Schatten Class Estimates for Paraproducts in Multi-parameter setting and applications

Let $\Pi _{b}$ be a bounded $n$ parameter paraproduct with symbol $b$. We demonstrate that this operator is in the Schatten class $\mathbb S_ p$, $0<p<\infty$, if the symbol is in the $n$ parameter Besov space $B_p$. Our result covers both the dyadic and continuous version of the paraproducts in the multiparameter setting. A direct method of proof is given and an application to the commutator of the double Hilbert transforms is provided

Besov Space, Schatten Classes And Commutators Of Riesz Transforms Associated With The Neumann Laplacian

This article provides a deeper study of the Riesz transform commutators associated with the Neumann Laplacian operator $\Delta_N$ on $\mathbb R^n$. Along the line of singular value estimates for Riesz transform commutators established by Janson--Wolff and Rochberg--Semmes, we establish a full range of Schatten-$p$ class characterization for these commutators

Expression and Regulation of Osteoprotegerin in Adipose Tissue

PURPOSE: Osteoprotegerin (OPG), a potent inhibitor of osteoclastic bone resorption, has a variety of biological functions that include anti-inflammatory effects. Adipocytes and osteoblasts share a common origin, and the formation of new blood vessels often precedes adipogenesis in developing adipose tissue microvasculature. We examined whether OPG is secreted from adipocytes, therefore contributing to the prevention of neovascularization and protecting the vessels from intimal inflammation and medial calcification. MATERIALS AND METHODS: The mRNA expression of OPG and receptor activator of NF-kappaB ligand (RANKL) was measured in differentiated 3T3L1 adipocytes and adipose tissues. RESULTS: OPG mRNA expression increased with the differentiation of 3T3L1 adipocytes, while RANKL expression was not significantly altered. OPG mRNA was expressed at higher levels in white adipose tissue than in brown adipose tissue and was most abundant in the epididymal portion. In differentiated 3T3L1 adipocytes, Rosiglitazone and insulin reduced the OPG/RANKL expression ratio in a dose- and time- dependent manner. In contrast, tumor necrosis factor-alpha (TNF-alpha) increased the expression of both OPG and RANKL in a time-dependent manner. The OPG/RANKL ratio was at a maximum two hours after TNF-alpha treatment and then returned to control levels. Furthermore, OPG was abundantly secreted into the media after transfection of OPG cDNA with Phi C31 integrase into 3T3L1 cells. CONCLUSION: Our results indicate that OPG mRNA is expressed and regulated in the adipose tissue. Considering the role of OPG in obesity-associated inflammatory changes in adipose tissue and vessels, we speculate that OPG may have both a protective function against inflammation and anti-angiogenic effects on adipose tissue.ope

Measurement of the mass difference and the binding energy of the hypertriton and antihypertriton

According to the CPT theorem, which states that the combined operation of charge conjugation, parity transformation and time reversal must be conserved, particles and their antiparticles should have the same mass and lifetime but opposite charge and magnetic moment. Here, we test CPT symmetry in a nucleus containing a strange quark, more specifically in the hypertriton. This hypernucleus is the lightest one yet discovered and consists of a proton, a neutron, and a $\Lambda$ hyperon. With data recorded by the STAR detector{\cite{TPC,HFT,TOF}} at the Relativistic Heavy Ion Collider, we measure the $\Lambda$ hyperon binding energy $B_{\Lambda}$ for the hypertriton, and find that it differs from the widely used value{\cite{B_1973}} and from predictions{\cite{2019_weak, 1995_weak, 2002_weak, 2014_weak}}, where the hypertriton is treated as a weakly bound system. Our results place stringent constraints on the hyperon-nucleon interaction{\cite{Hammer2002, STAR-antiH3L}}, and have implications for understanding neutron star interiors, where strange matter may be present{\cite{Chatterjee2016}}. A precise comparison of the masses of the hypertriton and the antihypertriton allows us to test CPT symmetry in a nucleus with strangeness for the first time, and we observe no deviation from the expected exact symmetry