Anisotropic Hardy Spaces Research Articles

LITTLEWOOD-PALEY CHARACTERIZATION OF WEIGHTED ANISOTROPIC HARDY SPACES

We obtain the weighted anisotropic Hardy space estimate for anisotropic singular integrals of convolution type, and apply it to derive Littlewood-Paley characterization of weighted anisotropic Hardy spaces.

MOLECULAR DECOMPOSITION OF WEIGHTED ANISOTROPIC HARDY SPACES

In this paper, the weighted anisotropic Hardy spaces associated with the general discrete group dilations, the weighted anisotropic atoms and molecules are introduced. Then the molecular decomposition of the weighted anisotropic Hardy spaces is obtained. As an application, the boundedness of Calderón-Zygmund operators on the weighted anisotropic Hardy spaces is discussed.

On dual spaces of anisotropic Hardy spaces

AbstractIn this paper we generalize the analysis of the Campanato‐type dual spaces 1, 3, for the highly anisotropic Hardy spaces on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^n$\end{document} introduced in 10. These Hardy spaces are constructed over multi‐level ellipsoid covers of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^n$\end{document}, where the ellipsoids can change shape rapidly from point to point and from level to level.

Hardy Spaces on ℝ n with Pointwise Variable Anisotropy

In this work we develop highly geometric Hardy spaces, for the full range 0<p≤1. These spaces are constructed over multi-level ellipsoid covers of ℝn that are highly anisotropic in the sense that the ellipsoids can change shape rapidly from point to point and from level to level. This generalizes previous work on anisotropic Hardy spaces where the geometry of the space was ‘fixed’ over ℝn and extends Hardy spaces over spaces of homogeneous type, where the theory holds for p values that are ‘close’ to 1.

Weakly strongly singular integral operators on anisotropic Hardy spaces and their dual operators

Let A be an expansive dilation. We define weakly strongly singular integral kernels and study the action of the operators induced by these kernels on anisotropic Hardy spaces associated with A.

Anisotropic weak Hardy spaces and interpolation theorems

In this paper, the authors establish the anisotropic weak Hardy spaces associated with very general discrete groups of dilations. Moreover, the atomic decomposition theorem of the anisotropic weak Hardy spaces is also given. As some applications of the above results, the authors prove some interpolation theorems and obtain the boundedness of the singular integral operators on these Hardy spaces.

New hardy spaces associated with some anisotropic Herz spaces and their applications

In this paper, a class of anisotropic Herz-type Hardy spaces associated with a non-isotropic dilation on ℝn are introduced, and the central atomic and molecular decomposition characterizations of those spaces are established. As some applications of the decomposition theory, the authors study the interpolation problem and the boundedness of the central δ-Calderon-Zygmund operators on the anisotropic Herz-type Hardy spaces.

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 \(m \in {\mathbb{N}}\). 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 Calderon-Zygmund singular integrals. Finally, we apply the above results to give the boundedness of the commutators of T and a BMO function.

Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators

In this paper we introduce and study weighted anisotropic Hardy spaces H p w (R n ; A) associated with general expansive dilations and A ∞ Muckenhoupt weights. This setting includes the classical isotropic Hardy space theory of Fefferman and Stein, the parabolic theory of Calderon and Torchinsky, and the weighted Hardy spaces of Garcia-Cuerva, Stromberg, and Torchinsky. We establish characterizations of these spaces via the grand maximal function and their atomic decompositions for p ∈ (0,1 ]. Moreover, we prove the existence of finite atomic decompositions achieving the norm in dense subspaces of H p w (R n ; A). As an application, we prove that for a given admissible triplet (p, q, s) w , if T is a sublinear operator and maps all (p, q, s) w -atoms with q < oo (or all continuous (p, q, s) w -atoms with q = ∞) into uniformly bounded elements of some quasi-Banach space B, then T uniquely extends to a bounded sublinear operator from H p w (R n ;A) to B. The last two results are new even for the classical weighted Hardy spaces on R n .

Anisotropic wavelet bases and thresholding

AbstractWe consider thresholding with respect to anisotropic wavelet bases measuring the approximation error in anisotropic Hardy spaces $ H^a_p $ for p &gt; 0, which are known to be equal to Lp for p &gt; 1. In particular, we characterize those functions that provide a specific convergence rate by intrinsic smoothness properties. To this end we apply restricted nonlinear approximation, see [3], which is a generalization of n ‐term approximation in which a weight function is used to control the terms of the approximations. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Some multiplier theorems for anisotropic hardy spaces

Let A be a symmetric expansive matrix and Hp (ℝ n) be the anisotropic Hardy space associated with A. For a function m in L∞ (ℝ n), an appropriately chosen function η in C ∞c (ℝ n) and j ∈ ℤ define mj(ξ) = m(Ajξ)η(ξ). The authors show that if 0 < p < 1 and \(\hat m_j \) belongs to the anisotropic non-homogeneous Herz space K 1/p−1,p 1 (ℝ n), then m is a Fourier multiplier from Hp (ℝ n) to Lp (ℝ n). For p = 1, a similar result is obtained if the space K 0,1 1 (ℝ n) is replaced by a slightly smaller space K(ω). Moreover, the authors show that if 0 < p ≤ 1 and if the sequence {(mj)∨} belongs to certain mixednorm space, depending on p, then m is also a Fourier multiplier from Hp (ℝ n) to Lp (ℝ n).

