Discrete Calderón’s Identity,Atomic Decomposition and Boundedness Criterion of Operators on Multiparameter Hardy Spaces

In this paper we establish a discrete Calderón’s identity which converges in both L ^q (ℝ ^n+m ) (1<q<∞) and Hardy space H ^p (ℝ ⁿ ×ℝ ^m ) (0<p≤1). Based on this identity, we derive a new atomic decomposition into (p,q)-atoms (1<q<∞) on H ^p (ℝ ⁿ ×ℝ ^m ) for 0<p≤1. As an application, we prove that an operator T, which is bounded on L ^q (ℝ ^n+m ) for some 1<q<∞, is bounded from H ^p (ℝ ⁿ ×ℝ ^m ) to L ^p (ℝ ^n+m ) if and only if T is bounded uniformly on all (p,q)-product atoms in L ^p (ℝ ^n+m ). The similar result from H ^p (ℝ ⁿ ×ℝ ^m ) to H ^p (ℝ ⁿ ×ℝ ^m ) is also obtained.     

Parabolic Marcinkiewicz integrals along surfaces on product domains

In this paper, parabolic Marcinkiewicz integral operators along surfaces on the product domain ℝ ⁿ × ℝ ^m (n,m ⩾ 2) are introduced. L ^p bounds of such operators are obtained under weak conditions on the kernels.     

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 ＜∞.     

Boundedness of Marcinkiewicz integrals and their commutators in H1 (?n × ?m)

The authors establish the boundedness of Marcinkiewicz integrals from the Hardy space H ¹ (ℝ ⁿ × ℝ ^m ) to the Lebesgue space L ¹(ℝ ⁿ × ℝ ^m ) and their commutators with Lipschitz functions from the Hardy space H ¹ (ℝ ⁿ × ℝ ^m ) to the Lebesgue space L ^q (ℝ ⁿ × ℝ ^m ) for some q > 1.     

On a singular integral on product spaces

We show that if K(x,y)=Ω(x,y)/|x|ⁿ|y|^m is a Calder n-Zygmund kerned on Rⁿ×R^m, where Ω∈L²(Sⁿ⁻¹×S^m−1) and b(x,y) is any bounded function which is radial with x∈Rⁿ and y∈R^m respectively, then b(x,y)K(x,y) is the kernel of a convolution operator which is bounded on L^p(Rⁿ×R^m) for 1<p<∞ and n≧2, m≧2. Project supported by NSFC     

Maximum principles for the polyharmonic equation on Lipschitz domains

The Agmon-Miranda maximum principle for the polyharmonic equations of all orders is shown to hold in Lipschitz domains in ℝ³. In ℝn,n≥4, the Agmon-Miranda maximum principle andL ^p-Dirichlet estimates for certainp>2 are shown to fail in Lipschitz domains for these equations. In particular if 4≤n≤2m+1 theL ^p Dirichlet problem for Δ ^m fails to be solvable forp>2(n−1)/(n−3). Supported in part by the NSF.     

Generalized commutators for Marcinkiewicz type integrals with variable kernels

Let A be a function with derivatives of order m and D γ A ∈■β (0 < β < 1, |γ| = m). The authors in the paper prove that if Ω(x, z) ∈ L ∞ (R n ) × L s (S n 1 ) (s ≥ n/(n β)) is homogenous of degree zero and satisfies the mean value zero condition about the variable z, then both the generalized commutator for Marcinkiewicz type integral μ A Ω and its variation μ A Ω are bounded from L p (R n ) to L q (R n ), where 1 < p < n/β and 1/q = 1/p β/n. The authors also consider the boundedness of μ A Ω and its variation μ A Ω on Hardy spaces.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Bounds for the Commutator of Parabolic Singular Integral with Rough Kernels

In this paper, the authors give the L ^p (1 < p < ∞ ) boundedness of the k-th order commutator of parabolic singular integral with the kernel function Ω ∈ L(log⁺ L) ^k + 1(S ^n − 1). The result in this paper is an extension of some known results. The research was supported by NSF of China (Grant: 10571015) and SRFDP of China (Grant: 20050027025).     

The Estimates for Sharp Maximal Functions of Multilinear Strongly Singular Integral Operators

In this paper, the author obtains that the multilinear operators of strongly singular integral operators and their dual operators are bounded from some L^p（R^n） to L^p（R^n） when the m-th order derivatives of A belong to L^p（R^n） for r large enough. By this result, the author gets the estimates for the Sharp maximal functions of the multilinear operators with the m-th order derivatives of A being Lipschitz functions. It follows that the multilinear operators are （L^p, L^p）-type operators for 1 〈 p 〈 ∞.     

Lp (?m × ?n) boundedness for the Marcinkiewicz integral on product spaces

L^p(R^m × Rⁿ) boundedness is considered for the multiple Marcinkiewicz integral. Some size conditions implying the L^p(R^m × Rⁿ) boundedness of the multiple Marcinkiewicz integral for some fixed 1 > p > ∞ are obtained.     

