supremum operator on its associate space that leads to a self-explanatory characterization of the above-mentioned extra condition. This part of the paper is one of the most innovative ones.

We finally consider two other classical operators of harmonic analysis, namely the Hilbert transform and the Riesz potential. The importance of these operators is very well known, and their properties have been deeply studied. Our contribution is the characterization of the optimality of the spaces involved. In case of the Hilbert transform we use the Stieltjes transform as the appropriate tool and obtain characterizations for it as well.

For each of the operators considered, we are also able to nail down the optimal domain partner when the range space is fixed, this task being in general slightly simpler than the converse one. To establish all this, a combination of new techniques developed here with those from [@CPS; @EKP] and [@T2] is used.

We illustrate the results obtained with variety of nontrivial examples. For instance, we recover the well-known fact that $$\begin{aligned}
M\colon L(\log L)^{\alpha }(Q)\to L(\log L)^{\alpha -1}(Q)\end{aligned}$$ when $\alpha \geq 1$, $Q\subset \mathbb{R}^{n}$ is a cube of finite measure and $L(\log L)^{\alpha }(Q)$ is the classical Zygmund class defined as 