On Nash and Carlson’s Inequalities for Symmetric <Emphasis Type="Italic">q</Emphasis>-Integral Transforms

The aim of this paper is to prove an \(\mathcal {L}_q^1 \cap \mathcal {L}_q^2\) versions of Nash and Carlson’s inequalities for a class of q-integral operator \(\mathcal {T}_q\) with a bounded kernel. As applications, we give q-analogues of Nash and Carlson’s inequalities for the q-Fourier-cosine, q-Fourier-sine, q-Dunkl and q-Bessel Fourier transforms.