such questions are of interest and importance. The optimality of r.i. spaces on which the Laplace transform $\mathcal{L}$ acts boundedly was studied in a recent paper [@BEP]. A special case of the results obtained is that if $p\in (1,\infty )$ and $q\in \lbrack 1,\infty ]$, then $\mathcal{L}$ maps the Lorentz space $L^{p,q}(0,\infty )$ boundedly into $L^{p^{\prime },q}(0,\infty )$, a fact which we denote by $
\mathcal{L}\colon L^{p,q}(0,\infty )\rightarrow L^{p^{\prime },q}(0,
\infty )$. Moreover, both the domain and target spaces are optimal: there is no r.i. space smaller than $L^{p^{\prime },q}(0,\infty )$ into which $\mathcal{L}$ maps $L^{p,q}(0,\infty )$, and there is no r.i. space larger than $L^{p,q}(0,\infty )$ mapped by $\mathcal{L}$ into $L^{p^{\prime },q}(0,\infty )$. Thus in particular $\mathcal{L}
\colon L^{p}(0,\infty )\rightarrow L^{p^{\prime },p}(0,\infty )$ and the spaces involved form an optimal pair; if $p>2$, there is no $q$ for which $\mathcal{L}\colon L^{p}(0,\infty )\rightarrow L^{q}(0,\infty )$.

In the present paper we discuss such problems for classical operators of great interest in analysis and its applications, namely the Hilbert and Stieltjes transforms, the Riesz potential and various versions of the maximal operator. The action of these operators on specific classes of function spaces has been extensively studied over several decades. Classical results 