In this paper,the weak Orlicz space wL Φ is introduced and its applications to the martingale theory are discussed.In particular,a series of martingale inequalities including the maximal function inequality in weak O...In this paper,the weak Orlicz space wL Φ is introduced and its applications to the martingale theory are discussed.In particular,a series of martingale inequalities including the maximal function inequality in weak Orlicz spaces are established;the relationships between these spaces are investigated.Moreover,the boundedness of several sublinear operators from one weak Orlicz space to another is proved;their vector-valued analogues are also considered.展开更多
Let￠ ： Nn x [0,∞） -4 [0, ∞） be a function such that ￠（x,.） is an Orlicz function and ￠（.,t） ∈ Aloc∞（Nn） （the class of local weights introduced by Rychkov）. In this paper, the authors introduce a loca...Let￠ ： Nn x [0,∞） -4 [0, ∞） be a function such that ￠（x,.） is an Orlicz function and ￠（.,t） ∈ Aloc∞（Nn） （the class of local weights introduced by Rychkov）. In this paper, the authors introduce a local Musielak-Orlicz Hardy space h￠（Nn） by the local grand maximal function, and a local BMO-type space bmo￠（Nn） which is further proved to be the dual space of h￠（Nn）. As an application, the authors prove that the class of pointwise multipliers for the local BMO-type space bmo￠（Nn）, characterized by Nakai and Yabuta, is just the dual of LI（Rn） ＋ hФ0 （Rn）, where Ф is an increasing function on （0, co） satisfying some additional growth conditions and Ф0 a Musielak-Orlicz function induced by Ф. Characterizations of h￠（Rn）, including the atoms, the local vertical and the local nontangential maximal functions, are presented. Using the atomic char- acterization, the authors prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of h￠（Rn）, from which, the authors further deduce some criterions for the boundedness on h￠（Rn） of some sublinear operators. Finally, the authors show that the local Riesz transforms and some pseudo-differential operators are bounded on h￠（Rn）.展开更多
Let L be a linear operator in L^2（R^n） and generate an analytic semigroup {e^-tL}t≥0 with kernel satisfying an upper bound estimate of Poisson type, whose decay is measured by θ（L） ∈ （0, ∞）. Let 4） be a pos...Let L be a linear operator in L^2（R^n） and generate an analytic semigroup {e^-tL}t≥0 with kernel satisfying an upper bound estimate of Poisson type, whose decay is measured by θ（L） ∈ （0, ∞）. Let 4） be a positive, continuous and strictly increasing function on （0, ∞）, which is of strictly critical lower type pФ （n/（n ＋ θ（L））, 1]. Denote by HФ, L（R^n） the Orlicz-Hardy space introduced in Jiang, Yang and Zhou＇s paper in 2009. If Ф is additionally of upper type 1 and subadditive, the authors then show that the Littlewood-Paley g-function gL maps HФ, L（R^n） continuously into LФ（R^n） and, moreover, the authors characterize HФ, L（R^n） in terms of the Littlewood-Paley gλ^＊-function with λ ∈ （n（2/pФ ＋ 1）, ∞）. If Ф is further slightly strengthened to be concave, the authors show that a generalized Riesz transform associated with L is bounded from HФ, L（R^n） to the Orlicz space L^Ф（R^n） or the Orlicz-Hardy space HФ （R^n）; moreover, the authors establish a new subtle molecular characterization of HФ, L （R^n） associated with L and, as applications, the authors then show that the corresponding fractional integral L^-γ for certain γ∈ E （0,∞） maps HФ, L（R^n） continuously into HФ, L（R^n）, where Ф satisfies the same properties as Ф and is determined by Ф and λ and also that L has a bounded holomorphic functional calculus in HФ, L（R^n）. All these results are new even when Ф（t） = t^p for all t ∈ （0, ∞） and p ∈ （n/（n ＋ θ（L））, 1].展开更多
