Anisotropic Singular Integrals in Product Spaces

Let $A_i$ for $i=1, 2$ be an expansive dilation, respectively, on ${\mathbb R}^n$ and ${\mathbb R}^m$ and $\vec A\equiv(A_1, A_2)$. Denote by {\mathcal A}_\infty(\rnm; \vec A) the class of Muckenhoupt weights associated with $\vec A$. The authors introduce a class of anisotropic singular integrals on $\mathbb R^n\times\mathbb R^m$, whose kernels are adapted to $\vec A$ in the sense of Bownik and have vanishing moments defined via bump functions in the sense of Stein. Then the authors establish the boundedness of these anisotropic singular integrals on $L^q_w(\mathbb R^n\times\mathbb R^m)$ with $q\in(1, \infty)$ and $w\in\mathcal A_q(\mathbb R^n\times\mathbb R^m; \vec A)$ or on $H^p_w(\mathbb R^n\times\mathbb R^m; \vec A)$ with $p\in(0, 1]$ and $w\in\mathcal A_\infty(\mathbb R^n \times\mathbb R^m; \vec A)$. These results are also new even when $w=1$.Comment: Sci. China Math., to appea

Local Hardy Spaces of Musielak-Orlicz Type and Their Applications

Let \phi: \mathbb{R}^n\times[0,\fz)\rightarrow[0,\fz) be a function such that $\phi(x,\cdot)$ is an Orlicz function and $\phi(\cdot,t)\in A^{\mathop\mathrm{loc}}_{\infty}(\mathbb{R}^n)$ (the class of local weights introduced by V. S. Rychkov). In this paper, the authors introduce a local Hardy space $h_{\phi}(\mathbb{R}^n)$ of Musielak-Orlicz type by the local grand maximal function, and a local $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{bmo}_{\phi}(\mathbb{R}^n)$ which is further proved to be the dual space of $h_{\phi}(\mathbb{R}^n)$. As an application, the authors prove that the class of pointwise multipliers for the local $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{bmo}^{\phi}(\mathbb{R}^n)$, characterized by E. Nakai and K. Yabuta, is just the dual of L^1(\rn)+h_{\Phi_0}(\mathbb{R}^n), where $\phi$ is an increasing function on $(0,\infty)$ satisfying some additional growth conditions and $\Phi_0$ a Musielak-Orlicz function induced by $\phi$. Characterizations of $h_{\phi}(\mathbb{R}^n)$, including the atoms, the local vertical and the local nontangential maximal functions, are presented. Using the atomic characterization, the authors prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of $h_{\phi}(\mathbb{R}^n)$, from which, the authors further deduce some criterions for the boundedness on $h_{\phi}(\mathbb{R}^n)$ of some sublinear operators. Finally, the authors show that the local Riesz transforms and some pseudo-differential operators are bounded on $h_{\phi}(\mathbb{R}^n)$.Comment: Sci. China Math. (to appear

Frame Theory for Signal Processing in Psychoacoustics

This review chapter aims to strengthen the link between frame theory and signal processing tasks in psychoacoustics. On the one side, the basic concepts of frame theory are presented and some proofs are provided to explain those concepts in some detail. The goal is to reveal to hearing scientists how this mathematical theory could be relevant for their research. In particular, we focus on frame theory in a filter bank approach, which is probably the most relevant view-point for audio signal processing. On the other side, basic psychoacoustic concepts are presented to stimulate mathematicians to apply their knowledge in this field

Biomarkers of disease differentiation: HCV recurrence versus acute cellular rejection

The wound-healing process induced by chronic hepatitis C virus (HCV) infection triggers liver damage characterized by fibrosis development and finally cirrhosis. Liver Transplantation (LT) is the optimal surgical treatment for HCV-cirrhotic patients at end-stage liver disease. However, acute cellular rejection (ACR) and HCV recurrence disease represent two devastating complications post-LT. The accurate differential diagnosis between both conditions is critical for treatment choice, and similar histological features represent a challenge for pathologists. Moreover, the HCV recurrence disease severity is highly variable post-LT. HCV recurrence disease progression is characterized by an accelerated fibrogenesis process, and almost 30% of those patients develop cirrhosis at 5-years of follow-up. Whole-genome gene expression (WGE) analyses through well-defined oligonucleotide microarray platforms represent a powerful tool for the molecular characterization of biological process. In the present manuscript, the utility of microarray technology is applied for the ACR and HCV-recurrence biological characterization in post-LT liver biopsy samples. Moreover, WGE analysis was performed to identify predictive biomarkers of HCV recurrence severity in formalin-fixed paraffin-embedded liver biopsies prospectively collected