What is the simplest oscillatory integral for which sharp bounds are unknown?

Question

I have either heard or read that sharp asymptotics and bounds for oscillatory integrals of the form

$ \int e^{i \lambda \Phi(x)} \psi(x) dx \quad \lambda \to \infty $

are unknown when the critical points for the phase function are not isolated. If this impression is correct, what are the simplest / most important integrals of this form for which the optimal decay rate and asymptotic have not been proven? E.g. are there examples with $\Phi$ being a polynomial? (I would also appreciate recommendations of references for estimating multidimensional oscillatory integrals if anyone has them.)

Answer 1

As soon as the Hessian is not full rank, the problem becomes quickly messy:

• if the hessian has rank $n-1$, then one can treat the one direction separately since we have explicit bound for a one-dimensional integral where the taylor expansion of $\Phi$ near a critical point $x_0$ looks like $(x-x_0)^p$ for any $p\ge 2$, the other directions will always give you $\lambda^{-\frac{1}{2}}$.
• when the rank is less, then one must first identify those directions where the phase isn't quadratic, and look at the next terms in the expansion. V.I. Arnold then classifies the simple jets of functions in terms of their corresponding maximal decay in $\lambda$ in the following paper:

V.I. Arnold, Remarks on the stationary phase method and coxeter numbers, Russian Math. Surveys, 28 (1973), p. 19

See also J.J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities, CPAM vol XXVII, 207-281 (1974)

The classification is algebraic and does not rely on estimating integrals, so it never tells you how to obtain the estimate corresponding to that optimal decay. For the simpler classes of degenerate critical points, Popov has worked on estimating the oscillatory integrals:

D.A. Popov, Estimates with constants for some classes of oscillatory integrals, Russian Math. Surveys, 52, pp. 73–145.

D.A. Popov, Remarks on uniform combined estimates oscillatory integrals with simple singularities, Izv. Math., 72, pp. 793–816.

These papers helped me for the following problem, where the oscillatory integral I had to study had an interesting degenerate behavior.

F. Monard-G. Bal "Inverse transport with isotropic time-harmonic sources", SIAM J. Math. Anal., Vol. 44, No. 1, pp. 134-161 (2012).

Along the way, another paper I found interesting for direct estimates: G.I. Arkhipov, A.A. Karatsuba, and V.N. Chubarikov, Trigonometric integrals, Izv. Akad. Nauk SSSR Ser. Mat., 43 (1979), pp. 971–1003 (in Russian); Math. USSR-Izv., 15 (1980), pp. 211–239 (in English). (I think they also have a multidimensional counterpart).

Answer 2

On multidimensional oscillatory integral, A. Averbuch, E. Braverman, M. Israeli, R. Coifman, On Efficient Computation of Multidimensional Oscillatory Integrals with Local Fourier Bases, Preprint submited to Elsevier, 2001.

Also, S. Olver, Numerical Approximation of Highly Oscillatory Integrals, PhD Thesis, University of Cambridge, 2008.