Abstract. We show that if a linear operator T is bounded on weighted Lebesgue space L 2 (w) and obeys a linear bound with respect to the A 2 constant of the weight, then its commutator [b, T ] with a function b in BM O will obey a quadratic bound with respect to the A 2 constant of the weight. We also prove that the kth-order] will obey a bound that is a power (k + 1) of the A 2 constant of the weight. Sharp extrapolation provides corresponding L p (w) estimates. In particular these estimates hold for T any Calderón-Zygmund singular integral operator. The results are sharp in terms of the growth of the operator norm with respect to the A p constant of the weight for all 1 < p < ∞, all k, and all dimensions, as examples involving the Riesz transforms, power functions and power weights show.

No abstract

No abstract

We derive transformation formulas for the generalized polarization tensors under rigid motions and scaling in three dimensions, and use them to construct an infinite number of invariants under those transformations. These invariants can be used as shape descriptors for dictionary matching.

Three kinds of hydrophilic derivatives of b-sitosterol (HPS) were synthesized from poly(ethylene glycol) (PEG) with molecular weight 1000, 2000, and 4000 by two-step reactions. The chemical structure and the distribution of isomers in HPSs were investigated by 1 H-NMR and Maldi-Tof, respectively. The solubility of HPS in water increased as the molecular weight of the parent PEG increased. However, when the solubility of each HPS was calculated based on the weight ratio of the b-sitosterol moiety in HPS, HPS-1000 prepared from PEG 1000 showed the highest solubility. Both the CMC and surface tension of HPS surfactant system have been found to increase with an increase in molecular weight of the parent PEG. Dynamic surface tension measurement using a maximum bubble pressure tensiometer showed that much longer time was required to reach an equilibrium value presumably due to the lower mobility of a surfactant molecule with high molecular weight. The interfacial tensions measured between HPS system and n-decane at 25 C were in the same order of magnitude as those exhibited between micellar solutions and nonpolar hydrocarbon oils and also found to decrease with a decrease in molecular weight of the parent PEG. The most stable foams were observed with HPS-1000 system which is consistent with that of surface tension measurement where HPS-1000 system exhibited a lower surface tension value at CMC than other two systems. V C 2011 Wiley Periodicals, Inc. J Appl Polym Sci 125: [888][889][890][891][892][893][894][895] 2012