Composition operators on W 1 X are necessarily induced by quasiconformal mappings

Let Ω ? ? ⁿ be an open set and X(Ω) be any rearrangement invariant function space close to L ^q (Ω), i.e. X has the q-scaling property. We prove that each homeomorphism f which induces the composition operator u ? u ? f from W ¹ X to W ¹ X is necessarily a q-quasiconformal mapping. We also give some new results for the sufficiency of this condition for the composition operator.     

Novel 2-phosphabicyclo[2.2.2]oct-5-ene derivatives and their use in phosphinylations

The [4 + 2] cycloaddition of the double-bond isomers ( A and B ) of dihydrophosphinine oxide 1 afforded novel phosphabicyclo[2.2.2]oct-5-ene derivates ( 2–4 ), formation of which was justified by PM3 semiempirical calculations. The compounds of dimer type ( 2–4 ) were utilized in the UV light-mediated fragmentation-related phosphinylation of nucleophiles, especially in that of alcohols. To explore the role of structural modifications on the fragmentation ability, disulfide 5 , phosphabicyclooctane 7 obtained by the hydrogenation of 2 , and the adduct of dihydrophosphinine oxide 1 with benzoquinone ( 7 ) were also synthesized and tested in fragmentation. © 2004 Wiley Periodicals, Inc. Heteroatom Chem 15:97–106, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/hc.10221