.

It is the purpose of the present paper to obtain analogues of and for the Heckman–Opdam transform associated to a triple $({\mathfrak{a}},\Sigma,m)$, where ${\mathfrak{a}}$ is an Euclidean $n$-dimensional vector space, $\Sigma$ a root system in ${\mathfrak{a}}^*$ and $m$ a positive multiplicity function. In order to place the contributions of the present paper in perspective, the reader is reminded that some classical aspects of the $L^2$-theory for hypergeometric Fourier analysis in root systems (that is, Plancherel and Paley–Wiener theorems and an inversion formula) was already obtained in [@Opdam-acta], whereas the $L^p$-analysis is much more recent. As far as we can ascertain, the first decisive contribution was given in the recent publication [@Narayanan-Pasquale-Pusti], and the results we obtain should be seen as natural contributions to the general theme of classical harmonic analysis in a root system framework,

The details pertaining to harmonic analysis in root systems will be presented in section \[sec.root1\]. There are several standard references but we follow closely the presentation in [@Narayanan-Pasquale-Pusti] as far as the Heckman–Opdam theory is concerned. Section \[sec.root1\] also summarizes the interpolation theorems for Lorentz spaces. An immediate consequence is a generalized Hausdorff–Young inequality of Paley-type. Section \[sec.HY-ineqs\] presents several versions of the Hardy–Littlewood 