The heat semigroups and uncertainty principles related to canonical Fourier–Bessel transform

The aim of this paper is to introduce the heat semigroups \(\left( {\mathcal {S}}_{\nu }^{{\textbf{m}}^{-1}}(t)\right) _{t\ge 0}\) ${\left({\mathcal{S}}_{\nu }^{{\mathbf{m}}^{-1}},(t)\right)}_{t\ge 0}$ related to \(\Delta _{\nu }^{{\textbf{m}}^{-1}}\) ${\mathrm{\Delta }}_{\nu }^{{\mathbf{m}}^{-1}}$ given by \(\begin{aligned} \Delta _{\nu }^{{\textbf{m}}^{-1}}=\frac{d^{2}}{dx^{2}}+\left( \frac{2\nu +1}{x}+2i\frac{a}{b} x\right) \frac{d}{dx}-\left( \frac{a^{2}}{b^{2}}x^{2}-2i\left( \nu +1\right) \frac{a}{b}\right) \end{aligned}\) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{\nu }^{{\mathbf{m}}^{-1}}=\frac{d^2}{d{x}^2}+\left(\frac{2\nu +1}{x},+,2,i,\frac{a}{b},x\right)\frac{d}{\mathit{dx}}-\left(\frac{a^2}{b^2},x^2,-,2,i,\left(\nu ,+,1\right),\frac{a}{b}\right)\end{array}$$ and we study some of its important properties. In the present paper, several uncertainty principles for the canonical Fourier–Bessel transform are given, including the Beurling, Gelfand–Shilov and Cowling–Price uncertainty principles.