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On compact manifolds (M, g) without boundary, the gradient estimates for unit band spectral projection operators Χλ are proved for a second order elliptic differential operator L. A new proof of the Hormander Multiplier Theorem (first proved by A. Seeger and C.D. Sogge in 1989) is given in this setting by using the gradient estimates and the Calderon-Zygmund argument.

New proof of the Hörmander multiplier theorem on compact manifolds without boundary