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$\boldsymbol {L}^{\boldsymbol {p}}$–
$\boldsymbol {L}^{\boldsymbol {q}}$ MULTIPLIERS ON COMMUTATIVE HYPERGROUPS
The main purpose of this paper is to prove Hörmander’s 
$L^p$–
$L^q$ boundedness of Fourier multipliers on commutative hypergroups. We carry out this objective by establishing the Paley inequality and Hausdorff–Young–Paley inequality for commutative hypergroups. We show the 
$L^p$–
$L^q$ boundedness of the spectral multipliers for the generalised radial Laplacian by examining our results on Chébli–Trimèche hypergroups. As a consequence, we obtain embedding theorems and time asymptotics for the 
$L^p$–
$L^q$ norms of the heat kernel for generalised radial Laplacian.
