L^{p}-boundedness of oscillating spectral multipliers on Riemannian manifolds

Abstract

We prove endpoint estimates for operators given by oscillating spectral multipliers on Riemannian manifolds with C ∞ -bounded geometry and nonnegative Ricci curvature.

Central oscillating multipliers on compact Lie groups

“…By an estimate from [20] (see also p. 140 in [16]), we have | ξ | Let {J k } be a sequence of sets defined by…”

“…The sufficient parts of (1), (2) in Theorem 1 have been obtained in [16] for the special case in which ≡ 1. 2.…”

“…The operator S α,β was also generalized to many different settings of Lie groups and manifolds. In [3], Giulini and Meda studied the oscillating multiplier on a rank one symmetric space of noncompact type; Alexopoulos [1] and Marias [16] studied analogous multiplier theorems on Lie groups of polynomial volume growth and on Riemannian manifolds of nonnegative curvatures. It is known that these multipliers on general Lie groups and manifolds are defined based on the eigenfunction expansions of the Laplacian operators (see [1,16]).…”

“…Also, we notice that the results in [1] and [16] are not as elegant as their Euclidean analogues, because the underlying spaces are of too wide a scope. For instance, on a general Riemannian manifold H p theory has only been well-established when p is close to 1; the kernel estimates in [1] and [16] are not precise enough to obtain the necessity of the range of p for the L p boundedness; the result of [1] for the case β = 1 is slightly weaker than its Euclidean analog (see [17]); etc. Motivated by these observations, this paper fills some of these gaps on compact Lie groups.…”

Optimal boundedness of central oscillating multipliers on compact Lie groups

“…Thus the operator T γ,β has the Fourier expansion It is well known that the operator S γ,β is bounded on H p (R d ) if and only if |1/2 − 1/p| ≤ γ/(dβ) for all 0 < p < ∞ (see [14,19,20,22]). We notice that when 1 < p < ∞, the boundedness of S γ,β has been generalized to many different settings of Lie groups and manifolds (see [1,5,15,18]). In a recent paper [5], we established the following optimal L p (G) boundedness of T γ,β on a compact Lie group.…”