Improved Hörmander’s theorem and new methods for oscillatory integral operators

Abstract

This paper proves a theorem on the decay rate of the oscillatory integral operator with a degenerate C∞ phase function, thus improving a classical theorem of Hormander. The proof invokes two new methods to resolve the singularity of such kind of operators: a delicate method to decompose the operator and balance the L2 norm estimates; and a method for resolution of singularity of the convolution type. The operator is decomposed into four major pieces instead of infinite dyadic pieces, which reveals that Cotlar’s Lemma is not essential for the L2 estimate of the operator. In the end the conclusion is further improved from the degenerate C∞ phase function to the degenerate C4 phase function.