  We study the behaviour on rearrangement-invariant spaces of such classical
operators of interest in harmonic analysis as the Hardy-Littlewood maximal
operator (including the fractional version), the Hilbert and Stieltjes
transforms, and the Riesz potential. The focus is on sharpness questions, and
we present characterisations of the optimal domain (or range) partner spaces
when the range (domain) is fixed. When a rearrangement-invariant partner space
exists at all, a complete characterisation of the situation is given. We
illustrate the results with a variety of examples of sharp particular results
involving customary function spaces.
