Duality of Hardy and BMO spaces associated with operators with heat kernel bounds on product domains

Let L be the infinitesimal generator of an analytic semigroup on L² (?) with suitable upper bounds on its heat kernels, and L has a bounded holomorphic functional calculus on L² (?). In this article, we introduce new function spaces H _L ¹ (? × ?) and BMO_L(? × ?) (dual to the space H _L* ¹ (? × ?) in which L* is the adjoint operator of L) associated with L, and they generalize the classical Hardy and BMO spaces on product domains. We obtain a molecular decomposition of function for H _L ¹ (? × ?) by using the theory of tent spaces and establish a characterization of BMO_L (? × ?) in terms of Carleson conditions. We also show that the John-Nirenberg inequality holds for the space BMO_L (? × ?). Applications include large classes of differential operators such as the magnetic Schrödinger operators and second-order elliptic operators of divergence form or nondivergence form in one dimension.     

Cesaro Summability with Respect to Two-Parameter Walsh Systems

 The one- and two-parameter Walsh system will be considered in the Paley as well as in the Kaczmarz rearrangement. We show that in the two-dimensional case the restricted maximal operator of the Walsh–Kaczmarz (C, 1)-means is bounded from the diagonal Hardy space H ^p to L ^p for every . To this end we consider the maximal operator T of a sequence of summations and show that the p-quasi-locality of T implies the same statement for its two-dimensional version T _α. Moreover, we prove that the assumption is essential. Applying known results on interpolation we get the boundedness of T _α as mapping from some Hardy–Lorentz spaces to Lorentz spaces. Furthermore, by standard arguments it will be shown that the usual two-parameter maximal operators of the (C, 1)-means are bounded from L ^p spaces to L ^p if . As a consequence, the a.e. convergence of the (C, 1)-means will be obtained for functions such that their hybrid maximal function is integrable. Of course, our theorems from the two-dimensional case can be extended to higher dimension in a simple way. (Received 20 April 2000; in revised form 25 September 2000)     

Hardy spaces and the<Emphasis Type="Italic">Tb</Emphasis> theorem

It is well-known that Calderón-Zygmund operators T are bounded on H^p for\(\frac{n}{{n + 1}}< p \leqslant 1\) provided T*(1) = 0. In this article, it is shown that if T*(b) = 0, where b is a para-accretive function, T is bounded from the classical Hardy space H^p to a new Hardy space H _b ^p . To develop an H _b ^p theory, a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequalities associated to a para-accretive function are established. Moreover, David, Journé, and Semmes’ result [9] about the L^P, 1 < p < ∞, boundedness of the Littlewood-Paley g function associated to a para-accretive function is generalized to the case of p ≤ 1. A new characterization of the classical Hardy spaces by using more general cancellation adapted to para-accretive functions is also given. These results complement the celebrated Calderón-Zygmund operator theory.     

Two distinguished subspaces of product BMO and Nehari-AAK theory for Hankel operators on the torus

In this paper we show that the theory of Hankel operators in the torus ^d , ford>1, presents striking differences with that on the circle , starting with bounded Hankel operators with no bounded symbols. Such differences are circumvented here by replacing the space of symbolsL ( ) by BMOr( ^d ), a subspace of product BMO, and the singular numbers of Hankel operators by so-called sigma numbers. This leads to versions of the Nehari-AAK and Kronecker theorems, and provides conditions for the existence of solutions of product Pick problems through finite Picktype matrices. We give geometric and duality characterizations of BMOr, and of a subspace of it, bmo, closely linked withA ₂ weights. This completes some aspects of the theory of BMO in product spaces.Sadosky was partially supported by NSF grants DMS-9205926, INT-9204043 and GER-9550373, and her visit to MSRI is supported by NSF grant DMS-9022140 to MSRI.     

