(f~) (pa) = 0 (p -n/2) as p

Transcription

1 45 OSCILLA~ORY INTEGRALS Michael Cowling and Shaun Diney We are intereted in the behaviour at infinity of the Fourier tranform ~ of the urface meaure ~ living on a mooth compact ~n Rnl, hyperurface S and in the tranform (f~) of the product of certain function f on S and ~ The aim of thi work i to ee how the decay of ~ and (f~) reflect the geometry of S. To implify the tatement of reult, we aume that S i analytic. The earliet and mot important reult on the decay of (f~) at infinity come from the principle of tationary phae: in almot every direction G in Sn, (f~) (pa) = 0 (p -n/2) a p -... More preciely, if a i a generic direction, which mean that the (finitely many) point of to which a i normal are point of non-zero Gauian curvature ~ then for mooth enough f (C 1 will do), (1) (f~) (pa) -ipa.k -n/2 e P o (p -n/2) a p - oo. The contant c(k) depend on the dimenion n of S, and on whether i an inward or outward normal at relative to the principal curvature. If a i a non-generic direction, o that there i a point

2 46 with normal vector a where 0, then (f!l) (pcr) decay lower that -n/2 p a at leat if due to the preence of an aymptotic term of the form (2) -ipo.k -o: 13 c(k) f(kl e p log (p) the indice o: (a poitive rational) and ~ (a non-negative integer) depend on the nature of S near k. Amongt the important paper on thi problem, we mention the work of B. Malgrange [5] and A.N. Varchenko [7], where the exitence of an aymptotic expanion i proved, and a and 13 computed for many example. It can be hown that, if f i mooth enough and if S i convex and ha no point of Gauian curvature 0, then there exit a contant C o that (3) I I f!i l (pcr)! :;; -n/2 C(lp) (ee C.S. Herz [3] and W. Littman [4]); it would appear from (1) that, in the general cae, if f = 19(1 112, then the inequality (3) hould till hold, a then in each generic direction the aymptotic decay i uniformly controlled. However, thi i fale, for at leat two reaon (ee Cowling and G. Mauceri [2], for one of thee). We now ummarie everal problem about the decay of (f~) at infinity which we conider important: (a) Decribe the decay of (f!ll (pcr) a p -> oo in term of the geometry of the point on S having normal vector cr, a in (2) aboveo (b) Find uniform etimate (a cr varie) for the decay of (f!j.) (pcr) a p --> =, a in (3) above.

3 47 (c) What i the relation, if any, between the lowet decay (a a varie) in the aymptotic expanion for (f~) (pcr), and the decay of (f~) (pcr) in a uniform etimate? Are thee the ame? (d) If we take f to be a power of the Gauian curvature, 1~1 9 ay, find etimate for (f~l, and find the mallet value of e for which (fj.t) (pcr) decay uniformly a -n/2 p a a in (3). There ha been ome progre on thee quetion recently: C.D. Sogge and E.M. Stein [6] howed that, if = 2n, then there i a contant C o that e A -n/2 I (lxi ~l (pal I :::; Cp n \:fa E S, itp E R, while M. Cowling and G. Mauceri [2] obtained thi inequality when a= [n/2] 2 ([ ] denote the integer part function), under the additional hypothei of the convexity of S. We believe, however, that e = 1 uffice for convex S, and perhap e = 2 uffice in general. J. Bruna, A. Nagel, and S. Wainger [1] conidered the urface meaure ~ on a convex hyperurface S, and etablihed the exitence of a contant C o that ltj. (pcr) I :::; C[ I Cap (p,cr)l!cap (p,cr) I J itcr e n, '<lp E R, where ITI indicate the urface area of a ubet T of S, and Cap (A.,cr) (repectively Cap (A.,cr)) denote the cap of S at the point (repectively ) of S where cr i an outward (repectively inward) normal, of height A.:- Cap (A.,a) { S E S: cr. - cr. e Cap (A.,cr) { E S: cr. - cr. E [O,A.]},

4 48 where of coure (repectively ) are the point on S where cr. i maximied (repectively minimied). Diagram 1: and Cap (A.,cr). Since the leading term of the aymptotic expanion for ~(pcr) are -1 I Cap (p,cr) I and - -1!Cap (p,a) I (ee Varchenko [7]), it follow that the lowet rate of decay in an aymptotic expanion will control II! (pcr) I, uniformly in a. The contribution which we offer here may or may not be ignificant in the long run. We believe that a modification of the argument of Bruna, Nagel and Wainger hould hot~ that (4) l!~l (p u l I :;;; C [ ICap 1 (p,cr) I ICap 1 (p,cr) I J, t,rhere 1Cap 1 (fo.,cr) I denote the area of Cap (lc,cr) relative to the ne ~ urface meaure ~~ and ICap 1 (A.,cr) I i defined analogouly. If (4) doe hold, then, for ome poibly different C, I(~) -n/2 (p cr l I :;;; c p a a conequence of the following theorem.

