Abstract. We develop a method to decompose functions with mean value zero that are defined on a (possibly unbounded) John domain into a countable sum of functions with mean value zero and support in cubes or balls. This method enables us to generalize results known for simple domains to the class of John domains and domains satisfying a certain chain condition. As applications we present the solvability of the divergence equation div u = f , the negative norm theorem, Korn's inequality, Poincaré's inequality and a localized version of the Fefferman-Stein inequality. We present the results for weighted Lebesgue spaces and Orlicz spaces.

Ajoene ((E,Z)-4,5,9-trithiadodeca-1,6,11-triene 9-oxide), a garlic-derived natural compound, is a covalent inhibitor as well as a substrate of human glutathione reductase (GR) and Trypanosoma cruzi trypanothione reductase (TR). The 2.1-A resolution crystal structure of GR inhibited by (E)-ajoene revealed a mixed disulfide between the active site Cys58 and the CH2=CH-CH2-SO-CH2-CH=CH-S moiety of ajoene. The modified enzyme has a markedly increased oxidase activity when compared to free GR. GR reduces (Z)-ajoene with a kcat/Km of 6.8 x 10(3) M-1 s-1 yielding 4,5,9-trithiadodeca-1, 6,11-triene (deoxyajoene) and 4,8,9,13-tetrathiahexadeca-1,6,10, 15-tetraene as stable reaction products. The reaction leads also to the formation of single-electron reduced products and concomitantly superoxide anion radicals as shown by coupling the reaction to the reduction of cytochrome c. The interactions between the flavoenzymes and ajoene are expected to increase the oxidative stress of the respective cell. The antiparasitic and cytostatic actions of ajoene may at least in part be due to the multiple effects on key enzymes of the antioxidant thiol metabolism.

We investigate very weak solutions to the stationary Stokes and Stokes resolvent problem in function spaces with Muckenhoupt weights. The notion used here is similar but even more general than the one used in Amann (Nonhomogeneous Navier-Stokes equations with integrable low-regularity data. Int. Math. Ser., pp. 1-26. Kluwer Academic/Plenum Publishing, New York, 2002) or Galdi et al. (Math. Ann. 331, 41-74, 2005). Consequently the class of solutions is enlarged. To describe boundary conditions we restrict ourselves to more regular data. We introduce a Banach space that admits a restriction operator and that contains the solutions according to such data.