Anisotropic Hardy spaces and wavelets

In this paper, motivated in part by the role of discrete groups of dilations in wavelet theory, we introduce and investigate the anisotropic Hardy spaces associated with very general discrete groups of dilations. This formulation includes the classical isotropic Hardy space theory of Fefferman and Stein and parabolic Hardy space theory of Calderon and Torchinsky. Given a dilation A, that is an n × n matrix all of whose eigenvalues λ satisfy |λ| > 1, define the radial maximal function M φf(x) := sup k∈Z |(f ∗ φk)(x)|, where φk(x) = | detA|φ(Ax). Here φ is any test function in the Schwartz class with ∫ φ 6= 0. For 0 < p < ∞ we introduce the corresponding anisotropic Hardy space H A as a space of tempered distributions f such that M φf belongs to L (R). Anisotropic Hardy spaces enjoy the basic properties of the classical Hardy spaces. For example, it turns out that this definition does not depend on the choice of the test function φ as long as ∫ φ 6= 0. These spaces can be equivalently introduced in terms of grand, tangential, or nontangential maximal functions. We prove the Calderon-Zygmund decomposition which enables us to show the atomic decomposition of H A. As a consequence of atomic decomposition we obtain the description of the dual to H A in terms of Campanato spaces. We provide a description of the natural class of operators acting on H A, i.e., Calderon-Zygmund singular integral operators. We also give a full classification of dilations generating the same space H A in terms of spectral properties of A. In the second part of this paper we show that for every dilation A preserving some lattice and satisfying a particular expansiveness property there is a multiwavelet in the Schwartz class. We also show that for a large class of dilations (lacking this property) all multiwavelets must be combined minimally supported in frequency, and thus far from being regular. We show that r-regular (tight frame) multiwavelets form an unconditional basis (tight frame) for the anisotropic Hardy space H A. We also describe the sequence space characterizing wavelet coefficients of elements of the anisotropic Hardy space. 2000 Mathematics Subject Classification. Primary 42B30, 42C40; Secondary 42B20, 42B25.

$H^1$ boundedness of determinants of vector fields

We consider multilinear operators T (f1, . . . , fl) given by determinants of matrices of the form (Xkfj)1≤j,k≤l, where the Xk’s are C∞ vector fields on Rn. We give conditions on the Xk’s so that the corresponding operator T map products of Lebesgue spaces L p1×· · ·×Lpl into some anisotropic space H1, when 1 p1 + · · · + 1 pl = 1. 0. Introduction and statement of results. A well known Theorem of Coifman, Lions, Meyer, and Semmes [CLMS] states that the Jacobian J(F ) of a map F = (f1, . . . , fn) from R into itself maps the product of Sobolev spaces L1 1 × · · · × Ln 1 into the Hardy space H, when 1 p1 + · · · + 1 pn = 1. J(F ) is given by the determinant of the matrix ( ∂ ∂xk fj)1≤j,k≤n, where { ∂ ∂xk }1≤k≤n is the usual basis of the tangent space of R at every point. Replacing the standard basis { ∂ ∂xk } by general vector fields {Xk}, we form the multilinear operator T ({fj}) = det(Xkfj). We consider the following question: Under what conditions on the Xk’s do we have that T maps products of Lebesgue spaces into some Hardy space H as before? The purpose of this paper is to give a satisfactory answer to the question posed above. If the Xk are taken from the usual basis of the Heisenberg group in R, Rochberg and the author [GR] prove that the corresponding T maps into the group space H. In this work we show that if the vector fields Xk satisfy Hormander’s condition, then the corresponding T maps suitable products of Lebesgue spaces into the local anisotropic Hardy space H with respect to the metric associated with the vector fields defined by Nagel, Stein, and Wainger [NSW]. Precise statements of results are given in Theorems A and B. Research partially supported by the NSF and the University of Missouri Research Board 1991 Mathematics Classification. Primary 42B30 Typeset by AMS-TEX 1 Suppose that S = {Y1, Y2, . . . , Yl} is a set of smooth vector fields defined on a bounded open connected subset Ω of R for some n ≥ 2. Assume that S is a Hormander system. This means that there exists an integer s such that the vector fields Y1, . . . , Yl together with their commutators of order at most s span the tangent space of Ω at every point x. [NSW] define a (quasi)metric ρ on Ω by setting ρ(x, y) = inf{t : ∃ piecewise smooth curve γ : [0, t] → R, γ(0) = x, γ(t) = y and γ′(s) = ∑l j=1 βj(s)Yj(γ(s)) with ∑l j=1 ∣∣βj(s)∣∣2 ≤ 1 for all s ∈ [0, t]}, for all x, y in Ω. Intuitively, ρ(x, y) is the least time taken to move from x to y along a path pointing in the directions of the Yj ’s. For x ∈ Ω and δ > 0, let B(x, δ) = {y ∈ Ω : ρ(x, y) 0 small enough, the push-forward of φδ by any of the coordinate maps constructed in [NSW] gives a smooth bump ψ0 δ supported in the ball B(x0, δ). One can check that for any compact subset K of Ω and for all j = 1, . . . , l and x ∈ K (0.2) |ψ0 δ (x)| ≤ CK |B(x0, δ)|−1 and |Yj(ψ δ )(x)| ≤ CKδ|B(x0, δ)|−1 when 0 0, let (0.3) (Mδf)(x0) = sup 0 0 such that Mδ0(K)f is in L(K). We define the Hardy-1 space norm of f on K by setting (0.4) ‖f‖H1(K) = ‖Mδ0(K)f‖L1(K). 2 For a C map F = (f1, . . . , fl) : Ω → R, define (0.5) Jac(F ) = Jac{Y1,...,Yl}(F ) = det   Y1f1 Y1f2 . . . Y1fl Y2f1 Y2f2 . . . Y2fl .. .. .. Ylf1 Ylf2 . . . Ylfl   . Our results are: Theorem A. For 1 ≤ j ≤ l, let 1 0 such that for all C functions F = (f1, . . . , fl) : Ω → R, we have: (0.6) ‖Jac{Y1,...,Yl}(F )‖H1(K) ≤ CK l ∏