ar X iv : m at h / 06 09 02 4 v 1 [ m at h . A P ] 1 S ep 2 00 6 L p - L q regularity of Fourier integral operators with caustics

The caustics of Fourier integral operators are defined as caustics of the corresponding Schwartz kernels (Lagrangian distributions on X × Y). The caustic set Σ(C) of the canonical relation C is characterized as the set of points where the rank of the projection π : C → X × Y is smaller than its maximal value, dim(X × Y) − 1. We derive the L p (Y) → L q (X) estimates on Fourier integral operators with caustics of corank 1 (such as caustics of type Am+1, m ∈ N). For the values of p and q outside of certain neighborhood of the line of duality, q = p ′ , the L p → L q estimates are proved to be caustics-insensitive. We apply our results to the analysis of the blow-up of the estimates on the half-wave operator just before the geodesic flow forms caustics. Stretching his stiff legs, Persikov got up, returned to his laboratory, yawned, rubbed his permanently inflamed eyelids, sat down on the stool and looked into the microscope. .. With his right eye Persikov saw the cloudy white plate and blurred pale amoebas on it, but in the middle of the plate sat a coloured tendril, like a female curl. The coloured streak of light merely got in the way and indicated that the specimen was out of focus. The zoologist's long fingers had already tightened on the knob, when suddenly they trembled and let go. .. He noticed that one particular ray in the coloured tendril stood out more vividly and boldly than the others. .. This strip of red was teeming with life. The old amoebas were forming pseudopodia in a desperate effort to reach the red strip, and when they did they came to life, as if by magic. They split into two in the ray, and each of the parts became a new, fresh organism in a couple of seconds.