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abstract: 'We establish Hardy–Littlewood inequalities for the Heckman–Opdam transform associated to a general root datum $({\mathfrak{a}},\Sigma,m)$ that generalizes an analogous result for the spherical Fourier transform on a Riemannian symmetric space of the non-compact type due to Eguchi and Kumahara. In particular we obtain a more precise Hausdorff–Young inequality that generalizes a recent result due to Narayanan, Pasquale, and Pusti.'
address: |
     Mathematisches Seminar\
    Chr.-Albrechts–Universität zu Kiel\
    Ludewig-Mey–Str. 4, DE-24098 Kiel\
    Germany 
author:
- Troels Roussau Johansen
title: 'Hardy–Littlewood inequalities for the Heckman–Opdam transform'
---

Introduction
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The classical Hausdorff–Young inequality $\|{\hat{f}}\|_q\leq c_p\|f\|_p$, $1\leq p\leq 2$, $\frac{1}{p}+\frac{1}{q}=1$, for the Euclidean Fourier transform can be viewed as a partial extension of the Plancherel theorem to $L^p$-functions. More generally, the Fourier transform extends to a continuous mapping from $L^p({\mathbb{R}}^n)$ into the Lorentz space $L^{p',p}({\mathbb{R}}^n)$, a result that is due to Paley. A variation on this theme is provided by the Hardy–Littlewood inequality which may be stated as follows: Let $f$ be a measurable function on ${\mathbb{R}}^n$ such that $x\mapsto f(x)\|x\|^{n(1-2/q)}$ belongs to $L^q({\mathbb{R}}^n)$, where $q\geq 2$. Then $f$ has a well-defined Fourier transform ${\hat{f}}$ in $L^q({\mathbb{R}}^n)$ and there exists a positive constant $A_q$ independent of $f$ such that $$\label{HL-ineq1}
\Bigl(\int_{{\mathbb{R}}^n}\vert{\hat{f}}(\xi)\vert^qd\xi\Bigr)^{1/q}\leq A_q\Bigl(\int_{{\mathbb{R}}^n}\vert 