inequality for the Heckman–Opdam transforms, corresponding to different weights. The last section briefly outlines a generalization of the Eguchi–Kumahara result for the Cartan motion groups. On can introduce a ‘flat’ Heckman–Opdam transform $\mathcal{F}_0$ in analogy with generalized Bessel transform on the flat space $G_0/K$, and we obtain Hardy–Littlewood inequalities for $\mathcal{F}_0$ as well. This involves generalized Bessel-type functions associated with root systems that were already considered by Opdam in [@Opdam-Bessel Section 6]. The connection to spherical functions on the Cartan motion group was explicitly indicated in [@Opdam-Bessel Remark 6.12] and later established formally in [@deJeu-PW].

Harmonic analysis in root systems {#sec.root1}
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Let ${\mathfrak{a}}$ be an $n$-dimensional real Euclidean vector space with inner product $\langle\cdot,\cdot\rangle$ and let ${\mathfrak{a}}^*$ denote the linear dual of ${\mathfrak{a}}$. For $\lambda\in{\mathfrak{a}}^*$ let $x_\lambda$ be the unique vector in ${\mathfrak{a}}$ such that $\lambda(x)=\langle x,x_\lambda\rangle$ for every $x\in{\mathfrak{a}}$. Define an inner product on ${\mathfrak{a}}^*$ via $\langle \lambda,\mu\rangle=\langle x_\lambda,x_\mu\rangle$, and let ${\mathfrak{a}}_{\mathbb{C}}$ and ${\mathfrak{a}}_{\mathbb{C}}^*$ denote the complexifications of ${\mathfrak{a}}$ and ${\mathfrak{a}}^*$. The inner products on ${\mathfrak{a}}$ and ${\mathfrak{a}}^*$ extend by ${\mathbb{C}}$-linearity to inner products on ${\mathfrak{a}}_{\mathbb{C}}$ and ${\mathfrak{a}}_{\mathbb{C}}^*$ that will denoted by the same symbol. Set $\lambda_\alpha=\frac{\langle\lambda,\alpha\rangle}{\langle\alpha,\alpha\rangle}$ and $|x|=\langle x,x\rangle^{1/2}$ for $x\in{\mathfrak{a}}$.

Let $\Sigma$ be a root system 