BOUNDEDNESS OF VECTOR-VALUED CALDERON-ZYGMUND OPERATORS ON HERZ SPACES WITH NON-DOUBLING MEASURES

In the paper we obtain vector-valued inequalities for Calderon-Zygmund operator,simply CZO On Herz space and weak Herz space.In particular,we obtain vector-valued inequalities for CZO on Lq(Rd,│x│αdμ)space,with 1＜q＜∞,-n＜α＜n(q-1),and on L1,∞(Rd,│x│αdμ)space,with -n＜α＜0.     

Continuous embeddings of Besov-Morrey function spaces

We study embeddings of spaces of Besov-Morrey type, M Bp1,q1s1,r1(Rd ) → M Bp2 ,q2s2 ,r2 (R d ), and obtain necessary and sufficient conditions for this. Moreover, we can also characterise the special weighted situation Bp1 ,r1s1 (R d , w) → M Bp2 ,q2s2 ,r2 (Rd ) for a Muckenhoupt A ∞ weight w, with wα(x) = |x|α , α -d1, as a typical example.     

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).     

Vector-valued Inequalities for Commutators of Singular Integrals on Herz Spaces with Variable Exponents

Based on the theory of variable exponents and BMO norms,we prove the vector-valued inequalities for commutators of singular integrals on both homogeneous and inhomogeneons Herz spaces where the two main indices are variable exponents.Furthermore,we show that a wide class of commutators generated by BMO functions and sublinear operators satisfy vector-valued inequalities.     

ON THE DECAY AND SCATTERING FOR THE KLEIN-GORDON-HARTREE EQUATION WITH RADIAL DATA

In this paper,we study the decay estimate and scattering theory for the Klein-Gordon-Hartree equation with radial data in space dimension d≥3.By means of a compactness strategy and two Morawetz-type estimates which come from the linear and nonlinear parts of the equation,respectively,we obtain the corresponding theory for energy subcritical and critical cases.The exponent range of the decay estimates is extended to 0<γ≤4 and γ相似文献   

Endpoint Estimates for Hardy Operator＇s Conjugate Operator with Power Weight on n-Dimensional Space

In this paper, we establish two integral inequalities for Hardy operator＇s conjugate operator at the endpoint on n-dimensional space. The operator Hn is bounded from Lxα1 （Gn） to Lxβq （Gn） with the bound explicitly worked out and the similar result holds for Hn＊.     

Lq Inequalities and Operator Preserving Inequalities

Let Pn be the class of polynomials of degree at most nRather and Shah [15]proved that if P∈Pnand P(z)≠0 in|z| 1, then for every R 0 and 0≤q ∞,|B[ P( Rz)]|q ≤|RnB[zn] + λ0 |q|1 + zn|q|P(z)|q,where B is a Bn-operator.In this paper, we prove some generalization of this result which in particular yields some known polynomial inequalities as specialWe also consider an operator Dαwhich maps a polynomial P(z) into DαP(z) := n P(z) +(α-z) P′(z) and obtain extensions and generalizations of a number of well-known Lq inequalities     

局部紧Vilenkin群上Herz空间的一个分解定理

In this paper, by applying a vector-valued inequality we obtained a decomposition theorem on Herz spaces over locally compact Vilenkin groups with new range 0 < q≤1.     

The Gleason''''s problem on mixed norm spaces in convex domains

Let Ω be a bounded convex domain with C2 boundary in C2 and for given 0 < p, q ≤∞ and normal weight function (r) let Hp,q, be the mixed norm space on Ω. In this paper we prove that the Gleason's problem (Ω, a, Hp,q,) is solvable for any fixed point a ∈ Ω. While solving the Gleason's problem we obtain the boundedness of certain integral operator on Hp,q,.     

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...     

Boundedness of an oscillating multiplier on Triebel-Lizorkin spaces

Abstract In this paper, we study Triebel-Lizorkin space estimates for an oscillating multiplier mΩ,α,β. This operator was initially studied by Wainger and by Fefferman-Stein in the Lebesgue spaces. We obtain the boundedness results on the Triebel-Lizorkin space Fpα,q（R^n） for different p, q.     

A Weighted Trudinger–Moser Inequality on R^N and Its Application to Grushin Operator

Let x=(x',x')with x'∈Rk and x'∈R^N-k andΩbe a x'-symmetric and bounded domain in R^N(N≥2).We show that if 0≤a≤k-2,then there exists a positive constant C>0 such that for all x'-symmetric function u∈C0^∞(Ω)with∫Ω|■u(x)|^N-a|x'|^-adx≤1,the following uniform inequality holds1/∫Ω|x|^-adx∫Ωe^βa|u|N-a/N-a-1|x'|^-adx≤C,whereβa=(N-a)(2πN/2Γ(k-a/2)Γ(k/2)/Γ(k/2)r(N-a/2))1/N-a-1.Furthermore,βa can not be replaced by any greater number.As the application,we obtain some weighted Trudinger–Moser inequalities for x-symmetric function on Grushin space.     

