Boundedness of classical operators on rearrangement-invariant spaces.

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.