Positions of convex bodies associated to extremal problems and isotropic measures

We show that there are close relations between extremal problems in dual Brunn-Minkowski theory and isotropic-type properties for some Borel measures on the sphere. The methods we use allow us to obtain similar results in the context of Firey-Brunn-Minkowski theory. We also study reverse inequalities for dual mixed volumes which are related with classical positions, such as ?-position or isotropic position.     

An integral inequality concerning isotropic measures on the unit circle

We prove a trigonometric integral inequality involving isotropic measures in the plane which can be applied to characterize the solution of extremal problems of convex bodies in in terms of properties of measures. The methods used include new estimates of hypergeometric functions and some cancellation lemmas.     

Rearrangement of Hardy-Littlewood maximal functions in Lorentz spaces

For the classical Hardy-Littlewood maximal function , a well known and important estimate due to Herz and Stein gives the equivalence . In the present note, we study the validity of analogous estimates for maximal operators of the form

where denotes the Lorentz space -norm.

     

On best interpolation in Orlicz spaces

For given data, an interpolant is sought, so that a certain convex functional defined by a Young's function in the corresponding Orlicz space is minimized. The freedom gained in considering more general kind of spaces can be used for selecting the interpolants in an adequate class of functions. Both authors were partically supported by DGICYT:PS90/0120     

l q -subspaces of stablep-Banach spaces, 0<p≦1

Each infinite dimensional subspace ofL ^p (0<p≦1) is shown to contain a copy of somel ^q p≦q<∞, using arguments similar to the ones that appearin Krivine and Maurey's paper concerning stable Banach spaces. Generally speaking, ifX is a stable infinite dimensionalp-Banach space, with 0<p≦1, then, there exists aq(p≦q<∞), such that,X contains (1+ε)-isomorphic copies ofl ^q , for all ε>0. Moreover, it is possible to prove that if a stablep-Banach space, 0<p≦1, contains an isomorphic copy ofl ^q,p≦q<∞, then, it also contains (1+ε) -isomorphic copies ofl ^q , for all ε>0.     

Elementary reverse Hölder type inequalities with application to operator interpolation theory

We give a very elementary proof of the reverse Hölder type inequality for the classes of weights which characterize the boundedness on of the Hardy operator for nonincreasing functions. The same technique is applied to Calderón operator involved in the theory of interpolation for general Lorentz spaces. This allows us to obtain further consequences for intermediate interpolation spaces.

     

Insights into CO/styrene copolymerization by using Pd(II) catalysts containing modular pyridine-imidazoline ligands

Continuing our studies into the effect that N-N' ligands have on CO/styrene copolymerization, we prepared new C(1)-symmetrical pyridine-imidazoline ligands with 4',5'-cis stereochemistry in the imidazoline ring (5) and 4',5'-trans stereochemistry (6-10) and compared them with our previously reported ligands (1-4). Their coordination to neutral methylpalladium(II) (5 a-10 a) and cationic complexes (5 b-10 b), investigated in solution by NMR spectroscopy, indicates that both the electronic and steric properties of the imidazolines determine the stereochemistry of the palladium complexes. The crystal structures of two neutral palladium precursors [Pd(Me)(2-n)Cl(n)(N-N')] (n=1 for 8 a; n=2 for 9 a') show that the Pd-N coordination distances and the geometrical distortions in the imidazoline ring depend on the electronic nature of the substituents in the imidazoline fragment. Density functional calculations performed on selected neutral and cationic palladium complexes compare well with NMR and X-ray data. The calculations also account for the formation of only one or two stereoisomers of the cationic complexes. The performance of the cationic complexes as catalyst precursors in CO/4-tert-butylstyrene copolymerization under mild pressures and temperatures was analyzed in terms of the productivity and degree of stereoregularity of the polyketones obtained. Insertion of CO into the Pd-Me bond, which was monitored by multinuclear NMR spectroscopy, shows that the N ligand influences the stereochemistry of the acyl species formed.     

Assessing the suitability of 1,2,3-triazole linkers for covalent immobilization of chiral ligands: application to enantioselective phenylation of aldehydes

Alkynyl-functionalized amino alcohols have been covalently supported on azidomethylpolystyrene resins with different levels of functionalization through Cu(I)-catalyzed 1,3-dipolar cycloadditions ("click chemistry"). The resulting 1,2,3-triazole-substituted resins, characterized by different levels of ligand loading and, depending on the nature of the alkynyl-functionalized amino alcohol, the presence of a one-carbon, four-carbon, or eight-carbon linear spacer, have been tested as catalysts in the enantioselective phenyl transfer from zinc to aldehydes. High catalytic activities and enantioselectivities (up to 82% ee) have been recorded. The influence of structural characteristics of the resin on enantioselectivity are discussed, and the limitations in enantiocontrol inherent to the use of a 1,2,3-triazole linker have been rationalized with the help of DFT calculations on model systems.