Equality in Minkowski inequality and a characterization of Lp-norm

Abstract

For a measure space (Q, S,/z) denote by S — S(£l, S,/z) the linear space of all p-integrable simple functions x : Q R, and by S+ = S+(£l, the set of all nonnegative x e S(Sl, It is easy to see that for an arbitrary bijection <j>: (0, oo) >-> (0, oo) the functional : S [0, oo) given by P<t>(x) ■= </> 1 ( 5 </> ° |z| d/z fi(r) ' .0 if /z(Q(a:)) > 0 if /z(Q(a:)) = 0 x £ S(£l, S,/z), where Q(a:) := {w G Q : ir(u>) 0}, is correctly defined i.e. for every x G S(Q, S, /z) the function </>o|x| is S-measurable and the integral <£o|x|dp is finite (cf. [4], Remark 5) (Fragment tekstu).