In this paper, we consider the operator \(\Delta ^{\mathcal {M}}\) defined on \((0,+\infty )\) by \(\Delta ^{\mathcal {M}}=\frac{d^2}{dx^2}+\Bigg (\frac{2ixd}{b}+\frac{A^{\prime }(x)}{A(x)}\Bigg )\frac{d}{dx}+\Bigg (\frac{ixd}{b}\frac{A^{\prime }(x)}{A(x)}+\frac{id}{b}-\frac{d^2x^2}{b^2}+\rho ^2\Bigg ),\, d,b\in \mathbb {R},\,b\ne 0\) , where A is a nonnegative function satisfying certain conditions. We develop nice harmonic analysis associated with the operator \(\Delta ^{\mathcal {M}}\) . Firstly, we define and study the linear canonical Lions transform and we derive some of its basic properties, such as inversion formula and Plancherel formula. Secondly, we introduce the translation operator associated with \(\Delta ^{\mathcal {M}}\) and we discuss some of its important properties. Next, we derive a convolution product for this transform. Finally, using the aforesaid results we define and study the linear canonical Lions wavelet transform and we establish their properties. Also, some inequalities of this transform are proved.
