Orlicz-Hardy spaces and their duals

We establish the theory of Orlicz-Hardy spaces generated by a wide class of functions.The class will be wider than the class of all the N-functions.In particular,we consider the non-smooth atomic decomposition.The relation between Orlicz-Hardy spaces and their duals is also studied.As an application,duality of Hardy spaces with variable exponents is revisited.This work is different from earlier works about Orlicz-Hardy spaces H(Rn)in that the class of admissible functions is largely widened.We can deal with,for example,(r)≡(rp1(log(e+1/r))q1,0r 6 1,rp2(log(e+r))q2,r1,with p1,p2∈(0,∞)and q1,q2∈(.∞,∞),where we shall establish the boundedness of the Riesz transforms on H(Rn).In particular,is neither convex nor concave when 0p11p2∞,0p21p1∞or p1=p2=1 and q1,q20.If(r)≡r(log(e+r))q,then H(Rn)=H(logH)q(Rn).We shall also establish the boundedness of the fractional integral operators I of order∈(0,∞).For example,I is shown to be bounded from H(logH)1./n(Rn)to Ln/(n.)(log L)(Rn)for 0n.     

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 < ∞.     

Necessary and sufficient conditions for boundedness of some commutators with weighted Lipschitz functions

In this paper we show that b∈Lipβ,μ if and only if the commutator [b,T] of the multiplication operator by b and the singular integral operator T is bounded from Lp(μ) to Lq(μ1−q), where 1<p<q<∞, 0<β<1 and 1/q=1/p−β/n. Also we will obtain that b∈Lipβ,μ if and only if the commutator [b,Iα] of the multiplication operator by b and the fractional integral operator Iα is bounded from Lp(μ) to Lr(μ1−(1−α/n)r), where 1<p<∞, 0<β<1 and 1/r=1/p−(β+α)/n with 1/p>(β+α)/n.     

Weighted Weak Behaviour of Fourier-Jacobi Series

Let w(x) = (1 - x)^α (1 + x)^β be a Jacobi weight on the interval [-1, 1] and 1 < p < ∞. If either α > ?1/2 or β > ?1/2 and p is an endpoint of the interval of mean convergence of the associated Fourier-Jacobi series, we show that the partial sum operators S_n are uniformly bounded from L^p,1 to L^p,∞, thus extending a previous result for the case that both α, β > ?1/2. For α, β > ?1/2, we study the weak and restricted weak (p, p)-type of the weighted operators f→uS_n(u^?1f), where u is also Jacobi weight.     

SOME MULTIPLIER THEOREMS FOR ANISOTROPIC HARDY SPACES ——In Memory of Professor Yongsheng Sun

Let A be a symmetric expansive matrix and Hp(Rn) be the anisotropic Hardy space associated with A. For a function m in L∞(Rn), an appropriately chosen function η in Cc∞(Rn) and j ∈ Z define mj(ξ) = m(Ajξ)η(ξ). The authors show that if 0 ＜ p ＜ 1 and (m)j belongs to the anisotropic nonhomogeneous Herz space K11/p-1,p(Rn), then m is a Fourier multiplier from Hp(Rn) to Lp(Rn). For p = 1, a similar result is obtained if the space K10,1(Rn) is replaced by a slightly smaller space K(w).Moreover, the authors show that if 0 ＜ p ≤ 1 and if the sequence {(mj)V} belongs to a certain mixednorm space, depending on p, then m is also a Fourier multiplier from Hp(Rn) to Lp(Rn).     

Boundedness of multilinear commutators of generalized fractional integrals

Let L be the infinitesimal generator of an analytic semigroup on L²(?ⁿ ) with Gaussian kernel bound, and let L^{–α /2} be the fractional integral of L for 0 < α < n. Suppose that b = (b₁, b₂, …, b_m ) is a finite family of locally integral functions, then the multilinear commutator generated by b and L^{–α /2} is defined by L^{–α /2} _b f = [b_m , …, [b₂, [b₁, L^{–α /2}]], …, ] f, where m ∈ ?⁺. When b₁, b₂, …, b_m ∈ BMO or b_j ∈ Λ (0 < β_j < 1) for 1 ≤ j ≤ m, the authors study the boundedness of L^{–α /2} _b . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Endpoint estimates for the commutator of pseudo-differential operators

