Stability of Oscillatory Integral Asymptotics in Two Dimensions

Abstract

The stability under phase perturbations of the decay rate of local scalar oscillatory integrals in two dimensions is analyzed. For a smooth phase S(x,y) and a smooth perturbation function f(x,y), the decay rate for phase S(x,y)+tf(x,y) is shown to be the same for all but finitely many t and given an explicit description. The decay rate of the generic S(x,y)+tf(x,y) is always at least as fast as that of S(x,y), and the “good” cases where it is the same as that of S(x,y) are explicitly described. Uniform stability of the decay rate is proven for S(x,y)+f(x,y) for small enough such good f(x,y), and the coefficient of the leading term of the asymptotics is shown to be Lipschitz of some order alpha, again for small enough good perturbations f(x,y).

Appendix: Formulas for the Coefficient of the Principal Term of the Asymptotics

In this appendix, we show that the formulas for the coefficients A _S,ϕ given in the real-analytic case in [5] carry over to the smooth situation when S(x,y) is in Case 1 or Case 2 superadapted coordinates. When S(x,y) has a Morse critical point at the origin, explicit formulas are well known for the general smooth case (see [13] pp. 344–347), so throughout we will assume S(x,y) does not have a Morse critical point at the origin. In [5], one defines

$$ I_{S, \phi}(\epsilon) = \int_{\{(x,y): 0 < S(x,y) < \epsilon\}} \phi(x,y) \,dx\,dy. $$

(A.1)

It is shown in [5] that for Case 1 or Case 2 in the smooth case, and in Case 3 for the real-analytic case only, that if S(x,y) is in superadapted coordinates then as ϵ→0 one has asymptotics

$$ I_{S,\phi}(\epsilon) = B_{\phi,S}\,\, \epsilon^{\delta}|\ln(\epsilon)|^p + o\bigl( \epsilon^{\delta}|\ln(\epsilon)|^p\bigr). $$

(A.2)

Here (−δ,p) is the oscillatory type of S and explicit formulas for B _ϕ,S are shown in [5]. In the real-analytic case, one can use well-known methods (cf. Chap. 7 of [1]) to get explicit formulas for the A _ϕ,S in terms of these formulas for B _ϕ,S . Namely, in Case 1 and Case 3 superadapted coordinates we have

$$ A_{\phi,S} = {\varGamma({1 \over d}) \over d} \bigl(e^{i {\pi\over2d}} B_{\phi,S} + e^{-i{ \pi\over2d}} B_{\phi,-S} \bigr). $$

(A.3)

In Case 2, one has

$$ A_{\phi,S} = -{\varGamma({1 \over d}) \over d} \bigl(e^{i {\pi\over2d}} B_{\phi,S} + e^{-i{ \pi\over2d}} B_{\phi,-S} \bigr). $$

(A.4)

In this section, we will explain why formulas (A.3) and (A.4) still hold for any smooth S(x,y) if it is in Case 1 or Case 2 superadapted coordinates. In both cases our arguments resemble those used in [11] for the Case 1 situation. (Case 2 is analyzed differently in that paper.)

We first suppose S(x,y) is in Case 1 superadapted coordinates. As in Sect. 4, we denote by e the edge of N(S) intersecting the bisectrix in its interior, and the equation of the line containing this edge by x+my=c. Again we take a small enough rectangle [−r,r]×[−r ^m,r ^m] and divide it into 4 regions via the curves y=±|x|^m. As before we focus our attention on the region U _r consisting of points where 0<x<r and −x ^m<y<x ^m as the other three regions are dealt with similarly. We also again use the wedges W _i used in Sect. 4. Recall each wedge W _i is of the form {(x,y):0<x<r, a _i x ^m<y<b _i x ^m}, and on a given wedge we either have an estimate ∂ _y S(x,y)>Cx ^c−lm for some 1≤l<d(S), or an estimate ∂ _x S(x,y)>Cx ^c−1 (possibly after a coordinate change of the form $(x,y) \rightarrow(x,y + {1 \over2}x^{m})$ in the case where the wedge was centered along the x-axis).

Let ϵ>0 be a small number, to be determined by our arguments. We write W _i =X _i ∪Y _i , where

$$ X_i = \bigl\{(x,y): 0 < x < \lambda^{-\epsilon}, \,a_ix^m < y < b_ix^m\bigr\}, $$

(A.5a)

$$ Y_i = \bigl\{(x,y): \lambda^{-\epsilon} < x < r, \,a_ix^m < y < b_ix^m\bigr\}. $$

(A.5b)

We first estimate $\int_{Y_{i}}e^{i\lambda S(x,y)}\phi(x,y)\,dx\,dy$. To do this, if ∂ _y S(x,y)>Cx ^c−lm on W _i then we use the Van der Corput lemma for measures (see [2]) in the y direction and integrate the result with respect to x, and if ∂ _x S(x,y)>Cx ^c−1 we use the Van der Corput lemma for measures in the x direction and integrate the result with respect to y. In the former case, for a given x we get the estimate

$$ \biggl|\int_{a_i x^m}^{b_ix^m}e^{i\lambda S(x,y)} \phi(x,y)\,dy\biggr| \leq C\lambda^{-{1 \over l}} x^{-{c \over l} + m}. $$

(A.6)