Fractional Integral Operators on Anisotropic Hardy Spaces

In this paper, we introduce the fractional integral operator T of degree α of order m with respect to a dilation A for 0 < α < 1 and . First we establish the Hardy-Littlewood-Sobolev inequalities for T on anisotropic Hardy spaces associated with dilation A, which show that T is bounded from H ^p to H ^q , or from H ^p to L ^q , where 0 < p ≤ 1/(1 + α) and 1/q = 1/p − α. Then we give anisotropic Hardy spaces estimates for a class of multilinear operators formed by fractional integrals or Calderón-Zygmund singular integrals. Finally, we apply the above results to give the boundedness of the commutators of T and a BMO function. Research supported by NSF of China (Grant: 10571015) and SRFDP of China (Grant: 20050027025).     

The maximal (C, α, β) operator of two-parameter walsh-fourier series

The two-parameter dyadic martingale Hardy spacesH _p are introduced and it is proved that the maximal operator of the (C, α, β) means of a two-dimensional Walsh-Fourier series is bounded from H_p to L_p (1/(α+1), 1/(β+1)<p<∞) and is of weak type (H ₁ ^# , L₁), where the Hardy space H ₁ ^# is defined by the hybrid maximal function. As a consequence, we obtain that the (C, α, β) means of a function f∈H ₁ ^# converge a.e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on H_p whenever 1/(α+1), 1/(β+1)<p<∞. Thus in case f∈H_p, the (C, α, β) means converge to f in H_p norm. The same results are proved for the conjugate (C, α, β) means, too.     

Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols

We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators, for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the theory of modulation spaces contains some modifications of the scalar-valued theory, depending on the Banach space. In the Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued theory on continuity on certain modulation spaces when the symbol belongs to an L^p,q space and M^∞, respectively. The first result is true for any Banach space as range space, and the second result is true for any Hilbert space as range space.     

Conical square function estimates in UMD Banach spaces and applications to H ∞-functional calculi

We study conical square function estimates for Banach-valued functions and introduce a vector-valued analogue of the Coifman-Meyer-Stein tent spaces. Following recent work of Auscher-M^cIntosh-Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator A with certain off-diagonal bounds such that A always has a bounded H ^∞-functional calculus on these spaces. This provides a new way of proving functional calculus of A on the Bochner spaces L ^p (ℝ ⁿ ; X) by checking appropriate conical square function estimates and also a conical analogue ofBourgain’s extension of the Littlewood-Paley theory to the UMD-valued context. Even when X = ℂ, our approach gives refined p-dependent versions of known results.     

Characterization of BMO in terms of rearrangement-invariant Banach function spaces

The atomic decomposition of Hardy spaces by atoms defined by rearrangement-invariant Banach function spaces is proved in this paper. Using this decomposition, we obtain the characterizations of BMO and Lipschitz spaces by rearrangement-invariant Banach function spaces. We also provide the sharp function characterization of the rearrangement-invariant Banach function spaces.     

Boundedness of Convolution Operators and Input–Output Maps between Weighted Spaces

We ask when convolution operators with scalar- or operator-valued kernel functions map between weighted L² spaces of Hilbert space-valued functions. For a certain class of decreasing weights, including negative powers (t + a)^−m for example, we solve the one-weight problem completely by using Laplace transforms and Bergman-type spaces of vector-valued analytic functions. For a much more general class of decreasing weights, we solve the one-weight problem for all positive real kernels (also for L^p(w) with p > 1), by results on Steklov operators which generalise the weighted Hardy inequality. When the kernel function is a strongly continuous semigroup of bounded linear Hilbert space operators, which arises from input–output maps of certain linear systems, then the most obvious sufficient condition for boundedness, obtained by taking norm signs inside the integrals, is also necessary in many cases, but not in general. Submitted: July 15, 2007.,Revised: November 19, 2007.,Accepted: December 14, 2007.     

Wavelet characterization of weighted spaces

We give a characterization of weighted Hardy spaces H ^p (w), valid for a rather large collection of wavelets, 0 <p ≤ 1,and weights w in the Muckenhoupt class A _∞ We improve the previously known results and adopt a systematic point of view based upon the theory of vector-valued Calderón-Zygmund operators. Some consequences of this characterization are also given, like the criterion for a wavelet to give an unconditional basis and a criterion for membership into the space from the size of the wavelet coefficients.     