It is well known that the commutator Tb of the Calderón-Zygmund singular integral operator is bounded on L^p(Rⁿ) for 1 < p < +∞ if and only if b ∈ BMO [1]. On the other hand, the commutator T_b is bounded from H¹(Rⁿ) into L¹(Rⁿ) only if the function b is a constant [2]. In this article, we will discuss the boundedness of commutator of certain pseudo-differential operators on Hardy spaces H¹. Let T^σ be the operators that its symbol is S⁰_1,δ with 0 ≤ δ < 1, if b ∈ LMO_∞, then, the commutator [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) and from L¹(Rⁿ) into BMO(Rⁿ); If [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) or L¹(Rⁿ) into BMO(Rⁿ), then, b ∈ LMO_loc.     

Multilinear Calderón-Zygmund operators on the product of Lebesgue spaces with non-doubling measures

Under the assumption that μ is a non-doubling measure on Rd, the author proves that for the multilinear Calderón-Zygmund operator, its boundedness from the product of Hardy space H¹(μ)×H¹(μ) into L1/2(μ) implies its boundedness from the product of Lebesgue spaces Lp₁(μ)×Lp₂(μ) into Lp(μ) with 1<p₁,p₂<∞ and p satisfying 1/p=1/p₁+1/p₂.     

Boundedness of parabolic singular integrals and Marcinkiewicz integrals on Triebel-Lizorkin spaces

In this paper,we obtain the boundedness of the parabolic singular integral operator T with kernel in L(logL)1/γ(Sn-1) on Triebel-Lizorkin spaces.Moreover,we prove the boundedness of a class of Marcinkiewicz integrals μΩ,q(f) from ∥f∥ F˙p0,q(Rn) into Lp(Rn).     

A Note on the Embedding of Subspaces of Lp into ℓNr, 0 < r < 1, r ≦ p < 2

In this paper we extend a result by Bourgain-Lindenstrauss-Milman (see [1]). We prove: Let 0 < ? < 1/2, 0< r < 1, r< p < 2. There exists a constant C = C(r,p,?) such that if X is any n-dimensional subspace of L_p(0, l), then there exists Y ? ?^N_r with d(X, Y) ≦ 1 + ?, whenever N > Cn. As an application, we obtain the following partial result: Let 0 < r < 1. There exist constants C = C(r) and C' = C' (r) such that if X is any n-dimensional subspace of L_r(0,1), then there exists Y ? ^N_r with d(X, Y) ≦ C (logn)^l/^r, whenever N ≧ C'n.     

Commutators of generalized Hardy operators

LetH_a,b be the commutator generated by the generalized Hardy operator and the CMO function. The (L^p, L^p) boundedness of H_a,b is discussed in this paper. Furthermore, the authors consider the boundedness of H_a,b on the weighted homogeneous Herz spaces (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lattice paths in regions with the catalan property

If α = {α₀, α₁,…, α_n} and β = {β₀, β₁,…, β_n} are two non-decreasing sets of integers such that α₀ = 0 < β₀, α_n < β_n = n, and α_i < i < β_i for 1 ? i ? n ? 1, let L denote the set of lattice points (p, q) such that 0 ? p ? n and α_p ? q ? β_p. We determine all such regions L with the property that the number of lattice paths from (0, 0) to (p, p) in L is the Catalan number$\text{(p + 2)}{}^{\mathrm{?1}}\text{(}\text{2p+2}\text{p+1}\text{)}$ for 0 ? p ? n.     

L p − Lq estimates for convolution operators with n-Dimensional singular measures

Let S ⊂ ℜⁿ⁺¹ be the graph of the function ϕ :[−1, 1] ⁿ → ℜ defined by ϕ (x ₁ , …, x_n) = ∑ _j=1 ⁿ |x_j|^αj, with1<α ₁ ≤ … ≤ α_n, let σ the Euclidean area measure on S. In this article we study the L^p − L^q boundedness of convolution operators with the singular Borel measure on Rⁿ⁺¹ given by μ (E)=σ (E ∩ S)     