粗糙平方函数及极大算子的弱型估计

§ 1　 PreliminariesWe considerψ( x)∈ L1 ( Rn) satisfying the mean valuezero,i.e.∫Rnψdx=0 ,and definethe square function g( f) on Rnbyg( f) ( x) =( k|ψk* f|2 ) 1 2 ( x)for f∈ S( Rn) ,the Schwartz space,whereψk( x) =ψ2 k( x) .　　 Whenψ has some smooth property,one can obtain the weak type estimate by viewingthe square function g( f) as the vector-valued singularintegrals,which the readercan referto [1 ,2 ] .As for the results aboutthe Lp-estimates,see [3,4 ] .In this paper,we sha…     

L~p INEQUALITIES AND ADMISSIBLE OPERATOR FOR POLYNOMIALS

Let p(z) be a polynomial of degree at most n. In this paper we obtain some new results about the dependence of p(Rz)-βp(rz) + α (R+1/r+1)n-|β | p(rz) s on p(z) s for every α, β∈ C with |α|≤ 1, |β | ≤ 1, R > r 1, and s > 0. Our results not only generalize some well known inequalities, but also are variety of interesting results deduced from them by a fairly uniform procedure.     

Finite time extinction of super-Brownian motions with deterministic catalyst

In this paper we consider a super-Brownian motion X with branching mechanism k(x)zα, where k(x) > 0 is a bounded Holder continuous function on Rd and infx∈Rd k(x) = 0. We prove that if k(x) ≥ //x// -l(0 ≤l < ∞) for sufficiently large x, then X has compact support property, and for dimension d = 1, if k(x) ≥exp(-l‖x‖)(0≤l < ∞) for sufficiently large x, then X also has compact support property. The maximal order of k(x) for finite time extinction is different between d = 1, d = 2 and d ≥ 3: it is O(‖x‖-(α+1)) in one dimension, O(‖x‖-2(log‖x‖)-(α+1) ) in two dimensions, and O(‖x‖2) in higher dimensions. These growth orders also turn out to be the maximum order for the nonexistence of a positive solution for 1/2Δu =k(x)uα.     

Sharp Estimates of p-Adic Hardy and Hardy-Littlewood-Pólya Operators

In this paper we get the sharp estimates of the p-adic Hardy and Hardy-Littlewood-Pólya operators on Lq(|x|αpdx). Also, we prove that the commutators generated by the p-adic Hardy operators(Hardy-Littlewood-Pólya operators) and the central BMO functions are bounded on Lq(|x|αpdx), more generally, on Herz spaces.     

Poincaré and Sobolev Inequalities for Vector Fields Satisfying Hrmander's Condition in Variable Exponent Sobolev Spaces

In this paper,we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces.The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type.We obtain the first order Poincare inequalities for vector fields satisfying Hrmander's condition in variable non-isotropic Sobolev spaces.We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups.Moreover,we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups.These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p(x)-subLaplacian.Our results are only stated and proved for vector fields satisfying Hrmander's condition,but they also hold for Grushin vector fields as well with obvious modifications.     

STOCHASTIC HEAT EQUATION WITH FRACTIONAL LAPLACIAN AND FRACTIONAL NOISE:EXISTENCE OF THE SOLUTION AND ANALYSIS OF ITS DENSITY

In this paper we study a fractional stochastic heat equation on R~d(d≥1) with additive noise ?/?t u(t,x) = Dα/δu(t,x) + b(u(t,x)) +W~H(t,x) where D α/δ is a nonlocal fractional differential operator and W~H is a Gaussian-colored noise. We show the existence and the uniqueness of the mild solution for this equation. In addition,in the case of space dimension d=1,we prove the existence of the density for this solution and we establish lower and upper Gaussian bounds for the density by Malliavin calculus.     

Weighted inequalities for fractional type operators with some homogeneous kernels

In this paper, we study integral operators of the form Tαf(x)=∫Rn|x-A1y|-α1 ··· |x-Amy|-αmf(y)dy,where Ai are certain invertible matrices, αi 0, 1 ≤ i ≤ m, α1 + ··· + αm = n-α, 0 ≤α n. For 1/q = 1/p-α/n , we obtain the Lp (Rn, wp)-Lq(Rn, wq) boundedness for weights w in A(p, q) satisfying that there exists c 0 such that w(Aix) ≤ cw(x), a.e. x ∈ Rn , 1 ≤ i ≤ m.Moreover, we obtain theappropriate weighted BMO and weak type estimates for certain weights satisfying the above inequality. We also give a Coifman type estimate for these operators.     

Universal inequalities for lower order eigenvalues of self-adjoint operators and the poly-Laplacian

In this paper, we first establish an abstract inequality for lower order eigenvalues of a self-adjoint operator on a Hilbert space which generalizes and extends the recent results of Cheng et al. （Calc. Var. Partial Differential Equations, 38, 409-416 （2010））. Then, making use of it, we obtain some universal inequalities for lower order eigenvalues of the biharmonic operator on manifolds admitting some speciM functions. Moreover, we derive a universal inequality for lower order eigenvalues of the poly-Laplacian with any order on the Euclidean space.