Uncertainty principles and characterization of the heat kernel for certain differential-reflection operators

International audienceWe prove various versions of uncertainty principles for a certain Fourier transform $\mathcal F_A.$ Here $A$ is a Chébli function (i.e. a Sturm-Liouville function with additional hypotheses). We mainly establish an analogue of Beurling's theorem, and its relatives such as theorems of type Gelfand-Shilov, Morgan's, Hardy's, and Cowling-Price, for $\mathcal F_A,$ and relating them to the characterization of the heat kernel corresponding to $\mathcal F_A.$ Heisenberg's and local uncertainty inequalities were also proved

LpL^p-Fourier analysis associated to a family of differential-reflection operators

In a previous paper we introduced a family of differential-reflection operators$\Lambda_{A, \varepsilon}$ acting on smooth functions defined on $\mathbb R,$ where the spectral problem for the operators $\Lambda_{A, \varepsilon}$ has been discussed. Here $A$ is a Sturm-Liouville function with additional hypotheses and $\varepsilon\in \mathbb R.$ Via the eigenfunctions of $\Lambda_{A,\varepsilon},$ we introduce in this paper a generalized Fourier transform $\mathcal F_{A,\varepsilon}.$ An $L^p$-harmonic analysis for $\mathcal F_{A,\varepsilon}$ is developed when $0< p \leq \frac{2}{1+\sqrt{1-\varepsilon^2}}$ and $-1\leq \varepsilon\leq 1.$ In particular, an $L^p$-Schwartz space isomorphism theorem for $\mathcal F_{A,\varepsilon}$ is proved

Lp harmonic analysis for differential-reflection operators

International audienceWe introduce and study differential-reflection operators$\Lambda_{A, \varepsilon}$ acting on smooth functions defined on $\mathbb R.$ Here $A$ is a Sturm-Liouville function with additional hypotheses and $\varepsilon\in \mathbb R.$ For special pairs $(A,\varepsilon),$ we recover Dunkl's, Heckman's and Cherednik's operators (in one dimension). As, by construction, the operators $\Lambda_{A, \varepsilon}$ are mixture of ${\rm d}/{\rm d}x$and reflection operators, we prove the existence of an operator $V_{A,\varepsilon}$ so that $\Lambda_{A, \varepsilon}\circ V_{A,\varepsilon}=V_{A,\varepsilon}\circ {\rm d}/{\rm d}x.$ The positivity of the intertwining operator $V_{A,\varepsilon}$ is also established. Via the eigenfunctions of $\Lambda_{A,\varepsilon},$ we introduce a generalized Fourier transform $\mathcal F_{A,\varepsilon}.$ For $-1\leq \varepsilon\leq 1$ and $0 < p \leq \frac{2}{1+\sqrt{1-\varepsilon^2}},$ we develop an $L^p$-Fourier analysis for $\mathcal F_{A,\varepsilon},$ and then we prove an $L^p$-Schwartz space isomorphism theorem for $\mathcal F_{A,\varepsilon}$.Details of this paper will be given in another article

Leptin decreases BC cell susceptibility to NK lysis via PGC1A pathway

International audienceLarge prospective studies established a link between obesity and breast cancer (BC) development. Yet, the mechanisms underlying this association are not fully understood. Among the diverse adipocytokine secreted by hypertrophic adipose tissue, leptin is emerging as a key candidate molecule linking obesity and cancer, since it promotes proliferation and invasiveness of tumors. However, the potential implication of leptin on tumor escape mechanisms remains unknown. This study aims to explore the effect of leptin on tumor resistance to NK lysis and the underlying mechanism. We found that leptin promotes both BC resistance to NK92-mediated lysis and β oxidation on MCF-7, by the up-regulation of a master regulator of mitochondrial biogenesis, the peroxisome proliferator activated receptor coactivator-1 α (PGC1A). Using adenoviral approaches, we show that acute elevation of PGC1A enhances the fatty acid oxidation pathway and decreases the susceptibility of BC cells to NK92-mediated lysis. Importantly, we identified the involvement of PGC1A and leptin in the regulation of hypoxia inducible factor-1 alpha (HIF1A) expression by tumor cells. We further demonstrate that basal BC cells MDA-MB-231 and BT-20 exhibit an increased PGC1A mRNA level and an enhanced oxidative phosphorylation activity; in comparison with luminal BC cells MCF7 and MDA-361, which are associated with more resistance NK92 lysis. Altogether, our results demonstrate for the first time how leptin could promote tumor resistance to immune attacks. Reagents blocking leptin or PGC1A activity might aid in developing new therapeutic strategies to limit tumor development in obese BC patients

Early onset breast cancer: differences in risk factors, tumor phenotype, and genotype between North African and South European women

International audiencePURPOSE: This report compares the risk factors, the tumor phenotypes, and the BRCA1/BRCA2 genotype of early onset breast cancer (EOBC) patients between Southern Europe and North Africa. METHODS: Four hundred and fifty six women with invasive EOBC (<=40~years) were prospectively included from four centers in France (n~=~270) and four centers in North Africa (Algeria, Egypt, Morocco, Tunisia; n~=~186). Life style, tumor phenotype, familial history, BRCA1/BRCA2 genotype were compared between the two populations. RESULTS: We found an older age at menarche, a higher number of childbearing, a more frequent breastfeeding, a higher body mass index, a lower use of oral contraceptives in North African women compared to French women. TNM stage at diagnosis was higher in North African women than in French women. North African women had a lower incidence of triple negative and proliferative (Ki 67 index~\textgreater~20%) tumors. There was a lower rate of BRCA1 mutation in North Africa (7 vs. 15%, P~=~0.02). Three putative BRCA1/2 founder mutations were identified in North Africa. CONCLUSIONS: In EOBC, we found~significant differences in risk factors, phenotype and a higher incidence of BRCA1 mutations in Southern Europe as compared to North Africa. The worst prognosis previously reported for EOBC in North Africa is more likely due to a higher stage at diagnosis than to a more aggressive phenotype, since triple negative tumors are more common in Southern Europe and advanced tumors in North Africa