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数学学报（英文版）
数学学报（英文版）

月刊

1439-8516

010-62551910

100080

北京中关村中科院数学所

数学学报（英文版）/Journal Acta Mathematica SinicaCSCDCSTPCD北大核心SCI
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    Musielak-Orlicz-Lorentz Hardy Spaces:Maximal Function,Finite Atomic,and Littlewood-Paley Characterizations with Applications to Dual Spaces and Summability of Fourier Transforms

    Hongchao JiaDer-Chen ChangFerenc WeiszDachun Yang...
    1-77页
    查看更多>>摘要：Let q ∈(0,∞]and ψ be a Musielak-Orlicz function with uniformly lower type p-ψ ∈(0,∞)and uniformly upper type p+ψ ∈(0,∞).In this article,the authors establish various real-variable characterizations of the Musielak-Orlicz-Lorentz Hardy space Hψ,q((R)n),respectively,in terms of various maximal functions,finite atoms,and various Littlewood-Paley functions.As applications,the authors obtain the dual space of Hψ,q((R)n)and the summability of Fourier transforms from Hψ,q((R)n)to the Musielak-Orlicz-Lorentz space Lψ,q((R)n)when q ∈(0,∞)or from the Musielak-Orlicz Hardy space Hψ((R)n)to Lψ,∞((R)n)in the critical case.These results are new when q ∈(0,∞)and also essentially improve the existing corresponding results(if any)in the case q=∞ via removing the original assumption that ψ is concave.To overcome the essential obstacles caused by both that ψmay not be concave and that the boundedness of the powered Hardy-Littlewood maximal operator on associated spaces of Musielak-Orlicz spaces is still unknown,the authors make full use of the obtained atomic characterization of Hψ,q((R)n),the corresponding results related to weighted Lebesgue spaces,and the subtle relation between Musielak-Orlicz spaces and weighted Lebesgue spaces.
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    A Weighted Maximal L2 Estimate of Operator-valued Bochner-Riesz Means

    Guixiang HongLiyuan Zhang
    78-98页
    查看更多>>摘要：In this paper,we establish a weighted maximal L2 estimate of operator-valued Bochner-Riesz means.The proof is based on noncommutative square function estimates and a sharp weighted noncommutative Hardy-Littlewood maximal inequality.
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    A Quantitative Second Order Sobolev Regularity for(inhomogeneous)Normalized p(·)-Laplace Equations

    Yuqing WangYuan Zhou
    99-121页
    查看更多>>摘要：Let Ω be a domain of(R)n withn ≥ 2 and p(•)be a local Lipschitz funcion in Ω with 1＜p(x)＜∞ in Q.We build up an interior quantitative second order Sobolev regularity for the normalized p(·)-Laplace equation-△Np.)u=0 in Ω as well as the corresponding inhomogeneous equation-△Np(·)u=f in Ω with f E C0(Q).In particular,given any viscosity solution u to △Np(.)u=0 in Q,we prove the following:(ⅰ)in dimension n=2,for any subdomain U(∈)Ω and any β≥ 0,one has|Du|βDu ∈ L2+cδ(U)with a quantitative upper bound,and moreover,the map(x1,x2)→|Du|β(ux1,-ux2)is quasiregular in U in the sense that|D[|Du|β Du]|2 ≤-C det D[|Du|βDu]a.e.in U.(ⅱ)in dimension n ≥ 3,for any subdomain U∈Ω with infU p(x)1 and supu p(x)＜3+2/n-2 one has D2u ∈ L2+δloc(U)with a quantitative upper bound,and also with a pointwise upper bound|D2u|2 ≤-C ∑1≤i＜j≤n[uxixjuxjxi-uxixiuxjxj]a.e.in U.Here constants δ＞0 and C ≥ 1 are independent of u.These extend the related results obtaind by Adamowicz-Hästö[Mappings of finite distortion and PDE with nonstandard growth.Int.Math.Res.Not.IMRN,10,1940-1965(2010)]when n=2 andβ=0.
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    A Remark on Stein-Tomas Type Restriction Theorems

    Xiaochun Li
    122-130页
    查看更多>>摘要：A local Lp estimate is proved by using the σ-uniformity,which is motivated by the study of the Stein-Tomas type restriction theorems and Waring's problem.
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    Restricting Riesz-Logarithmic-Gagliardo-Lipschitz Potentials

    Xinting HuLiguang Liu
    131-148页
    查看更多>>摘要：For s ∈[0,1],b ∈(R)and p ∈[1,∞),let(B)s,bp,∞((R)n)be the logarithmic-Gagliardo-Lipschitz space,which arises as a limiting interpolation space and coincides to the classical Besov space when b=0 and s ∈(0,1).In this paper,the authors study restricting principles of the Riesz potential space(T)((B)s,bp,∞(R)n)into certain Radon-Campanato space.
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    Jump and Variational Inequalities for Singular Integral with Rough Kernel