Thus we have

$$ \biggl|\int_{Y_i}e^{i\lambda S(x,y)}\phi(x,y)\,dx\,dy\biggr| \leq C \lambda^{-{1 \over l}}\int_{\lambda^{-{\epsilon}}}^r x^{-{c \over l} + m}\,dx. $$

(A.7)

Since l<d, we may let δ>0 be the minimal possible value of ${1 \over l} - {1 \over d(S)}$. Thus if we choose ϵ small enough that the x integral here is bounded by $C\lambda^{{\delta \over2}}$, then since $-{1 \over l} + {\delta\over2} \leq-{1 \over d(S)} - {\delta\over2}$ we have

$$ \biggl|\int_{Y_i}e^{i\lambda S(x,y)}\phi(x,y)\,dx\,dy\biggr| \leq C' \lambda^{-{1 \over d(S)} - {\delta\over2}}. $$

(A.8)

Hence the Y _i integral is of order of magnitude less than $\lambda ^{-{1 \over d(S)}}$. We now turn to the X _i integral. Here we write S(x,y)=S _N (x,y)+(S(x,y)−S _N (x,y)), where S _N (x,y) is the sum of the terms of the Taylor expansion of S(x,y) to a high enough order N to be determined as a function of ϵ. Then

(A.9)

(A.10)

Since on X _i we have 0<x<λ ^−ϵ and |y|<x ^m, by choosing N sufficiently large (depending on ϵ) we can ensure that on X _i we have

$$ \bigl|S(x,y) - S_N(x,y)\bigr| \leq C_N \lambda^{-3}. $$

(A.11)

By the mean value theorem this in turn implies that

$$ \bigl|e^{i\lambda S_N(x,y)} \bigl(e^{i\lambda(S(x,y) - S_N(x,y))} - 1\bigr)\bigr| \leq C_N'\lambda^{-2}. $$

(A.12)

Thus we have

(A.13)

As a result,

(A.14)

Furthermore, by applying the Y _i integral argument to S _N in place of S for N sufficiently large, we have

(A.15)

We conclude that

(A.16)

Adding this over all W _i and over all 4 pieces of [−r,r]×[−r ^m,r ^m], we get that if N is sufficiently large, then

$$ J_{S, \phi}(\lambda) = J_{S_N,\phi}(\lambda) + O\bigl( \lambda^{-{1 \over d(S)} - {\delta\over2}}\bigr). $$

(A.17)

But $J_{S_{N},\phi}(\lambda)$ has asymptotics with an $O(\lambda^{-{1 \over d(S)}})$ term, whose formula is given explicitly in terms of finitely many terms of the Taylor expansion of S _N (x,y). Since these terms are the same for S and S _N if N is sufficiently large, these formulas will hold for S(x,y) as well. This completes the proof for Case 1 superadapted coordinates.

We now move on to Case 2 superadapted coordinates. We argue similarly to the Case 1 situation, using the Case 2 wedges of Sect. 4. Specifically, we let m be such that the line x+my=c intersects N(S) at the point (d(S),d(S)) only, and we divide the rectangle [−r,r]×[−r ^m,r ^m] into 4 regions V _r via the curves y=±|x|^m. As in Sect. 4, we focus our attention on W _r , the region where 0<x<r and −r ^m<y<r ^m as the other three regions are done the same way. As in (4.18), on W _r we have an estimate

$$ \bigl|\partial_y^dS(x,y)\bigr| > Cx^d. $$

(A.18)

Similar to in Case 1, for small ϵ>0 we subdivide into regions X and Y defined by

$$ X= \bigl\{(x,y): 0 < x < \lambda^{-\epsilon}, -x^m < y < x^m\bigr\}, $$

(A.19a)

$$ Y= \bigl\{(x,y): \lambda^{-\epsilon} < x < r, -x^m < y < x^m\bigr\}. $$

(A.19b)

One may use the Van der Corput lemma in the y direction to obtain

$$ \biggl|\int_{-x^m}^{x^m}e^{i\lambda S(x,y)} \phi(x,y)\,dy\biggr| \leq C\lambda^{-{1 \over d}} x^{-1}. $$

(A.20)

Thus we have

(A.21)

(A.22)

So while this does contribute to the main term of the asymptotics, it does so in a way that shrinks linearly with ϵ. As for the X integral, one can argue exactly as in the Case 1 situation and say that if one replaces S by S _N in ∫ _X e ^iλS(x,y) ϕ(x,y) dx  dy, the difference is bounded in absolute value by C _ϵ λ ⁻². (Since N is a function of ϵ so is the constant here.) Furthermore, the argument used to show (A.22) works for S _N in place of S, so $|\int_{Y} e^{i\lambda S_{N}(x,y)}\phi(x,y)\,dx\,dy|$ is also bounded by (A.22). We conclude that

(A.23)

Adding (A.23) over all four regions, we conclude that

$$ \bigl|J_{S,\phi}(\lambda) - J_{S_N,\phi}(\lambda)\bigr| \leq C_{\epsilon}\lambda^{-2} + C\epsilon\ln(\lambda) \lambda^{-{1 \over d}}. $$

(A.24)

Like in Case 1, the formulas of [5] are such that for N sufficiently large, the formula applied to S is the same as the formula applied to S _N . Thus we have shown that the formula for the leading term of the asymptotics in the real-analytic case also holds in the smooth case modulo a term bounded by $C\epsilon\ln(\lambda)\lambda^{-{1 \over d}}$ as λ→∞. Letting ϵ go to zero shows that the leading terms are in fact the same, and we are done.