Some properties of the pseudodifferential operator \(h_{\mu ,a}\) (Q996875)

The authors study pseudodifferential operators of the form \[ (h_{\mu,a}u)(x)= \int^\infty_0(x\xi)^{1/2}y_\mu(x\xi)a(x,\xi) (h_\mu u)(\xi)\,d\xi,\quad u\in H_\mu, \] where \(a\in{\mathcal C}^\infty(I\times I)\), \(I=(a,\infty)\), is the symbol of \(h_{\mu,a}\), \(h_\mu\) denotes the Hankel transformation and \(H_\mu\) is the Zemanian space. They prove that the kernel of \(h_{\mu,a}\) is a continuous function and \(h_{\mu,a}\) is pseudolocal. A global regularity result for an elliptic partial differential operator is obtained and an existence theorem for the weak solution of a pseudodifferential equation is established.