    Yanping ChenLiu YangMeng Qu
    149-168页
    查看更多>>摘要：In this paper,we consider the jump and variational inequalities of truncated singular integral operator with rough kernel TΩ,β,εf(x)=∫|y|＞εΩ(y)/|y|n-βf(x-y)dy,where the kernel Ω ∈(L(log+L)2)n/n-β(Sn-1)satisfies the vanishing condition and the homogeneous condition of degree 0.This kind of singular integral appears in the approximation of the surface quasi-geostrophic(SQG)equation from the generalized SQG equation.We establish the(Lp,Lq)estimate of the jump and variational inequalities of the families {TΩ,β,ε}ε＞0 for 1/q=1/p-β/n and 0＜β＜1.Moreover,one can get the Lp boundedness of the Calderón-Zygmund operator with the same kernel by letting β → 0+.
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    On Weighted Compactness of Commutators of Bilinear Vector-valued Singular Integral Operators and Applications

    Zhengyang LiLiu LuFanghui LiaoQingying Xue...
    169-190页
    查看更多>>摘要：Let T be a bilinear vector-valued singular integral operator satisfies some mild regularity conditions,which may not fall under the scope of the theory of standard Calderón-Zygmund classes.For any b=(b1,b2)∈(CMO(Rn))2,let[T,bj]ej(j=1,2),[T,b→]α be the commutators in the j-th entry and the iterated commutators of T,respectively.In this paper,for all p0/2＞1,＜p＜∞,and p0 ≤ p1,p2＜∞ with 1/p=1/p1+1/p2,we prove that[T,bj]ej and[T,b]α are weighted compact operators from Lp1(w1)x LP2(w2)to Lp(v(w)),where w=(w1,w2)∈ A(p)/p0 andvw=wp1/p1 wp2/P2.As applications,we obtain the weighted compactness of commutators in the j-th entry and the iterated commutators of several kinds of bilinear Littlewood-Paley square operators with some mild kernel regularity,including bilinear g function,bilinear g*λ function and bilinear Lusin's area integral.In addition,we also get the weighted compactness of commutators in the j-th entry and the iterated commutators of bilinear Fourier multiplier operators,and bilinear square Fourier multiplier operators associated with bilinear g function,bilinear g*λ function and bilinear Lusin's area integral,respectively.
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    Weighted Estimates for Generalised Conical Square Functions and Applications

    The Anh BuiXuan Thinh DuongJi Li
    191-208页
    查看更多>>摘要：Let {At}t＞0 be a family of bounded linear operator on L2(X)where(X,d,μ)is a metric space with metric d and doubling measure μ.Assume that the family {At}t＞0 satisfies suitable off-diagonal estimates from Lp0 to L2 for some p0＜2.This paper aims to prove weighted bound estimates for conical square functions and g-functions associated to the family {At}t＞0.Some applications such as weighted bounds for bilinear estimates associated to certain differential operators are also obtained.
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    Interpolation of Closed Ideals of Bilinear Operators

    Fernando CobosLuz M.Fernández-CabreraAntón Martínez
    209-230页
    查看更多>>摘要：We extend the(outer)measure γ(I)associated to an operator ideal(I)to a measure γ(＞)for bounded bilinear operators.If(T)is surjective and closed,and(⊃)is the class of those bilinear operators such that γ(⊃)(T)=0,we prove that(⊃)coincides with the composition bideal(I)o(B).If(I)satisfies the∑r-condition,we establish a simple necessary and sufficient condition for an interpolated operator by the real method to belong to(⊃).Furthermore,if in addition(I)is symmetric,we prove a formula for the measure γ(⊃)of an operator interpolated by the real method.In particular,results apply to weakly compact operators.
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    Mapping Properties of Fourier Transforms,Revisited

    Dorothee D.HaroskeLeszek SkrzypczakHans Triebel
    231-254页
    查看更多>>摘要：The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished Besov spaces Bsp((R)n)=Bsp,p(Rn),1 ≤ p ≤ ∞,and between Sobolev spaces Hsp((R)n),1＜p＜∞.In contrast to the paper H.Triebel,Mapping properties of Fourier transforms.Z.Anal.Anwend.41(2022),133-152,based mainly on embeddings between related weighted spaces,we rely on wavelet expansions,duality and interpolation of corresponding(unweighted)spaces,and(appropriately extended)Hausdorff-Young inequalities.The degree of compactness will be measured in terms of entropy numbers and approximation numbers,now using the symbiotic relationship to weighted spaces.
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