THE ENDPOINT ESTIMATE FOR FOURIER INTEGRAL OPERATORS

Let T_a,φbe a Fourier integral operator defined by the oscillatory integral T_a,φu(x)=1/(2π)ⁿ∫_Rⁿ^e^iφ(x,ξ)a(x,ξ)(u)(ξ)dξ,where a∈S_e,δ^mandφ∈Φ²,satisfying the strong non-degenerate condition.It is shown that if0<(e)≤1,0≤δ<1 and m≤e²-n/2,thenT_α,φis a bounded operator from L^∞(Rⁿ)to BMO(Rⁿ).     

MAXIMAL FUNCTION CHARACTERIZATIONS OF HARDY SPACES ASSOCIATED WITH BOTH NON-NEGATIVE SELF-ADJOINT OPERATORS SATISFYING GAUSSIAN ESTIMATES AND BALL QUASI-BANACH FUNCTION SPACES

Assume that L is a non-negative self-adjoint operator on L²(Rⁿ) with its heat kernels satisfying the so-called Gaussian upper bound estimate and that X is a ball quasiBanach function space on Rⁿ satisfying some mild assumptions.Let H_X,L(Rⁿ) be the Hardy space associated with both X and L,which is defined by the Lusin area function related to the semigroup generated by L.In this article,the authors establish various maximal function character...     

Fourier transform of anisotropic mixed-norm Hardy spaces

Let a:=(a1,…,an)∈[1,∞)n,p:=(p1,…,pn)∈(0,1]n,Hpa(Rⁿ)be the anisotropic mixed-norm Hardy space associated with adefined via the radial maximal function,and let f belong to the Hardy space Hpa(Rⁿ).In this article,we show that the Fourier transform fcoincides with a continuous function g on?n in the sense of tempered distributions and,moreover,this continuous function g,multiplied by a step function associated with a,can be pointwisely controlled by a constant multiple of the Hardy space norm of f.These proofs are achieved via the known atomic characterization of Hpa(Rⁿ)and the establishment of two uniform estimates on anisotropic mixed-norm atoms.As applications,we also conclude a higher order convergence of the continuous function gat the origin.Finally,a variant of the Hardy-Littlewood inequality in the anisotropic mixed-norm Hardy space setting is also obtained.All these results are a natural generalization of the well-known corresponding conclusions of the classical Hardy spaces Hp(Rⁿ)with p∈0,1],and are even new for isotropic mixed-norm Hardy spaces on∈n.     

SOME CONVERGENCE PROBLEMS REGARDING THE FRACTIONAL SCHR?DINGER PROPAGATOR ON NONCOMPACT MANIFOLDS

Let L be the Laplace-Beltrami operator.On an n-dimensional(n≥ 2),complete,noncompact Riemannian manifold M,we prove that if 0 <α <1,s> α/2 and f ∈ H^s(M),then the fractional Schr?dinger propagator e⁽it|L|^α/2)(f)(x)→f(x) a.e.as t→0.In addition,for when M is a Lie group,the rate of the convergence is also studied.These results are a non-trivial extension of results on Euclidean spaces and compact manifolds.     

本刊英文版Vol.30(2014),No.12论文摘要(英文)北大核心CSCD

Regularity of the Inverse of a Homeomorphism with Finite Inner Distortion Chang Yu GUO Abstract Let f:Ω→f（Ω）Rⁿ be a W^1,1-homeomorphism with L¹-integrable inner distortion.We show that finiteness of min{lip_f（x）,k_f（x）},for every x∈Ω\E,implies that f^-1∈W^1,nand has finite distortion,provided that the exceptional set E hasσ-finite H¹-measure.Moreover,f has finite distortion,differentiable a.e.and the Jacobian J_f>0 a.e.     

Potential space estimates in local Hardy spaces for Green potentials in convex domains

Let Ω be a bounded co.nvex domain in Rn(n≥3) and G(x,y) be the Green function of the Laplace operator -△ on Ω. Let hrp(Ω) = {f ∈ D'(Ω) :(E)F∈hp(Rn), s.t. F|Ω = f}, by the atom characterization of Local Hardy spaces in a bounded Lipschitz domain, the bound of f→(△)2(Gf) for every f ∈ hrp(Ω) is obtained, where n/(n 1)＜p≤1.     

