On Hermite pseudo-multipliers

In this article we deal with a variation of a theorem of Mauceri concerning the $L^p$ boundedness of operators $M$ which are known to be bounded on $L^2.$ We obtain sufficient conditions on the kernel of the operaor $M$ so that it satisfies weighted $L^p$ estimates. As an application we prove $L^p$ boundedness of Hermite pseudo-multipliers.Comment: 28 page

Sparse bounds for pseudo-multipliers associated to Grushin operators, I

In this article, we prove sharp quantitative weighted $L^p$-estimates for Grushin pseudo-multipliers satisfying H\"ormander's condition as an application of pointwise domination of Grushin pseudo-multipliers by appropriate sparse operators.Comment: We have removed the analysis pertaining to the family of operator-valued Fourier pseudo-multipliers from the original version, and we plan to submit those results elsewhere. Effectively, the introductory section is majorly revised, and as long as the mathematical results are concerned, this version is a proper subset of the first version, consisting of main results on Grushin pseudo-multiplier

p73: A Multifunctional Protein in Neurobiology

p73, a transcription factor of the p53 family, plays a key role in many biological processes including neuronal development. Indeed, mice deficient for both TAp73 and ΔNp73 isoforms display neuronal pathologies, including hydrocephalus and hippocampal dysgenesis, with defects in the CA1-CA3 pyramidal cell layers and the dentate gyrus. TAp73 expression increases in parallel with neuronal differentiation and its ectopic expression induces neurite outgrowth and expression of neuronal markers in neuroblastoma cell lines and neural stem cells, suggesting that it has a pro-differentiation role. In contrast, ΔNp73 shows a survival function in mature cortical neurons as selective ΔNp73 null mice have reduced cortical thickness. Recent evidence has also suggested that p73 isoforms are deregulated in neurodegenerative pathologies such as Alzheimer’s disease, with abnormal tau phosphorylation. Thus, in addition to its increasingly accepted contribution to tumorigenesis, the p73 subfamily also plays a role in neuronal development and neurodegeneration