L^p spaces for $1

L^p spaces for $1

Thank you. Actually, I am interested in "weak-Lp" or the Lorentz space L_{p,\infty}. Is there some equi-measurability condition similar to equi-integrability which would characterize a relatively weakly compact subset of L_{p,\infty}? In other words, I wonder if there is some explicit condition perhaps in terms of the distribution functions of the functions from the set. For example, if you have a compact convex subset S of L_{p,\infty} then I believe(I need to check the argument again) that there exists some f in S and some constant C so that D_g(t)\le C D_f(t) for all g in S. Is it possible that this goes the other way in a sense, i.e. if such a g and C exist for a convex set in L_{p,\infty} must the set be relatively compact or relatively weakly compact. Of course lets assume that 1

The answer is in my paper appeared this year at the first issue of St Peterssburg Math J

Thanks Yuri. This looks very interesting.

Dear Geoff DIESTEL

Since you are interested in the "weak-L^{p}" space (introduced by Marcinkiewicz, I suppose), which coincides with the Lorentz space L_{p,\infty} if 1

Also, look at the results of Joe Diestel. These are easy to find