Harmonic analysis on nilpotent groups and singular integrals I. Oscillatory integrals

JOURNAL

OF FUNCTIONAL

ANALYSIS

Harmonic

73, 179-194 (1987)

Analysis on Nilpotent and Singular Integrals I. Oscillatory Integrals*

Groups

FULVIO Rrccr AND E. M. STEIN Dipartimento di Matematica, Politecnico di Torino, 10129 Torino, Italy, and Department of Mathematics, Princeton University, Fine Hall, Princeton, New Jersey 08544 Communicated by L. Gross Received February 21, 1986

This paper is devoted to the study of the operator given by

T(IW=jI.

erP(r,J‘K(x - y)f(

y) dy,

where K is a standard Calderon-Zygmund kernel, and P is a real polynomial on Iw” x W. We show T is bounded on Lp to itself, when 1

1. INTRODUCTION This is the first of a series of papers dealing with oscillatory singular integrals and harmonic analysis on nilpotent groups. It is our intention to deal with the following three interrelated problems: (1) The study of singular integrals which also carry oscillatory tors that are exponentials of imaginary polynomials.

fac-

(2) The study of convolution operators with singular kernels that are supported on lower-dimensional sub-manifolds. (3) The extension of the properties of some basic operators (hitherto studied only in the context of groups carrying automorphic dilations) to the general setting of arbitrary nilpotent Lie groups. The present paper is devoted to the first problem, but we shall also indicate how our first results bear on the second and third problems. The * This research was partially supported Institute for Scientitic Interchange.

by the National

Science Foundation

and the

179 0022-1236187 53.00 CopyrIght (; 1987 by Academic Press, Inc All rights of reproductmn m any form reserved.

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reader may also wish to consult the announcement [S], where some of these matters are more fully sketched. The basic example of the operator we consider here is the one given by

(Tf)(x) = P.V.JR.e-J)K(X

--y)f( y) dy,

(1.1)

where K is a standard Calderon-Zygmund kernel in R” and P is a general real-valued polynomial in x and y. Our main result is then the boundedness of T on Lp(lR”) to itself, for 1

= @, with multiplication law (,~,,x~,z)~(x~,x~,z’)=(x,+x~, x,+x;, z+z’+ 2(x; x2 -xix,).

(i)

Consider the distribution on H’ given by M= K(x) 6(z), where kernel on Iw* and 6(z) is the Dirac delta function in the z-variable. Then the study of the convolution operator f-f * M, leads via the Fourier transform in the z-variable to the operator (1.1) when n = 2, and with P the nondegenerate skew-symmetric form attached to the Heisenberg group. Note that the distribution A4 is homogeneous of the critical degree -4 with respect to the automorphic dilations, (x, , x2, z) + (tx, , tx,, t’z), t > 0. However, it is significant that A4 is also homogeneous of critical degree (now -3) with respect to the nonautomorphic dilations (x,, x2, z) -+ (tx, , tx2, tz). This is an indication that there should be a theory of singular integrals relative to nonautomorphic dilations on H’. K is a standard Calderon-Zygmund

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181

(ii) A further confirmation of this arises when we take a standard Calderon-Zygmund kernel K(x,, x2, z) on H’, now homogeneous with respect to the isotropic dilations. Then if we consider the operator f + f * K (convolution with respect to the Heisenberg group!), one can easily show that it is bounded on L P,l
Then by the representation theory of H’, the L2 boundedness of H, is equivalent with the L* boundedness of the operator (1.1) when n = 1, K(x)= l/x, and P(x,y)=I{(x-y)b+4x(x-y)u-2(x-y)U’+’}, with the bound being uniform in 1, -co<;l
2. SOME INEQUALITIESFOR POLYNOMIALS

Our treatment of the oscillatory operators (1.1) will be based in part on some simple but useful inequalities concerning polynomials. The main result along these lines that we shall use can be stated as follows. Let W) = Cl,, Gda ?1xoLdenote a polynomial in R” of degree d, where we write xa=xy1...x2, a=(al ,..., a,), with Ial =a1 +a,... +a, ’ This methodcan also be adaptedto prove weak-type(1.1) inequalities;see[ 131

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PROPOSITION.

Suppose that E< l/d, then

I

1x1< I

IP(x)l-“dx
The bound A, depends on E (and the dimension n), but not on the coefficients bh >.

