Oscillatory integrals on unit square along surfaces

Let Q ² = [0, 1]² be the unit square in two-dimensional Euclidean space ℝ². We study the L ^p boundedness of the oscillatory integral operator T _α,β defined on the set ℒ(ℝ²⁺ⁿ ) of Schwartz test functions by

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε (ℝⁿ) is discussed from L ^p(ℝⁿ) to L ^q(ℝⁿ), , and from L ^p(ℝⁿ) to Triebel-Lizorkin space . We also obtain the boundedness of generalized Toeplitz operator Θ _α0 ^b from L ^p(ℝⁿ) to L ^q(ℝⁿ), . All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L ^p(ℝⁿ), 1 < p < ∞.     

Homogenization of two-phase metrics and applications

We consider two-phase metrics of the form ϕ(x, ξ) ≔ , where α,β are fixed positive constants and B _α, B _β are disjoint Borel sets whose union is ℝ^N, and prove that they are dense in the class of symmetric Finsler metrics ϕ satisfying

Moments of a statistic caused by random combinations or random matings

Summary Given two sets of sizek, {α ₁...,α _k} and {β ₁...,β _k} there arek! possible combinations of these two , and suppose there is apriori given a number corresponding to the partnership (α ₁,β _j}. The average of the numbers corresponding to is a random variable, and this paper presents the first five moments of the average, and an application in the study of an isolated human population is demonstrated.     

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T _A ,

Boundedness of Some Marcinkiewicz Integral Operators Related to Higher Order Commutators on Hardy Spaces

In this paper, the authors study the boundedness properties of μΩ↑m,b generated by the function b ∈Lipβ（R^n）（0 〈β≤ 1/m） and the Marcinkiewicz integrals operator μΩ. The boundednesses are established on the Hardy type spaces Hb^m^p,n（R^n） and the Herz Hardy type spaces Hbm Kq^α,p（R^b）.     

Oscillating multipliers for some eigenfunction expansions

Let P be a non-negative, self-adjoint differential operator of degree d on ℝⁿ. Assume that the associated Bochner-Riesz kernel s _R ^δ satisfies the estimate, |s _R ^δ (x, y)| ≤ C R^n/d(1+R^1/d|x - y|^-αδ+β)for some fixed constants a>0 and β. We study L^p boundedness of operators of the form m(P), m coming from the symbol class S _p ^−α . We prove that m(P) is bounded on L^P if . We also study multipliers associated to the Hermite operator H on ℝⁿ and the special Hermite operator L on ℂⁿ given by the symbols . As a special case we obtain L^p boundedness of solutions to the Wave equation associated to H and L.     

Regolarità hölderiana delle derivate dei minimi di funzionali con parte principale quadratica

Riassunto In questo lavoro si prova la regolarità h?lderiana delle derivate, fino all'ordinek, dei minimi locali dei funzionali sotto opportune ipotesi suA _ij ^αβ e sug.

---

Summary In this paper we prove h?lder-continuity of the derivates, up to orderk, of local minima of functionals under suitable hypotheses forA _ij ^αβ andg.

     

THE BOUNDEDNESS OF MULTILINEAR COMMUTATORS FROM L^P （R^n） TO TRIEBEL-LIZORKIN SPACES

Let (→b)=(b1,…,bm),bi∈Λβi(Rn),1≤i≤m,0＜βi＜β,0＜β＜1,[(→b),T]f(x)=∫Rn,(b1(x)-b1(y))…(bm(x)-bm(y)))K(x-y)f(y)dy where K is a Calder(o)n-Zygmund kernel.In this paper,we show that[(→b),T] is bounded from Lp (Rn) to Fβ,∞p(Rn),as well as[(→b,Iα)] from Lp(Rn) to Fβ,∞p(Rn),where 1/q=1/p-α/n.     

Mapping properties of certain oscillatory integrals on modulation spaces

Suppose β1 α1 ≥0,β2 α2 ≥ 0 and(k,j) ∈R2. In this paper, we mainly investigate the mapping properties of the operator T_αβf(x,y,z)=∫_Q~2f(x-t,y-s,z-t~ks~j)e~(-2πit-β1_s-β2)t~(-1-α1)s~(-1-α2)dtds on modulation spaces, where Q~2 = [0,1] x [0,1] is the unit square in two dimensions.     

Some Inequalities for Tree Martingales

In this paper we study tree martingales and proved that if 1≤α,β〈∞,1≤p〈∞ then for every predictable tree martingale f=（ft,t∞T）and E[σ^（P）（f）]〈∞,E[S^（P）（f）]〈∞,it holds that ‖（St^（p）（f）,t∈T）‖M^α∞≤Cαβ‖f‖p^αβ,‖（σt^（p）（f）,t∈T）‖M^α,β‖f‖P^αβ,where Cαβ depends only on α and β.     

Two Inequalities for Convex Functions

Let a ₀ < a ₁ < ··· < a _n be positive integers with sums $ {\sum\nolimits_{i = 0}^n {\varepsilon _{i} a_{i} {\left( {\varepsilon _{i} = 0,1} \right)}} } Let a ₀ < a ₁ < ··· < a _n be positive integers with sums distinct. P. Erd?s conjectured that The best known result along this line is that of Chen: Let f be any given convex decreasing function on [A, B] with α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n being real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , Then In this paper, we obtain two generalizations of the above result; each is of special interest in itself. We prove: Theorem 1 Let f and g be two given non-negative convex decreasing functions on [A, B], and α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n , α' ₀, α' ₁, ... , α' _n , β' ₀ , β' ₁ , ... , β' _n be real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , α' ₀ ≤ α' ₁ ≤ ··· ≤ α' _n , Then Theorem 2 Let f be any given convex decreasing function on [A, B] with k ₀, k ₁, ... , k _n being nonnegative real numbers and α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n being real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , Then      

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.     

Solvability of elliptic functional-differential equations with compressions of the arguments in weighted spaces

In the paper, the equation

On a conjectured inequality of Gautschi and Leopardi for Jacobi polynomials

Motivated by work on positive cubature formulae over the spherical surface, Gautschi and Leopardi conjectured that the inequality holds for α,β > − 1 and n ≥ 1, θ ∈ (0, π), where are the Jacobi polynomials of degree n and parameters (α, β). We settle this conjecture in the special cases where .      

Multipliers and Cyclic Vectors in Bloch Type Spaces

In this paper we study the coefficient multipliers, pointwise multipliers and cyclic vectors in the Bloch type spaces B ^α and little Bloch type spaces for 0 < α < ∞. We give several full characterizations of the coefficient multipliers (B ^α ,B ^β ) and for 0 < α, β < ∞ and pointwise multipliers M(B ^α ,B ^β ) and for 1 ≠ α, β ∈ (0,∞). We also obtain some properties of cyclic vectors for Bloch type spaces. Dedicated to Professor Yu Zan HE on the occasion of his 65^th birthday     

非Lipschitz条件下严格伪压缩映射不动点的逼近

Abstract. Without the Lipschitz assumption and boundedness of K in arbitrary Banach spaces,the Ishikawa iteration     

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> boundedness for multilinear fractional integral on product spaces

For 0 < α < mn and nonnegative integers n ≥ 2, m ≥ 1, the multilinear fractional integral is defined by

Solutions to Some Open Problems in Fluid Dynamics

Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0.     

Uniform boundedness of (<Emphasis Type="Italic">C</Emphasis>, 1) means of Jacobi expansions in weighted sup norms. II (Some necessary estimations)

The paper is concerned with bounds for integrals of the type