Let L be a linear operator in L 2 (? n ) and generate an analytic semigroup {e ?tL }t?0 with kernel satisfying an upper bound of Poisson type, whose decay is measured by θ(L) ∈ (0, ∞). Let ω on (0,∞) be of upper ...Let L be a linear operator in L 2 (? n ) and generate an analytic semigroup {e ?tL }t?0 with kernel satisfying an upper bound of Poisson type, whose decay is measured by θ(L) ∈ (0, ∞). Let ω on (0,∞) be of upper type 1 and of critical lower type p o (ω) ? (n/(n+θ(L)),1] and ρ(t) = t t1/ω ?1(t ?1) for t ∈ (0,∞). We introduce the Orlicz-Hardy space H ω, L (? n ) and the BMO-type space BMO ρ, L (? n ) and establish the John-Nirenberg inequality for BMO ρ, L (? n ) functions and the duality relation between H ω, L ((? n ) and BMO ρ, L* (? n ), where L* denotes the adjoint operator of L in L 2 (? n ). Using this duality relation, we further obtain the ρ-Carleson measure characterization of BMO ρ, L* (? n ) and the molecular characterization of H ω, L (? n ); the latter is used to establish the boundedness of the generalized fractional operator L ρ ?γ from H ω, L (? n ) to H L 1 (? n ) or L q (? n ) with certain q > 1, where H L (? n ) is the Hardy space introduced by Auscher, Duong and McIntosh. These results generalize the existing results by taking ω(t) = t p for t ∈ (0,∞) and p ∈ (n/(n + θ(L)), 1].展开更多
This paper discusses the approximation by reciprocals of polynomials with positive coefficients in Orlicz spaces and proved that if f（x） ∈ L^＊M[0, 1], changes its sign at most once in （0, 1）, then there exists ...This paper discusses the approximation by reciprocals of polynomials with positive coefficients in Orlicz spaces and proved that if f（x） ∈ L^＊M[0, 1], changes its sign at most once in （0, 1）, then there exists x0 ∈ （0, 1） and a polynomial Pn∈ Fin（＋） such that ｜｜f（x）-x-x0/Pn（x）｜｜M≤Cω（f,n-1/2）M, where Пn（＋） indicates the set of all polynomials of degree n with positive coefficients展开更多
In this paper, we apply function parameters to real interpolation of Lorentz- Orlicz martingale spaces. Some new interpolation theorems are formulated which generalize some known results in Lorentz spaces An introduce...In this paper, we apply function parameters to real interpolation of Lorentz- Orlicz martingale spaces. Some new interpolation theorems are formulated which generalize some known results in Lorentz spaces An introduced by Sharpley.展开更多
In this article we introduce the generalized lacunary difference sequence spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and [N_θ,M,Δ~m]_∞ using m^(th)- difference.We study their properties like completeness,solidness,symm...In this article we introduce the generalized lacunary difference sequence spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and [N_θ,M,Δ~m]_∞ using m^(th)- difference.We study their properties like completeness,solidness,symmetricity.Also we obtain some inclusion relations involving the spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and[N_θ,M,Δ~m]_∞ and the Cesàro summable and strongly Cesàro summable sequences.展开更多
基金supported by National Natural Science Foundation of China (Grant No.10671147)
文摘In this paper,the weak Orlicz space wL Φ is introduced and its applications to the martingale theory are discussed.In particular,a series of martingale inequalities including the maximal function inequality in weak Orlicz spaces are established;the relationships between these spaces are investigated.Moreover,the boundedness of several sublinear operators from one weak Orlicz space to another is proved;their vector-valued analogues are also considered.
基金supported by National Natural Science Foundation of China(Grant No.11171027)Program for Changjiang Scholars and Innovative Research Team in University of China
文摘Let￠ ： Nn x [0,∞） -4 [0, ∞） be a function such that ￠（x,.） is an Orlicz function and ￠（.,t） ∈ Aloc∞（Nn） （the class of local weights introduced by Rychkov）. In this paper, the authors introduce a local Musielak-Orlicz Hardy space h￠（Nn） by the local grand maximal function, and a local BMO-type space bmo￠（Nn） which is further proved to be the dual space of h￠（Nn）. As an application, the authors prove that the class of pointwise multipliers for the local BMO-type space bmo￠（Nn）, characterized by Nakai and Yabuta, is just the dual of LI（Rn） ＋ hФ0 （Rn）, where Ф is an increasing function on （0, co） satisfying some additional growth conditions and Ф0 a Musielak-Orlicz function induced by Ф. Characterizations of h￠（Rn）, including the atoms, the local vertical and the local nontangential maximal functions, are presented. Using the atomic char- acterization, the authors prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of h￠（Rn）, from which, the authors further deduce some criterions for the boundedness on h￠（Rn） of some sublinear operators. Finally, the authors show that the local Riesz transforms and some pseudo-differential operators are bounded on h￠（Rn）.
