A two-weight Sobolev inequality for Carnot-Caratheodory spaces
RICERCHE DI MATEMATICA
Authors: Alberico, Angela; Di Gironimo, Patrizia
Abstract
Let X = {X-1, X-2,..., X-m} be a system of smooth vector fields in R-n satisfying the Hormander's finite rank condition. We prove the following Sobolev inequality with reciprocal weights in Carnot-Caratheodory space G associated to system X (1/integral(BR) K(x) dx integral(BR) vertical bar u vertical bar(t) K (x) dx)(1/t) <= C R (1/integral(BR) 1/K(x) dx integral(BR) vertical bar Xu vertical bar(2)/K(x) dx)(1/2), where Xu denotes the horizontal gradient of u with respect to X. We assume that the weight K belongs to Muckenhoupt's class A(2) and Gehring's class G tau, where tau is a suitable exponent related to the homogeneous dimension.
Spearmint Extract Containing Rosmarinic Acid Suppresses Amyloid Fibril Formation of Proteins Associated with Dementia
NUTRIENTS
Authors: Ogawa, Kenjirou; Ishii, Ayumi; Shindo, Aimi; Hongo, Kunihiro; Mizobata, Tomohiro; Sogon, Tetsuya; Kawata, Yasushi
Abstract
Neurological dementias such as Alzheimer's disease and Lewy body dementia are thought to be caused in part by the formation and deposition of characteristic insoluble fibrils of polypeptides such as amyloid beta (A beta), Tau, and/or alpha-synuclein (alpha Syn). In this context, it is critical to suppress and remove such aggregates in order to prevent and/or delay the progression of dementia in these ailments. In this report, we investigated the effects of spearmint extract (SME) and rosmarinic acid (RA; the major component of SME) on the amyloid fibril formation reactions of alpha Syn, A beta, and Tau proteins in vitro. SME or RA was added to soluble samples of each protein and the formation of fibrils was monitored by thioflavin T (ThioT) binding assays and transmission electron microscopy (TEM). We also evaluated whether preformed amyloid fibrils could be dissolved by the addition of RA. Our results reveal for the first time that SME and RA both suppress amyloid fibril formation, and that RA could disassemble preformed fibrils of alpha Syn, A beta, and Tau into non-toxic species. Our results suggest that SME and RA may potentially suppress amyloid fibrils implicated in the progression of Alzheimer's disease and Lewy body dementia in vivo, as well.
COA
Publication 
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