5 49 THEOREM. Let S be a compact convex analytic hyperurface in anl. Then there i a contant C o that n/2 ICap 1 (A,a) I C A We hall ketch the main feature of the proof. Let g: S ~ gn denote the Gau map : for in S with outward unit normal a, g() =a. Gau howed that ICap 1 (A,a) I lg(cap (A,a)) I. We chooe coordinate in Rn 1 o that are coordinate for the tangent plane T (S) to S at, o that lie at the origin, and o that a = (0,...,0,-1). Then the part of the urface near i the graph of a convex analytic function f f ' defined on T (S), of radiu of convergence at leat one, ay, and atifying f (0) 0 and Vf (0) 0, i.e. {- X e Rn 1 : X n1 Now n lg(cap (A,a))l l{'t e S: 'tn 1 M(A)}I, where M(A) max{gn 1 () e Cap (A, a) }. For A mall enough, any element of Cap (A,a) which atifie gnl () M(A) can be written in the form (~,f(~)), for ome ~ in T (S) of length at mot 1/2, and then M(A)

6 50 Chooe 9 (1) in (O,n/21 o that co(b (l)) = (livflil l/ 2, then - 2-1/2 n c~ [arcco((livf(~) I I ll - n 5 c IVf (~) I. n In ummary, then, Cap (ll.,cr) g (Cap (ll., cr) ) 'nl ; M(A,)} Diagram 2: The geometry of Theorem 1 Thu it uffice to prove that, for ome contant C, (5) I Vf (x) I 5 - c (~)1/2 ' for all f in the family of function :r which arie by conidering the part of the urface near a the graph of a function f defined on T (S), a varie in, and for all ~ in T (S) with

7 51 11,2 11 < L For each unit vector ~ in T (S), 'ille may write f (tel -- 1ft E ( -1, 1) i vjhere, for ome (S-dependent) contant K > 0, li > 0, and P in N, (i) ao (~) = 0 It~ E T (S), (ii) (~) 0 'v'~ E T (S), (iii) Ia (e.) I m- ~ K 'v'~ E T (S), and (iv) max{! a (e) I 2 $ m P} <: \t~ E T (S). m- [Thi lat condition i etablihed by compactne: if it were fale, then for all P in Nl there \ lould be element of 1' (S), ~P ay, o that rn and then there would be ome ~ in T(S) with a (e) = 0 'v'm e N\.] m- We may make a further implification: by retricting attention to twodimenional ubpace of T (S) it uffice to conider a compact family ~ of convex analytic function of two real variable, centred at the origin, with radiu of convergence at leat one 1 atifying the tvm dimenional analogue of the above condition (i) - (iv); again, we mut prove that 2 lllf (.? ) I :::;; Cf (.? ) for all.2 in the unit ball in R 2 centred at the origin. The firt tep in proving thi i chooing a uitable coordinate ytem. For each 2-plane in we chooe unit vector and o that i orthogonal to ~1' and ~1 i the "direction of lowet

8 52 growth" of f 0 Thi direction i obtained a follow: we write, for any unit vector ~ in ~, f (te) - then either there are poitive integer p and q, ~lith p > q ;;, 2, and a direction o that min{m a (e) 0) m- or min{m a (e).. 0) p m~.. r q if e = ±e - -1 otherwie for all ~ in ~. In the former cae, i the "direction of hortet growth"; in the latter we chooe ~1 o that a (e) p- and being unit vector. Now for f in "#', we may write f(x~ 1 y~2l = 00 mn Lm,n=O a X y, mn where a 0 if rnn m/p n/q < 1, by convexity, and I a I, K rnn and max{ I amo I 2 m Pl e: i5 and Let p = min{m : lamo I ;;, a l and q = min{n We now apply induction on p q. One poibility i that 0 if m < P 0 if n < q and (by convexity) a rnn 0 if m/p n/q < 1.

9 53 The difficul ty in proving (5) lie when 111!:11 i mall; here we etimate [by majoriing the geometric mean by the arithmetic mean, for mall m and n]. To etimate f, we let :z: mn m/pn/q=l amnx y ; we how that by oberving that the quotient function mut be bounded away from 0 by convexity and homogeneity argument; and then it follow that for mall x and y. 'rhe other poibility i that ome a mn are non-zero for ome m and n with m/p n/q < 1. Now a dilation argument, like that employed by Cowling and Mauceri [2], (bu'c in 2-variable), enable u to reduce to a cae with maller pq. 0 REFERENCES [1] J. Bruna, A. Nagel and S. Wainger, Convex hyperurface and Fourier tranform, preprint, [2) M. Cowling and G. Mauceri, OcillatoJ:y integral and Fourier tranform of urface-carried meaure, Tran. Amer. Math. Soc., to appear. [3] C.S. Herz, Fourier tranform related to convex et, Ann. of I>iath. 75 (1961),

10 54 [4] W. Littman, Fourier tranform of urface carried meaure and differentiability of urface average, Bull. Amer. Math. Soc. 69 (1963), [5] B. Malgrange, Integrale aymptotique et monodromie, Ann. Sci. Ec. Norm Sup. 7 (1974), [6] C.D. Sogge and E.M. Stein, Average of function over hyperurface in Rn, Invent. Math. 82 (1985), [7] A.N. Varchenko, Newton polyhedra and etimation of ocillating integral, Funct. Anal. Appl. 10 (1976), School of Mathematic Univerity of New South Wale P.O. Box 1 Kenington 2033 Autralia