A Universal Inequality for Stability of Coarse Lipschitz Embeddings

Let X and Y be two pointed metric spaces.In this article,we give a generalization of the Cheng-Dong-Zhang theorem for coarse Lipschitz embeddings as follows:If f:X→Y is a standard coarse Lipschitz embedding,then for each x^*∈Lip₀(X) there exist α,γ> 0 depending only on f and Q_x*G Lip₀(Y) with ‖Q_x*‖_Lip ≤α‖x^*‖_Lip such that ■,for all x∈X.Coarse stability for a pair of metric spaces is studied.This can be consi...     

Riesz Transforms Associated with Schrdinger Operators Acting on Weighted Hardy Spaces

Let L =-? + V be a Schrdinger operator acting on L2(Rn), n ≥ 1, where V ≡ 0 is a nonnegative locally integrable function on Rn. In this article, we will intropduce weighted Hardy spaces H L(w) associated with L by means of the square function and then study their atomic decomposition theory. We will also show that the Riesz transform ?L-1/2associated with L is bounded from our new space Hp L(w) to the classical weighted Hardy space Hp(w) when n/(n +1) p 1 and w ∈ A1∩ RH(2/p)′.     

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...     

Boundedness of Commutators Generated by Campanato-type Functions and Riesz Transforms Associated with Schrdinger Operators

Let■=-△+V be a Schrdinger operator on R~n,n3,where △is the Laplacian on R~n and V≠0 is a nonnegative function satisfying the reverse Holder's inequality.Let[b,T]be the commutator generated by the Campanatotype function b∈■ and the Riesz transform associated with Schrdinger operator T=▽(-△+V)~(-1/2).In the paper,we establish the boundedness of[b,T]on Lebesgue spaces and Campanato-type spaces.     

Weighted Norm Inequalities for Multipliers and Littlewood-Paley Functions on Local Fields

Let K be a local field, w(x) be a A_p-weight on K (1≤p≤∞). We say that the measurable function m(x) is a multiplier on L~p(K,w), if (m)~v ∈L~p(K,w) for all f∈L~p(K,w) and there is a constant c>0,independent of f such that ‖(m     

THE WELL-POSEDNESS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN COMPLEX BANACH SPACES

Let X be a complex Banach space and let B and C be two closed linear operators on X satisfying the condition D(B) ? D(C),and let d ∈ L¹(R₊) and 0 ≤β <α ≤2.We characterize the well-posedness of the fractional integro-differential equations D^αu(t)+CD^βu(t)=Bu(t)+∫_-∞^td(t-s)Bu(s)ds+f(t),(0≤t ≤2π) on periodic Lebesgue-Bochner spaces L^p(T;X) and periodic Besov spaces B_p,q^s(T;X).     

GABOR ANALYSIS OF THE SPACES M(p,q,w)(R~d) AND S(p,q,r,w,ω)(R~d)

Let g be a non-zero rapidly decreasing function and w be a weight function. In this article in analog to modulation space, we define the space M(p, q, w)(Rd) to be the subspace of tempered distributions f ∈ S′(Rd) such that the Gabor transform Vg(f) of f is in the weighted Lorentz space L(p, q, wdμ) (R2d). We endow this space with a suitable norm and show that it becomes a Banach space and invariant under time frequence shifts for 1 ≤ p, q ≤ ∞. We also investigate the embeddings between these spaces and the...     

