Marcin Bownik

Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces

Weighted anisotropic Triebel-Lizorkin spaces are introduced and studied with the use of discrete wavelet transforms. This study extends the isotropic methods of dyadic φ-transforms of Frazier and Jawerth (1985, 1989) to non-isotropic settings associated with general expansive matrix dilations and A ∞ weights. In close analogy with the isotropic theory, we show that weighted anisotropic Triebel-Lizorkin spaces are characterized by the magnitude of the φ-transforms in appropriate sequence spaces. We also introduce non-isotropic analogues of the class of almost diagonal operators and we obtain atomic and molecular decompositions of these spaces, thus extending isotropic results of Frazier and Jawerth.

Boundedness of operators on Hardy spaces via atomic decompositions

An example of a linear functional defined on a dense subspace of the Hardy space H 1 (R n ) is constructed. It is shown that despite the fact that this functional is uniformly bounded on all atoms, it does not extend to a bounded functional on the whole H 1 . Therefore, this shows that in general it is not enough to verify that an operator or a functional is bounded on atoms to conclude that it extends boundedly to the whole space. The construction is based on the fact due to Y. Meyer which states that quasi-norms corresponding to finite and infinite atomic decompositions in H p , 0 < p < 1, are not equivalent.

Anisotropic Triebel-Lizorkin Spaces with Doubling Measures

We introduce and study anisotropic Triebel-Lizorkin spaces associated with general expansive dilations and doubling measures on ℝn with the use of wavelet transforms. This work generalizes the isotropic methods of dyadic ϕ-transforms of Frazier and Jawerth to nonisotropic settings.We extend results involving boundedness of wavelet transforms, almost diagonality, smooth atomic and molecular decompositions to the setting of doubling measures. We also develop localization techniques in the endpoint case of p = ∞, where the usual definition of Triebel-Lizorkin spaces is replaced by its localized version. Finally, we establish nonsmooth atomic decompositions in the range of 0 < p ≤ 1, which is analogous to the usual Hardy space atomic decompositions.

Tight Frames of Multidimensional Wavelets.

AbstractIn this paper we deal with multidimensional wavelets arising from a multiresolution analysis with an arbitrary dilation matrix A, namely we have scaling equations

Riesz wavelets and generalized multiresolution analyses

\varphi ^s (x) = \sum\limits_{k \in \mathbb{Z}^n } {h_k^s \sqrt {|\det A|} \varphi ^1 } (Ax - k) for s = 1, \ldots ,q,

The construction ofr-regular wavelets for arbitrary dilations

where ϕ1 is a scaling function for this multiresolution and ϕ2, …, ϕq (q=|det A |) are wavelets. Orthogonality conditions for ϕ1, …, ϕq naturally impose constraints on the scaling coefficients

Characterization of sequences of frame norms

\{ h_k^s \} _{k \in \mathbb{Z}^n }^{s = 1, \ldots ,q}