The product of pseudo-differential operators associated with Bessel operators (Q2581103)

The author considers Bessel pseudodifferential operators, i.e. operators \(P\) of the form \[ Pu(x)= \int^{+\infty}_0 (xy)^{1/2} J_\mu(xy)\, a(x,y)\,U_\mu(y)\,dy, \] where \(J_\mu\) denotes the Bessel function of the first kind of order \(\mu\), and \[ U_\mu(y)= \int^{+\infty}_0 (xy)^{1/2} J_\mu(xy)\,u(x)\,dx. \] The function \(u(x)\) is assumed to belong to the function spaces \(H_\mu\) defined by \textit{A. H. Zemanian} [Generalized integral transformations (Pure Applied Mathematics 18, New York: Interscience) (1968; Zbl 0181.12701)]. Results of continuity are proved in these spaces under suitable assumptions on the symbol \(a(x,y)\).