Estimate of the Fourier-Bessel multipliers for the poly-axially operator (Q2194750)

The authors prove the analogue of Hörmander-Mikhlin multiplier theorem for the multidimensional Fourier-Bessel transform \[ \mathcal{F}_\alpha f(\lambda)=\int_{{\mathbb{R}}_+^{n}}f(x_1, \dots , x_n)j_{\alpha_1}(\lambda_1x_1)\cdots j_{\alpha_n}(\lambda_nx_n)x_1^{2\alpha_1+1}\cdots x_n^{2\alpha_n+1}\,dx_1\cdots dx_n \] (\(j_\gamma\) represents the normalized Bessel function of first kind and order \(\gamma\)) associated with the poly-axially operator \(\sum_{i=1}^n \frac{\partial^2}{\partial x_i^2}+\frac{2\alpha_i+1}{x_i}\frac{\partial}{\partial x_i} \), \(x_i>0\), \(\alpha_i>-1/2\), \(i=1, \dots , n\).