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

Maximal and other operators in exponential Orlicz and Grand Lebesgue Spaces