Some results of pseudo-differential operators related to the spherical mean operator

In this paper, we prove a potentially useful \(L^p(d\nu )\) $L^p(d\nu )$ -boundedness result for the pseudo-differential operators associated with the spherical mean operator. Also, boundedness result for symmetrically global pseudo-differential operator on \(L^p(d\nu )\) $L^p(d\nu )$ -type Sobolev space \(\mathcal {H}^{u,v,p}\) ${\mathcal{H}}^{u,v,p}$ of order (u, v) are discussed. An application in solving a generalized heat equation is given.