Inequalities involving classical operators of harmonic analysis, such as maximal
functions, fractional integrals and singular integrals of convolutive type have been
extensively investigated in various function spaces. Results on weak and strong type
inequalities for operators of this kind in Lebesgue spaces are classical and can be
found for example in [4, 20, 24]. Generalizations of these results to Zygmund spaces
are presented in [4]. An exhaustive treatment of the problem of boundedness of such
operators in Lorentz and Lorentz–Zygmund spaces is given in [3]. See also [8, 9] for
further extensions in the framework of generalized Lorentz–Zygmund spaces.
As far as Orlicz spaces are concerned, a characterization of Young functions A
having the property that the Hardy–Littlewood maximal operator or the Hilbert and
Riesz transforms are of weak or strong type from the Orlicz space LA into itself is
known (see for example [13]). In [17, 23] conditions on Young functions A and B are
given for the fractional integral operator to be bounded from LA into LB under some
restrictions involving the growths and certain monotonicity properties of A and B.
The main purpose of this paper is to find necessary and sufficient conditions on
general Young functions A and B ensuring that the above-mentioned operators are
of weak or strong type from LA into LB. Our results for (fractional) maximal
operators are presented in Section 2, while Section 3 deals with fractional and singular
integrals. In particular, we re-cover a result concerning the standard Hardy–Littlewood
maximal operator which has recently been proved in [2, 11, 12]. Finally,
in Section 4, the resolvent operator of some differential problems is taken into
account and a priori bounds for Orlicz norms of solutions to elliptic boundary value
problems in terms of Orlicz norms of the data are established. Let us mention that
part of the results of the present paper were announced in [6].