基金supported by National Natural Science Foundation of China (Grant No. 10871025)Program for Changjiang Scholars and Innovative Research Team in Universities of China
文摘Let L be a linear operator in L^2（R^n） and generate an analytic semigroup {e^-tL}t≥0 with kernel satisfying an upper bound estimate of Poisson type, whose decay is measured by θ（L） ∈ （0, ∞）. Let 4） be a positive, continuous and strictly increasing function on （0, ∞）, which is of strictly critical lower type pФ （n/（n ＋ θ（L））, 1]. Denote by HФ, L（R^n） the Orlicz-Hardy space introduced in Jiang, Yang and Zhou＇s paper in 2009. If Ф is additionally of upper type 1 and subadditive, the authors then show that the Littlewood-Paley g-function gL maps HФ, L（R^n） continuously into LФ（R^n） and, moreover, the authors characterize HФ, L（R^n） in terms of the Littlewood-Paley gλ^＊-function with λ ∈ （n（2/pФ ＋ 1）, ∞）. If Ф is further slightly strengthened to be concave, the authors show that a generalized Riesz transform associated with L is bounded from HФ, L（R^n） to the Orlicz space L^Ф（R^n） or the Orlicz-Hardy space HФ （R^n）; moreover, the authors establish a new subtle molecular characterization of HФ, L （R^n） associated with L and, as applications, the authors then show that the corresponding fractional integral L^-γ for certain γ∈ E （0,∞） maps HФ, L（R^n） continuously into HФ, L（R^n）, where Ф satisfies the same properties as Ф and is determined by Ф and λ and also that L has a bounded holomorphic functional calculus in HФ, L（R^n）. All these results are new even when Ф（t） = t^p for all t ∈ （0, ∞） and p ∈ （n/（n ＋ θ（L））, 1].
基金supported by National Science Foundation for Distinguished Young Scholars of China (GrantNo. 10425106)
文摘Let L be a linear operator in L 2 (? n ) and generate an analytic semigroup {e ?tL }t?0 with kernel satisfying an upper bound of Poisson type, whose decay is measured by θ(L) ∈ (0, ∞). Let ω on (0,∞) be of upper type 1 and of critical lower type p o (ω) ? (n/(n+θ(L)),1] and ρ(t) = t t1/ω ?1(t ?1) for t ∈ (0,∞). We introduce the Orlicz-Hardy space H ω, L (? n ) and the BMO-type space BMO ρ, L (? n ) and establish the John-Nirenberg inequality for BMO ρ, L (? n ) functions and the duality relation between H ω, L ((? n ) and BMO ρ, L* (? n ), where L* denotes the adjoint operator of L in L 2 (? n ). Using this duality relation, we further obtain the ρ-Carleson measure characterization of BMO ρ, L* (? n ) and the molecular characterization of H ω, L (? n ); the latter is used to establish the boundedness of the generalized fractional operator L ρ ?γ from H ω, L (? n ) to H L 1 (? n ) or L q (? n ) with certain q > 1, where H L (? n ) is the Hardy space introduced by Auscher, Duong and McIntosh. These results generalize the existing results by taking ω(t) = t p for t ∈ (0,∞) and p ∈ (n/(n + θ(L)), 1].
基金Supported by Inner Mongolia Natural Science Foundations of China (200408020108).
文摘This paper discusses the approximation by reciprocals of polynomials with positive coefficients in Orlicz spaces and proved that if f（x） ∈ L^＊M[0, 1], changes its sign at most once in （0, 1）, then there exists x0 ∈ （0, 1） and a polynomial Pn∈ Fin（＋） such that ｜｜f（x）-x-x0/Pn（x）｜｜M≤Cω（f,n-1/2）M, where Пn（＋） indicates the set of all polynomials of degree n with positive coefficients
基金Li Baode is supported by NSFC(No.11461065,No.11161044)Scientific Research Projects in Colleges and Universities in Xinjiang Uyghur Autonomous Region(No.XJEDU2014S001)
基金supported by National Natural Science Foundation of China(Grant No.11201354)Hubei Province Key Laboratory of Systems Science in Metallurgical Process(Wuhan University of Science and Technology)(Y201321)National Natural Science Foundation of Pre-Research Item(2011XG005)
文摘In this paper, we apply function parameters to real interpolation of Lorentz- Orlicz martingale spaces. Some new interpolation theorems are formulated which generalize some known results in Lorentz spaces An introduced by Sharpley.
文摘In this article we introduce the generalized lacunary difference sequence spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and [N_θ,M,Δ~m]_∞ using m^(th)- difference.We study their properties like completeness,solidness,symmetricity.Also we obtain some inclusion relations involving the spaces [N_θ,M,Δ~m]_0,[N_θ,M,Δ~m]_1 and[N_θ,M,Δ~m]_∞ and the Cesàro summable and strongly Cesàro summable sequences.