On classification of orthogonal multiplications à la do carmo-wallach

The space of range-equivalence classes of full orthogonal multiplications F: ℝ ⁿ ×ℝ ⁿ →ℝ ^p , n≤p ≤n ², is shown to be a compact convex body lying in so(n)⊗so(n). Furthermore, the dimension of the space of equivalence classes is determined to be (n ²(n−1)²)/4−n(n−1).     

Riesz means for the sublaplacian on the Heisenberg group

The uniform boundedness of the Riesz means for the sublaplacian on the Heisenberg groupH ⁿ is considered. It is proved thatS _R ^α are uniformly bounded onL ^p(Hⁿ) for 1≤p≤2 provided α>α(p)=(2n+1)[(1/p)−(1/2)].     

L^p（R^n） Boundedness for the Maximal Operator of Multilinear Singular Integrals with Calderon-Zygmund Kernels

In this paper, the authors consider a class of maximal multilinear singular integral operators and maximal multilinear oscillatory singular integral operators with standard Calderón–Zygmund kernels, and obtain their boundedness on L ^p (ℝ ⁿ ) for 1 < p < ∞. Research supported by Professor Xu Yuesheng’s Research Grant in the program of "One hundred Distinguished Young Scientists" of the Chinese Academy of Sciences     

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 bilinear algorithm for sparse representations

We consider the following sparse representation problem: represent a given matrix X∈ℝ ^m×N as a multiplication X=AS of two matrices A∈ℝ ^m×n (m≤n<N) and S∈ℝ ^n×N , under requirements that all m×m submatrices of A are nonsingular, and S is sparse in sense that each column of S has at least n−m+1 zero elements. It is known that under some mild additional assumptions, such representation is unique, up to scaling and permutation of the rows of S. We show that finding A (which is the most difficult part of such representation) can be reduced to a hyperplane clustering problem. We present a bilinear algorithm for such clustering, which is robust to outliers. A computer simulation example is presented showing the robustness of our algorithm.     

Characterizations of BMO associated with Gauss measures via commutators of local fractional integrals

Let dγ(x) ≡ π ^−n/2 e ^{−|x| 2} dx for all x ∈ ℝ ⁿ be the Gauss measure on ℝ ⁿ . In this paper, the authors establish the characterizations of the space BMO(γ) of Mauceri and Meda via commutators of either local fractional integral operators or local fractional maximal operators. To this end, the authors first prove that such a local fractional integral operator of order β is bounded from L ^p (γ) to L ^p/(1−pβ)(γ), or from the Hardy space H ¹(γ) of Mauceri and Meda to L ^1/(1−β)(γ) or from L ^1/β (γ) to BMO(γ), where β ∈ (0, 1) and p ∈ (1, 1/β).     

Boundedness of Hausdorff operators on some product Hardy type spaces

We study Hausdorff operators on the product Besov space B01,1 (Rn × Rm) and on the local product Hardy space h1 (Rn × Rm).We establish some boundedness criteria for Hausdorff operators on these functio...     

L^2（R^n） Boundedness for the Commutators of Homogeneous Singular Integral Operators

The commutators of singular integral operators with homogeneous kernel Ω(x)/|x|^n are studied, where Ω is homogeneous of degree zero, and has mean value zero on the unit sphere. It is proved that Ω∈ L(logL)^(k 1)(S^(n-1)) is a sufficient condition such that the k-th order commutator is bounded on L^2(R^n).     

Pseudodifferential Operators on Local Hardy Spaces

In this work, we study the continuity of pseudodifferential operators on local Hardy spaces h ^p (ℝ ⁿ ) and generalize the results due to Goldberg and Taylor by showing that operators with symbols in S _1,δ ⁰(ℝ ⁿ ), 0≤δ<1, and in some subclasses of S _1,1⁰(ℝ ⁿ ) are bounded on h ^p (ℝ ⁿ ) (0<p≤1). As an application, we study the local solvability of the planar vector field L=∂ _t +ib(x,t)∂ _x , b(x,t)≥0, in spaces of mixed norm involving Hardy spaces. Work supported in part by CNPq, FINEP, and FAPESP.     

Surjective partial differential operators on real analytic functions defined on open convex sets

Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on a covex open set Ω⊂ℝ ⁿ . Let L(P _m ) denote the localizations at ∞ (in the sense of H?rmander) of the principal part P _m . Then Q(x+iτN)≠ 0 for (x,τ)∈ℝ ⁿ ×(ℝ\{ 0}) for any Q∈L(P _m ) if N is a normal to δΩ which is noncharacteristic for Q. Under additional assumptions this implies that P _m must be locally hyperbolic. Received: 24 January 2000