  The main purpose of this paper is to prove H\"ormander's $L^p$-$L^q$
boundedness of Fourier multipliers on commutative hypergroups. We carry out
this objective by establishing 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 Ch\'{e}bli-Trim\`{e}che 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. Finally, we present applications of
the obtained results to study the well-posedness of nonlinear partial
differential equations.
