Some multiplier theorems on the sphere (Q1585381)

The authors develop a sort of transference principle for zonal multipliers on the sphere \( S^n\). A zonal multiplier is any function of the spherical Laplacian, or equivalently it multiplies the spherical harmonics of degree \(l\) by a scalar \(m(l)\). Let \(L\) denote the multiplication operation \( m(l) \to l m(l)\), and \(D\) denote the differencing operation \( m(l) \to m(l) - m(l-1)\). The main result is that in order for the multiplier with symbol \(m\) to be bounded on \(L^p(S^n)\), it suffices for the multiplier with symbol \((DL)^N m \) to be bounded on \(L^p(S^1)\), for sufficiently large \( N \) (\(N \geq n/2 \) in the paper). This is achieved by reducing \(n\) to two dimensions at a time; the main technicalities have to do with analyzing the spherical harmonics of \( S^2\).