We extend the results in
[5] to several variables and to larger classes of domains. In particular it is
shown that if B is the unit ball in Cn and G ⊆ B is measurable then for any
p < 0
for all analytic functions f in Lp(B) if and only if there exist δ > 0 and
0 < r < 1 such that |G ∩ Q(B(0,r))|≥ δ|Q(B(0,r))| for all automorphisms
Q : B → B. This is actually done for weighted integrals in more general
domains. This is easily seen to be a criterion for the operation f → f|G to
have closed range. In addition, some partial results on the closed range of
f →{f(zn)} in some weighted lp spaces are obtained for sequences {zn} in certain
domains.
