Trace norm estimates for products of integral operators and diffusion semigroups

We give trace norm estimates for products of integral operators and for diffusion semigroups. These are applied to differences of heat semigroups. A natural example of an integral operator with finite trace which is not trace class is given.     

Regioselective palladium-catalyzed alkylation of allylic halides with benzylic grignard reagents. Two-step synthesis of abietane terpenes and tetracyclic polyprenoid compounds

A highly regioselective palladium-catalyzed alpha-alkylation of allylic bromides 1a,c-e and chloride 1b with substituted and unsubstituted benzylic Grignard reagents is reported. The resulting all-trans polyenehomobenzene derivatives were obtained in excellent yields and regioselectivity. These products were easily converted to abietane-type diterpenes (10-12) and tetracyclic polyprenoid compounds (13, 14) through a Lewis acid-promoted cascade polyene cyclization reaction.     

Novel Variant of the Tricyclooctanone Approach to Cyclopentanoid Natural Products. Synthesis of (±)-Coriolin. Preliminary Communication

A total synthesis of the antitumor sesquiterpene coriolin ( 9 ; racemic) in 11 steps from 3,3,6-trimethylbicyclo[2.2.2]oct-7-ene-2,5-dione ( 2a / 2b ) is described (yield 2a / 2b → 8: 28%). The sequence is unprecedentedly short and avoids difficult separation problems. The key step in the scheme is a novel facet of oxadi-π-methane photochemistry, i.e., the steering by subtle steric effects of the β,γ-unsaturated ?-diketone to undergo a regioselective photorearrangement involving one β,γ-enone partial chromophore. Furthermore, the overall phototransformation, which can be carried out at unusually high concentrations (≥20% solutions), involves also a Norrish type I process equilibrating the two epimeric starting enediones 2a and 2b in favour of the desired stereoisomer.