On the L2n 1– Extension Properties

We have proved that for all compact linear operator u from R into an L_p ([0,1], ν) (0 < p < 1) extends to L ₁ ([0,1], ν), where R denotes the closed linear subspace in L ₁ ([0,1], ν) of the Rademacher functions {r_n }_{n ? N}. In this paper, we study this type of extension for E_n ? L_2n ¹ where E_n is the n–dimensional subspace which appears in Kasin's theorem such that L_2n ¹ = E_n ⊕ E ^⊥ _n and the L_2n ¹ , L_2n ² norms are universally equivalent on both E_n , E ^⊥ _n. We show that, the precedent extension fails for the pair (E_n , L_2n ¹ ) and we generalize this to any E in an L ₁(Ω, A, P) by giving some conditions on E.     

On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line

In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the Dunkl-type fractional maximal operator Mβ, and the Dunkl-type fractional integral operator Iβ from the spaces Lp,α(R) to the spaces Lq,α(R), 1<p<q<∞, and from the spaces L1,α(R) to the weak spaces WLq,α(R), 1<q<∞. In the case , we prove that the operator Mβ is bounded from the space Lp,α(R) to the space L∞,α(R), and the Dunkl-type modified fractional integral operator is bounded from the space Lp,α(R) to the Dunkl-type BMO space BMOα(R). By this results we get boundedness of the operators Mβ and Iβ from the Dunkl-type Besov spaces to the spaces , 1<p<q<∞, 1/p−1/q=β/(2α+2), 1?θ?∞ and 0<s<1.     

l p n Superspaces of spans of independent random variables

We show that for 1 ≦p < ∞,p ≠ 2, ifɛ > 0 is small enough andX ≦L _p is the span ofn independent Rademacher functions orn independent Gaussian random variables, then any superspaceY ofX satisfyingd(Y,L _p ^m ) ≦ 1 +ɛ has dimension larger thanr ⁿ, wherer =r(ɛ, p) > 1. This forms part of the author’s doctoral dissertation prepared at Texas A&M University under the direction of Professor W. B. Johnson. Supported in part by NSF DMS-85 00764.     

乘子算子及其交换子在Morrey空间上的有界性

证明了乘子算子(M_p~q(R~n),Lip(β-n/q))的有界性和(M_p~q(R~n),BMO(R~n))的有界性.还得到乘子算子及其交换子在广义Morrey空问Lp,L_(p,φ)(R~n)上的有界性.     

非齐型空间中一类满足Hörmander条件的Marcinkiewicz交换子估计

记μ为上的非负Radon测度,且仅满足对固定的C₀>0和n∈(0,d],及所有的和r>0, μ(B(x,r))≤C₀ rⁿ.作者建立了一类核函数满足Hörmander条件的Marcinkiewicz积分与Lip_β(μ)(0<β)函数生成的交换子由L^p(μ)到L^q(μ),由L^p(μ) 到Lip_β-n/p(μ)及L^n/β(μ)到RBMO(μ)有界.部分结论对经典 Marcinkiewicz积分也是新的.       

Strongly singular convolution operators on the weighted Herz-type Hardy spaces

Let 0<p≤1<q<0, andw ₁ ,w ₂ ∈ A ₁ (Muckenhoupt-class). In this paper the authors prove that the strongly singular convolution operators are bounded from the homogeneous weighted Herz-type Hardy spacesH K^{α, p} _q(w₁; w₂) to the homogeneous weighted Herz spacesK ^{α, p} _q (w₁; w₂), provided α=n(1−1/q). Moreover, the boundedness of these operators on the non-homogeneous weighted Herz-type Hardy spacesH K ^{α, p} _q (w ₁;w ₂) is also investigated. Supported by the National Natural Science Foundation of China     

Sharp L 2 boundedness of the oscillatory hyper-Hilbert transform along curves

Consider the oscillatory hyper-Hilbert transform Hn,α,βf（x）=∫0^1 f（x-Г（t））e^it-βt^-1-α dt along the curve P（t） = （tp1, tP2,..., tpn）, where β 〉 α ≥ 0 and 0 〈 p1 〈 p2 〈 ... 〈 Pn. We prove that H n,α,β is bounded on L2 if and only if β ≥ （n ＋ 1）α. Our work extends and improves some known results.