LITTLEWOOD-PALEY THEOREM AND MULTIPLIERS ON THE QUANTUM TORUS＊

The Littlewood-Paley and Marcinkiewicz's multiplier theorems on the quantum torus are established.An key ingredient of the proof is vector-valued Littlewood-Paley and noncommutative Khinchin's inequali...     

L P boundedness of area function associated with operators on product spaces

Let L be the infinitesimal generator of an analytic semigroup on L2 (Rn) with suitable upper bounds on its heat kernels. Assume that L has a bounded holomorphic functional calculus on L2(Rn). In this paper,we define the Littlewood- Paley g function associated with L on Rn × Rn, denoted by GL(f)(x1, x2), and decomposition, we prove that ‖SL(f)‖p ≈‖GL(f)‖p ≈‖f‖p for 1 ＜ p ＜∞.     

L p bounds for singular integrals with rough kernels on product domains

This paper is concerned with singular integral operators on product domains with rough kernels both along radial direction and on spherical surface.Some rather weaker size conditions,which imply the Lp-boundedness of such operators for certain fixed p(1 p ∞),are given.     

Hardy space estimates for bi-parameter Littlewood-Paley square functions

Suppose that g(f) are bi-parameter Littlewood-Paley square functions which were introduced by H. Martikainen. It is known that the L^{\mathrm{2}}({ℝ}^n\times {ℝ}^m) boundedness and the H^{\mathrm{1}}({ℝ}^n\times {ℝ}^m)-L^{\mathrm{1}}({ℝ}^n\times {ℝ}^m) boundedness of g(f) have been proved by H. Martikainen and by Z. Li and Q. Xue, respectively. In this paper, we apply the vector-valued theory, the atomic decomposition of product Hardy spaces, and Journe's covering lemma to show that g(f) are bounded from H^p({ℝ}^n\times {ℝ}^m) to L^p({ℝ}^n\times {ℝ}^m) with p smaller than 1.     

Littlewood-Paley characterizations of anisotropic Hardy-Lorentz spaces

Let p ∈(0, 1], q ∈(0, ∞] and A be a general expansive matrix on R~n. Let H_A~(p,q )(R~n) be the anisotropic Hardy-Lorentz spaces associated with A defined via the nontangential grand maximal function. In this article, the authors characterize H_A~(p,q )(R~n) in terms of the Lusin-area function, the Littlewood-Paley g-function or the Littlewood-Paley g~*_λ-function via first establishing an anisotropic Fefferman-Stein vector-valued inequality in the Lorentz space L_(p,q)(R~n). All these characterizations are new even for the classical isotropic Hardy-Lorentz spaces on R~n. Moreover, the range of λ in the g~*_λ-function characterization of H_A~(p,q )(R~n) coincides with the best known one in the classical Hardy space H~p(R~n) or in the anisotropic Hardy space H_A~p (R~n).     

Littlewood-Paley g-函数交换子的加权估计

设g_(φ,b)是Littlewood-Paley g-函数与b生成的交换子,ω∈A_1.证明了若b属于加权BMO空间BMO(ω),则g_(φ,b)是L~p(ω)到L~p(ω~(1-p))(1p∞)有界的;若b属于加权Lipschitz空间Lip_β(ω)(0β1),则g_(φ,b)是L~p(ω)到L~q(ω~(1-q))的有界算子,其中1pq∞,1/q=1/p-β/n.     

齐型空间上的Lipschitz函数与Littlewood－Paley g－函数

在θ阶正规齐型空间上，如果算子列｛Ｓｋ｝ｋ∈Ｚ是恒等逼近，Ｄｋ＝Ｓｋ－Ｓｋ－１；本文给出一个用｛Ｄｋ｝ｋ∈Ｚ表达的ｆ∈Ｌｉｐα（Ｌｉｐｓｃｈｉｔｚ函数类，０＜α＜θ）的充分必要条件．作为其推论得到，对于ｆ∈ＬＩｐα，其Ｌｉｔｔｌｅｗｏｏｄ－Ｐａｌｅｙｇ函数ｇ（ｆ）（Ｘ）或者处处为无穷大，或者在Ｌｉｐα上有界．     

A two-parameter ``Bergman space' inequality

For , define , where is a tensor product of one-parameter Haar functions. Let and . We prove a sufficient condition, which is close to necessary, on double sequences of weights and non-negative , which ensures that the inequality

holds for all . We extend our result to an inequality concerning two-parameter wavelet families.

     

Hyperbolic g-function and Bloch pullback operators

The purpose of this paper is to obtain a hyperbolic version of Littlewood-Paley g-function equivalence and then use it to estimate the operator norm of Bloch-pullback operators.