  We derive in this preprint the exact up to multiplicative constant
non-asymptotical estimates for the norms of some non-linear in general case
operators, for example, the so-called maximal functional operators, in two
probabilistic rearrangement invariant norm: exponential Orlicz and Grand
Lebesgue Spaces.
  We will use also the theory of the so-called Grand Lebesgue Spaces (GLS) of
measurable functions.
