Estimates for Riesz kernels of eigenfunction expansions of elliptic differential operators on compact manifolds (Q804053)

The authors study the kernels and the operator norms of the Riesz means of order \(\delta\) (\(\geq 0)\) of f(x): \[ R^{\delta}_{\Lambda}f(x)=\int_{M}\sum_{\lambda <\Lambda}(I- \frac{\lambda^ 2}{\Lambda^ 2})^{\delta}\phi_{\lambda}(x)\overline{\phi_{\lambda}(y)}f(y)d\mu (y) \] on the spaces \(L^ p(M)\), where M is a smooth connected compact N-dimensional manifold without boundary, \(\{\lambda^ 2\}\) and \(\{\phi_{\lambda}\}\) are eigenvalues and an orthonormal system of eigenfunctions of a positive elliptic second order differential operator on M, with smooth coefficients and self-adjoint with respect to positive density \(d\mu\). The main results are formulated in three theorems. Theorems 1 and 2 contain sharp estimates for the norms of the operators \(\{R^{\delta}_{\Lambda}\}\) on the spaces \(L^ 1(M)\), \(L^{\infty}(M)\) and \(L^ p(M)\), \(1\leq p\leq 2(N+1)/(N+3).\) Theorem 3 is the following Theorem. If \(N=p=1\), or \(1\leq p<2(N+1)/(N+3)\), and if \(\delta =N/p-(N+1)/2,\) then the operators \(\{R^{\delta}_{\Lambda}\}\) are of weak type (p,p) uniformly with respect to \(\Lambda\) ; i.e., for every function f in \(L^ p(M)\) one has \[ \mu (\{x:\;| R^{\delta}_{\Lambda}f(x)| >\alpha \}\leq c\alpha^{-p}\| f\|^ p_ p. \]