Uncertainty principles for continuous wavelet transforms related to the Riemann-Liouville operator (Q1683917)

The authors consider the singular partial differential operators \[ \begin{aligned} D &= \frac{\partial}{\partial x},\\ \Xi &= \frac{\partial^2}{\partial r^2}+\frac{2\alpha +1}{r}\frac{\partial}{\partial r}-\frac{\partial^2}{\partial x^2}, \quad (r,x)\in ]0,+\infty[ \times \mathbb{R}, \;\alpha\geq 0, \end{aligned} \] and associate to them the so-called Riemann-Liouville operator, defined on the space of continuous functions on \(\mathbb{R}^2\) which are even with respect to the first variable, by \[ \mathcal{R}_\alpha(f)(r,x)= \begin{cases} \frac{\alpha}{\pi}\int_{-1}^1\int_{-1}^1 f\left(rs\sqrt{1-t^2},x+rt\right)(1-t^2)^{\alpha-1/2}(1-s^2)^{\alpha-1}\,dt\,ds, & \alpha>0\\ \frac{1}{\pi}\int_{-1}^1 f\left(r\sqrt{1-t^2},x+rt\right)\frac{dt}{\sqrt{1-t^2}}, & \alpha=0. \end{cases} \] In fact, for all \(\alpha\geq 0\) and \((\mu, \lambda)\in \mathbb{C}^2\), the unique solution \(\varphi_{\mu,\lambda}(r,x)\) to the problem \[ \begin{cases} (Du)(r,x)=-i\lambda u(r,x)\\ (\Xi u)(r,x)=-\mu^2u(r,x)\\ u(0,0)=1,\quad \frac{\partial u}{\partial x}(0,x)=0, \; x\in \mathbb{R}, \end{cases} \] can be expressed as \[ \varphi_{\mu,\lambda}(r,x)=\mathcal{R}_\alpha\left(\cos(\mu\cdot)e^{-i\lambda\cdot}\right)(r,x),\quad (r,x)\in [0,+\infty[ \times \mathbb{R}. \] After introducing the continuous wavelet transform connected with the Riemann-Liouville operator and establishing some of its basic properties, the authors prove an uncertainty principle for this transform and analyze its concentration on sets of finite measure.