Molecular Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

Let p→∈(0,∞)n be an exponent vector and A be a general expansive matrix on Rn. Let HAp→(Rn) be the anisotropic mixed-norm Hardy spaces associated with A defined via the non-tangential grand maximal function. In this article, using the known atomic characterization of HAp→(Rn), the authors characterize this Hardy space via molecules with the best possible known decay. As an application, the authors establish a criterion on the boundedness of linear operators from HAp→(Rn) to itself, which is used to explore the boundedness of anisotropic Calderón–Zygmund operators on HAp→(Rn). In addition, the boundedness of anisotropic Calderón–Zygmund operators from HAp→(Rn) to the mixed-norm Lebesgue space Lp→(Rn) is also presented. The obtained boundedness of these operators positively answers a question mentioned by Cleanthous et al. All of these results are new, even for isotropic mixed-norm Hardy spaces on Rn.

Smoothness of refinable function vectors on ℝn

Let [Formula: see text] be a dilation matrix, an [Formula: see text] expansive matrix that maps [Formula: see text] into itself. Let [Formula: see text] be a finite subset of [Formula: see text] and for [Formula: see text] let [Formula: see text] be [Formula: see text] complex matrices. The refinement equation corresponding to [Formula: see text] and [Formula: see text] is [Formula: see text] A solution [Formula: see text] if one exists, is called a refinable vector function or a vector scaling function of multiplicity [Formula: see text] This paper characterizes the higher-order smoothness of compactly supported solutions of the refinement equation, in terms of the [Formula: see text]-norm joint spectral radius of a finite set of finite matrices determined by the coefficients [Formula: see text]

Dimension Functions of Self-Affine Scaling Sets

In this paper, the dimension function of a self-affine generalized scaling set associated with an n×n integral expansive dilation A is studied. More specifically, we consider the dimension function of an A-dilation generalized scaling set K assuming that K is a self-affine tile satisfying BK = (K+d1)[ (K + d2), where B = At , A is an n×n integral expansive matrix with |det A| = 2, and d1, d2 ∊ ℝn. We show that the dimension function of K must be constant if either n = 1 or 2 or one of the digits is 0, and that it is bounded by 2|K| for any n.

The Existences of Frame Wavelet Set

In this paper, Let be a real expansive matrix, it mainly discusses the existences of frame wavelet set, we discuss the characterization of frame wavelet sets in, and several examples are presented, in order to deepen the understanding of frame wavelet set, which gives the two related theorems; we try to use an equivalent condition to describe frame scale sets, and give an equivalent description about a normalized frame scale set.

Anisotropic Weak Hardy Spaces and Wavelets

We characterize the anisotropic weak Hardy spacesHAp,∞(ℝn)associated with an expansive matrixAby using square functions involving wavelets coefficients.

GMRA-BASED CONSTRUCTION OF FRAMELETS IN REDUCING SUBSPACES OF L2(ℝd)

This paper develops GMRA-based construction procedures of Parseval framelets in the setting of reducing subspaces of L2(ℝd). A unitary extension principle is established; in particular, for a general expansive matrix A with | det A| = 2, an explicit construction of Parseval framelets is obtained. Some examples are also provided to illustrate the generality of our theory.

Characterization of Generators for Multiresolution Analyses with Composite Dilations

This paper introduces multiresolution analyses with composite dilations (AB-MRAs) and addresses frame multiresolution analyses with composite dilations in the setting of reducing subspaces ofL2(ℝn)(AB-RMRAs). We prove that an AB-MRA can induce an AB-RMRA on a given reducing subspaceL2(S)∨. For a general expansive matrix, we obtain the characterizations for a scaling function to generate an AB-RMRA, and the main theorems generalize the classical results. Finally, some examples are provided to illustrate the general theory.