particular, we do not impose any additional conditions on the damped functions bi （x） except the continuity. Several examples are given to illustrate the main results.     

Philos-type oscillation theorems for second order damped elliptic equations

Using general means, we establish several new Philos-type oscillation theorems for the second order damped elliptic differential equation
Σi,j=1 N Di[aij（x）Djy]＋Σi=1 N bi（x）Diy＋c（x）f（y）=0
under quite general assumptions. The obtained results are extensions of the well-known oscillation results due to Kamenev, Philos, Yan for second order linear ordinary differential equations and improve recent results of Xu, Jia and Ma.     

Uniform estimates of positive solutions to quasi-linear differential equations

One considers the differential inequality

多线性分数次积分算子交换子的有界性

Ibαf ( x) =∫R ∏mj=1( bj( x) - bj( y) ) 1| x - y| n-αf ( y) dyare considered.The following priori estimates are proved.For 1 01Φ1t| {y∈Rn:| Ibαf( y) | >t}| 1q ≤csupt>01Φ1t| {y∈Rn:ML( log L) 1r ,α(‖b‖f ) ( y) >t}| 1q,where‖b‖=∏mj=1‖bj‖Oscexp Lrj,Φ( t) =t( 1 + log+t) 1r,1r =1r1+ ...+ 1rm,ML(…     

On a theorem of existence for scaling problems

We study the question of the existence of the global minimum of the functional

Singular integrals in the Cesàro sense

The existence of the singular integral ∫K(x, y)f(y)dy associated to a Calderón-Zygmund kernel where the integral is understood in the principal value sense TF(x)=lim_ε→0+∫_|x−y|>εK(x, y)f(y)dy has been well studied. In this paper we study the existence of the above integral in the Cesàro-α sense. More precisely, we study the existence of

Periodic Solutions for a Class of Forced Liénard-type Equations

Abstract   By applying the topological degree theory, we establish some sufficient conditions for the existence on T-periodic solutions for the Liénard-type equation

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.     

An end point estimate for maximal commutators

In this paper we prove that the maximal commutator of singular integral operator [b, T]^* satisfies the inequality:

Boundedness of a class of super singular integral operators and the associated commutators

In this paper we give the (Lα p, Lp) boundedness of the maximal operator of a class of super singular integrals defined bywhich improves and extends the known result. Moreover, by applying an off-Diagonal T1 Theorem, we also obtain the (Lp, Lq) boundedness of the commutator defined by