Singularities of affine fibrations in the regularity theory of Fourier integral operators

Abstract

In the paper the regularity properties of Fourier integral operators are considered in different function spaces. The most interesting case are L p spaces, for which a survey of recent results is made. Thus, the sharp orders are known for operators, satisfying the so-called smooth factorization condition. Further in the paper this condition is analyzed in both real and complex settings. In the last case conditions for the continuity of Fourier integral operators are related to the singularities of affine fibrations in (subsets of) C n , defined by the kernels of Jacobi matrices of holomorphic mappings. Singularities of such fibrations are analyzed in this paper in the general case. In particular, it is shown that for small dimensions n or for small ranks of the Jacobi matrix, all singularities of an affine fibration are removable. Fourier integral operators lead to fibrations, given by the kernels of the Hessian of a phase function of the operator. Based on the analysis of singularities for operators, commuting with translations, in a number of case the factorization condition is shown to be satisfied, which leads to L p estimates for operators. In the other cases, the failure of the factorization condition is exhibited by a number of examples. Results are applied to derive L p estimates for solutions of the Cauchy problem for hyperbolic partial differential operators.

The weak-type (1,1) of Fourier integral operators of order –(n–1)/2

“…Of course, in the most extreme case, when 4> vanishes, then the Fourier integral operator collapses to a pseudo-differential operator, and one has weak-type (1, 1) for operators of order 0, and not just -(n -1)/2. In intermediate cases when V 2 <1> consistently has rank strictly between 0 and n -1, there are intermediate results (see [4,5]). Perhaps one can unify these results by introducing symbol classes adapted to the phase function Q{x, £); for instance one might study symbols a(x, £) which obeyed bounds such as together with some corresponding bounds on higher derivatives.…”

“…For instance, in the completely degenerate case when <t>(£) = xo • £ is linear in £, the convolution kernel K is essentially a fractional integral kernel \/\x -x o \, which is clearly integrable. More generally, when <J> is close to degenerate, then the error terms in a Taylor expansion of 4> become more favorable, and one can coarsen the standard 'second dyadic decomposition' (see for instance [4,5]) in order to improve the standard estimates [5] on the kernel K (which in the non-degenerate case, just barely fail by a logarithm to be integrable, because the Fourier integral operator has the critical order -(n -l)/2).…”

On the failure of the factorization condition for non-degenerate Fourier integral operators

“…For general properties of operators with ranks k and their relation to the singularity theory of affine fibrations we refer to [5] for the real valued phase functions, and to [6] for the complex valued phase functions, respectively. For the backgrounds on the L p theory of Fourier integral operators we refer to [8], [9], and a survey [5] for smaller ranks k.…”

“…For the backgrounds on the L p theory of Fourier integral operators we refer to [8], [9], and a survey [5] for smaller ranks k.…”

“…We will write κ(ξ) for κ(y, ξ). In [5], we have shown that if H is analytic, the smallest dimension when the factorization condition may fail, is n = 5. Moreover, the factorization condition holds if k ≤ 2.…”