Marcin Bownik

TL;DR: In this article, a simple characterization of frames and Riesz families generated by shifts of a countable set of generators in terms of their behavior on subspaces of l2(Zn) was given.

TL;DR: In this article, Calderon-Zygmund singular integral operators have been studied in the context of discrete groups of dilations, and they have been shown to be an unconditional basis for the anisotropic Hardy space H A.

TL;DR: Weighted anisotropic Triebel-Lizorkin spaces were introduced and studied with the use of discrete wavelet transforms in this article, where the authors extended the isotropic methods of dyadic φ-transforms of Frazier and Jawerth (1985) to non-isotropic settings associated with general expansive matrix dilations and A ∞ weights.

TL;DR: In this article, an example of a linear functional defined on a dense subspace of the Hardy space H 1 (R n ) is constructed, and 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 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.

TL;DR: In this paper, the theory of weighted anisotropic Besov spaces associated with general expansive matrix dilations and doubling measures with the use of discrete wavelet transforms was developed, and the isotropic Littlewood-Paley methods of dyadic φ-transforms of Frazier and Jawerth were extended to non-isotropic settings.