一类粗糙核参数型Marcinkiewicz积分

In this paper,the L2-boundedness of a class of parametric Marcinkiewicz integral μρΩ,h with kernel function Ω in Bq0.0 (Sn-1) for some q＞ 1,and the radial function h (x)∈ l∞ (Ls) (R+) for 1＜s≤∞ are given. The Lp(Rn) (2≤p＜∞) boundedness of μ*Ω,ph,λ and μρΩ,h,s with Ω in Bq0,0(Sn-1) and h(|x|)∈l∞(Ls)(R+) in application are obtained. Here μ*Ω,p h,λ and μpΩ,h,s are parametric Marcinkiewicz integrals corresponding to the Littlewood-Paley gλ* function and the Lusin area function S,respectively.     

BLO spaces associated with the Ornstein–Uhlenbeck operator

Let (Rn,|⋅|,dγ) be the Gauss measure metric space, where Rn denotes the n-dimensional Euclidean space, |⋅| the Euclidean norm and for all x∈Rn the Gauss measure. In this paper, for any a∈(0,∞), the authors introduce some BLOa(γ) space, namely, the space of functions with bounded lower oscillation associated with a given class of admissible balls with parameter a. Then the authors prove that the noncentered local natural Hardy–Littlewood maximal operator is bounded from BMO(γ) of Mauceri and Meda to BLOa(γ). Moreover, a characterization of the space BLOa(γ), via the local natural maximal operator and BMO(γ), is given. The authors further prove that a class of maximal singular integrals, including the corresponding maximal operators of both imaginary powers of the Ornstein–Uhlenbeck operator and Riesz transforms of any order associated with the Ornstein–Uhlenbeck operator, are bounded from L^∞(γ) to BLOa(γ).     

On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space

In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L ^{n +1} as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L ^{n +1} which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L ^{n +1} which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it. Received: 5 July 1999     

Boundedness of Commutators on Hardy-Type Spaces

In this paper, the boundedness of the commutator generated by strongly singular Calderón-Zygmund operator and a Lipschitz function is discussed on the classical Hardy spaces and the Herz-type Hardy spaces. The authors also get the boundedness of the strongly singular Calderón-Zygmund operator itself and the commutator generated by strongly singular Calderón-Zygmund operator and a BMO function on the Herz-type Hardy spaces.     

Factorization of Hardy spaces and Hankel operators on convex domains in ℂ n

In this article we study the (small) Hankel operator h_b on the Hardy and Bergman spaces on a smoothly bounded convex domain of finite type in ℂⁿ. We completely characterize the Hankel operators h_b that are bounded, compact, and belong to the Schatten ideal S_p, for 0 < p < ∞. In particular, if h_b denotes the Hankel operator on the Hardy space H² (Ω), we prove that h_b is bounded if and only if b ∈ BMOA, compact if and only if b ∈ VMOA, and in the Schatten class if and only if b ∈e B_p, 0 < p < ∞. This last result extends the analog theorem in the case of the unit disc of Peller [19] and Semmes [21]. In order to characterize the bounded Hankel operators, we prove a factorization theorem for functions in H¹ (Ω), a result that is of independent interest.     

Weak type estimates of intrinsic square functions on the weighted Hardy spaces

In this paper, by using the atomic decomposition theory of weighted Hardy spaces, we will give some weighted weak type estimates for intrinsic square functions including the Lusin area function, Littlewood–Paley g-function and g^*_l{g^*_\lambda}-function on these spaces.     

Gradient estimates for elliptic systems in non-smooth domains

We obtain the global W ^1,p , 1 < p < ∞, estimate for the weak solution of an elliptic system with discontinuous coefficients in non-smooth domains without using maximal function approach. It is assumed that the boundary of a bounded domain is well approximated by hyperplanes at every point and at every scale, and that the tensor coefficients belong to BMO space with their BMO semi-norms sufficiently small. S.-S. Byun was supported in part by KRF-2006-C00034 and L. Wang was supported in part by NSF Grant 0701392.