  The $(k,a)$-generalised Fourier transform is the unitary operator defined
using the $a$-deformed Dunkl harmonic oscillator.
  The main aim of this paper is to prove $L^p$-$L^q$ boundedness of $(k,
a)$-generalised Fourier multipliers. To show the boundedness we first establish
Paley inequality and Hausdorff-Young-Paley inequality for $(k, a)$-generalised
Fourier transform. We also demonstrate applications of obtained results to
study the well-posedness of nonlinear partial differential equations.