ON CONVERGENCE OF PAL-TYPE INTERPOLATION POLYNOMIALS

Let {x_k~*}_(k=1)~(n-1) be the zeros of the (n-1) -th Legendre polynomial p_(n-1)(x) and {x_k}_(a=1)~n be the zeros of the polynomial w(x)= (1-x2~)p_(n-1)~1(x). By the theory of the Pal interpolation, for afunction f ∈ C_([-1,1])~1, there exists a unique polynomial Q_n(f, x) of degree 2n-1 satisfying conditions Q_n(f, x_k)=f(x_k), Q'_n(f, x_k~*)=f'(x_k~*), where k=1, 2, …, n and x_n~*=-1. The main result of this paper is that if f ∈ C_([-1,1])~r, thenf(x)-Q_n(f, x)=O(1)W(x)w(f~(r), 1/n)n~((1/2)-r), -1≤x≤1.Hence, if f ∈ C_[-1,1])~1, then Q_n(f, x) converges to the function f(x)uniformly on the interval [-1, 1].     

GABOR ANALYSIS OF THE SPACES M（p,q,w）（R~d） AND S（p,q,r,w,ω）（R~d）

Let g be a non-zero rapidly decreasing function and w be a weight function. In this article in analog to modulation space, we define the space M(p, q, w)(Rd) to be the subspace of tempered distributions f ∈ S′(Rd) such that the Gabor transform Vg(f) of f is in the weighted Lorentz space L(p, q, wdμ) (R2d). We endow this space with a suitable norm and show that it becomes a Banach space and invariant under time frequence shifts for 1 ≤ p, q ≤∞. We also investigate the embeddings between these spaces and the dual space of M(p, q, w)(Rd). Later we define the space S(p, q, r, w, ω)(Rd) for 1 < p < ∞, 1 ≤ q ≤∞. We endow it with a sum norm and show that it becomes a Banach convolution algebra. We also discuss some properties of S(p, q, r, w, ω)(Rd). At the end of this article, we characterize the multipliers of the spaces M(p, q, w)(Rd) and S(p, q, r, w, ω)(Rd).     

Solutions of Equations Involving Analytic Functions in a Banach Algebra

By the properties of univalent analytic functions,we have discussed the exi-stence and uniqueness of eqation f(x)=a in a Banach algebra.We have the follo-wing fundamental lemmas. Lemma 1. Let A be a Banach algebra with identity e,W,U be two opensubsets of C, U W,f be an analytic function in W, and univalent in U, V=f(U),a∈A.If σ(a) V,then there exists unique x∈A such that σ(x) U {λ∈W|f(λ) V}and f(x)=a     

符号混合控制问题的某些NP-完全性结果

Let G =(V, E) be a simple graph with vertex set V and edge set E. A signed mixed dominating function of G is a function f:V∪E→ {-1, 1} such that ∑_(y∈N_m(x)∪{x})f(y)≥ 1for every element x∈V∪E, where N_m(x) is the set of elements of V∪E adjacent or incident to x. The weight of f is w(f) =∑_(x∈V∪E)f(x). The signed mixed domination problem is to find a minimum-weight signed mixed dominating function of a graph. In this paper we study the computational complexity of signed mixed domination problem. We prove that the signed mixed domination problem is NP-complete for bipartite graphs, chordal graphs, even for planar bipartite graphs.     

Morrey smoothness spaces: A new approach

In the recent years, the so-called Morrey smoothness spaces attracted a lot of interest. They can (also) be understood as generalisations of the classical spaces A_p,q^s(Rⁿ) with A ∈ {B, F } in Rⁿ, where the parameters satisfy s ∈ R(smoothness), 0 < p ∞(integrability) and 0 < q ∞(summability). In the case of Morrey smoothness spaces, additional parameters are involved. In our opinion, among the various approaches at least two scales enjoy special a...     

The Boundedness for Commutators of Multipliers

Let[b,T]be the commutator generated by a Lipschitz function b ∈ Lip(β)(0<β<1)and multiplierT.The authors studied the boundedness of[b,T]on the Lebesgue spaces and Hardy spaces.     

SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION

This paper is a continuation of recent work by Guo-Xiang-Zheng [10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Riviere equation △²u=△(V▽u)+div(w▽u)+(▽ω+F)·▽u+f in B⁴,under the smallest regularity assumptions of V,w,ω,F,where f belongs to some Morrey spaces.This work was motivated by many geometrical problems such as the flow of biharmonic mappings.Our results deepens the L^p type regularity theory of [10...