New Abstract Hardy Spaces

The aim of this paper is to propose an abstract construction of spaces which keep the main properties of the (already known) Hardy spaces H^1. We construct spaces through an atomic (or molecular) decomposition. We prove some results about continuity from these spaces into L^1 and some results about interpolation between these spaces and the Lebesgue spaces. We also obtain some results on weighted norm inequalities. Then we apply this abstract theory to the L^p maximal regularity. Finally we present partial results in order to understand a characterization of the duals of Hardy spaces.Comment: 53 page

Maximal regularity and Hardy spaces

In this work, we consider the Cauchy problem for $u' - Au = f$ with $A$ the Laplacian operator on some Riemannian manifolds or a sublapacian on some Lie groups or some second order elliptic operators on a domain. We show the boundedness of the operator of maximal regularity $f\mapsto Au$ and its adjoint on appropriate Hardy spaces which we define and study for this purpose. As a consequence we reobtain the maximal $L^q$ regularity on $L^p$ spaces for $p,q$ between 1 and $\infty$.Comment: 27 page

Addendum to "Maximal regularity and Hardy spaces"

We correct an inaccuracy in a previous article [Auscher, Pascal; Bernicot, Fr\'ed\'eric; Zhao, Jiman. Maximal regularity and Hardy spaces. Collect. Math. 59 (2008), no. 1, 103-127.

Abstract framework for John-Nirenberg inequalities and applications to Hardy spaces

22 pagesInternational audienceIn this paper, we develop an abstract framework for John-Nirenberg inequalities associated to BMO-type spaces. This work can be seen as the sequel of [5], where the authors introduced a very general framework for atomic and molecular Hardy spaces. Moreover, we show that our assumptions allow us to recover some already known John-Nirenberg inequalities. We give applications to the atomic Hardy spaces too

Real Clifford Windowed Fourier Transform

We study the windowed Fourier transform in the framework of Clifford analysis, which\ud we call the Clifford windowed Fourier transform (CWFT). Based on the spectral representation of the\ud Clifford Fourier transform (CFT), we derive several important properties such as shift, modulation,\ud reconstruction formula, orthogonality relation, isometry, and reproducing kernel. We also present an\ud example to show the differences between the classical windowed Fourier transform (WFT) and the\ud CWFT. Finally, as an application we establish a Heisenberg type uncertainty principle for the CWF

Clifford Algebra-Valued Wavelet Transform on Multivector Fields

This paper presents a construction of the n = 2 (mod 4) Clifford\ud algebra Cln,0-valued admissible wavelet transform using the admissible\ud similitude group SIM(n), a subgroup of the affine group of Rn. We\ud express the admissibility condition in terms of the Cln,0 Clifford Fourier\ud transform (CFT). We show that its fundamental properties such as inner\ud product, norm relation, and inversion formula can be established\ud whenever the Clifford admissible wavelet satisfies a particular admissibility\ud condition. As an application we derive a Heisenberg type uncertainty\ud principle for the Clifford algebra Cln,0-valued admissible wavelet\ud transform. Finally, we provide some basic examples of these extended\ud wavelets such as Clifford Morlet wavelets and Clifford Hermite wavelets