Power weighted \(L^p\) -inequalities for Laguerre-Riesz transforms (Q950719)

Given a real number \(\alpha> -1\), one defines the Laguerre second-order differential operator \[ L_\alpha= -y{d^2\over dy^2}-{d\over dy}+{y\over 4}+{\alpha^2\over 4y},\quad y> 0. \] Also, one defines the first-order derivatives, for \(\alpha> -1\) and \(\beta> 0\), \[ \delta^\alpha_y= \sqrt{y}{d\over dy}+ {1\over 2}\Biggl(\sqrt{y}- {\alpha\over\sqrt{y}}\Biggr) \] and \[ \delta^\beta_y= -\sqrt{y}{d\over dy}+{1\over 2}\Biggl(\sqrt{y}- {\beta\over\sqrt{y}}\Biggr). \] Then the Riesz transform for the Laguerre function expansions is defined by \[ {\mathfrak R}^\alpha= (R^\alpha_+, R^{\alpha+ 1}_-) \] and its norm is defined by \[ \|{\mathfrak R}^\alpha f\|= (|R^\alpha_+ f|^2+ |R^\alpha_- f|^2)^{1/2}, \] where \[ R^\alpha_+= \delta^\alpha_y(L_\alpha)^{-1/2},\quad R^\beta_-= \delta^\beta_y(L_\beta)^{-1/2}. \] In the paper, the authors give a complete description of the power weighted inequalities, of strong, weak and restricted weak type for the operator \(\|{\mathfrak R}^\alpha\|\).