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

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