LEMMA. The inequality (2.1) holds in the case of one dimension. To prove this write P(x) = cy=, ajxi, and we may assume the normalization J$‘=, laj 1= 1. One is tempted to factor P as P(x) = ad n,*_, (x - uj) where aj are the complex roots of P and estimate the integral of 1P( -’ directly from this. The difficulty that arises in this approach is that ad may be small. To get around this abstacle we argue as follows: First jL1 IP(x)(dxac>O since I’, IP(x)(dx and cy=, Ia,/ give equivalent norms on the polynomials of degreed. Thus there is an x,, E [ - 1, 11, so that IP( > c. Now write P(x) = (P(x)/P(xO)) P(x,), and split the roots {a,} into two sets; those of the form /?,, with IfijI < 2, and the remaining roots {y,}, with lrjl > 2. Hence [P(x)1 -E=

lFI(x - Pj)lIJ(x, - Pj)l P-ElH(x - Yj)llTI(x~- Yj)l -“IP(

e-E.Now the term

In(x,- flj)l” is bounded by 3”” since lx01 < 1 and I/I,1 Q 2. Next the term In(x-~)/n(x~-~)I-” is bounded by n((lyl + l)/(lrl - 1))“<3”“, since x, x0 E [ - 1, l] and IyI $2. Finally IP( +
-1

IXAPjIe”“dx)l’d~

IP(x)l~‘dxCC:n(~~, i

and the lemma follows since Ed < 1. We now come to the proof of the Proposition which is easily carried out by induction. Let us illustrate this in two dimensions which case is already entirely typical. Write P(x, y) = C a,,xky’, and assume again that xlak,,l = 1. Choose the pair k,, I, so that jako,$l is maximal; then of course lak,,rO12 0. Now by the lemma,

j-<,

IP(x,1.)l-‘dx~A.(~~~ak.,y’~)-’ k

I

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Integrate the resulting inequality in y to get

which proves the proposition. The above proposition will be applied via two immediate consequences which we now state. P(x) = &, =d u,x’ COROLLARY 1. Suppose polynomial of degree d in KY’, with E< l/d. Then

!‘.

,r,<, IP(x)l~“dx=(~,~~,=,

is

a

homogeneous

IP(x)l-‘do(x))Cr”“r”-‘dr,

and so the corollary follows. COROLLARY 2. Suppose P(x) = & Gdaax’ homogeneous) polynomial of degree d. Then IP(x-y)l-“dxdA, sup s b-t”I.rIi 1

is

a

(not

necessarily

(2.3)

YE

To seethis note that (x - y)” = xa + polynomial of degree less than Ial in x. So P(x - y) = x,,, = d a,x’ + polynomial of degree less than d and hence (2.3) again follows from (2.1). We shall state another consequence of our Proposition, which while not applied below does have an interest in its own right, because it gives general relation between arbitrary polynomials in R” on the one hand, and the weight classes A, and the space BMO on the other hand. (For an earlier, but less precise result of this kind see [lo].) COROLLARY 3.

Suppose P(x) is a polynomial of degree d in R”. Then,

(i) IPI belongs to the weight-class A,, as long as p > d+ 1, where its A, bound can be taken to depend only on p and d, and not on the coefficients of P.

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(ii) log IPI belongs to BMO with a norm which is similarly independent of the coefficients of P. The statement that IPI belongs to A, (with bound AP) is the assertion that (2.4) with B ranging over all balls in IR”, where r = (p - 1)) ‘. However, when B is the unit ball centered at the origin, (2.4) is the same as (2.1) with r = E since

c la,l. I/VI<1IP( dxz IZI d+ 1. The second conclusion of the corollary follows from the first by known arguments for A, weights (see, e.g., [ 11).

3. SOME ESTIMATES OF VAN DER CORPUT TYPE In addition to the above inequalities for polynomials, we shall need some estimates for one-dimensional oscillatory integrals whose phases are generalized polynomials. We shall consider real-valued phases @ of the form D(t) = to’ + pz tu2+ . . . + pn t”“,

(3.1)

where p*,..., pL, are real parameters, and a,, al,..., a, are distinct positive (not necessarily integral) exponents. Our objective is to find estimates for

I(n) = i” e”@(‘)&, 1

(3.2)

as II,+ co, which are independent of pLz,...,pn, as long as 0 < CI< /3d 1. PROPOSITION. IZ(l)l < CA-..“, 1,> 0, with .s= min(l/a,, not depend on p2 ,..., p, o A.

l/n), where C does

Simple examples show that the bound for E is sharp. To prove the proposition we shall need two lemmas.

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INTEGRALS

LEMMA 1. Suppose that f(t) is a real-valued Clk) function, that (f (k)(t)l > t’, for some nonnegative 1. Then <

CA

- ‘l(k

k > 1, and

+ 0.

(3.3)

For k = 1 a similar conclusion to (3.3) holds if we make the additional hypothesis that f ‘( t) is monotonic. Proof Write Jf = Jt + Jf, with h to be chosen in terms of II. (If h falls outside the interval [cc,/I] then only one term needs to be considered). Now since we have assumed that 0 < IX,the integral ji is trivially majorized by h. However, IfCk’(t)j > h’ in [h, /?I, so by a known van der Corput estimate (see, e.g., [ 11I), we have that the second term is O(lh’) - Ilk. We then choose h = A~ ‘lck+‘), and get the desired conclusion. We now return to phases @ of the form (3.1) and follow the line of argument in [ 11, II, Sect.21. 2. There exists a C, = Cl (a,, a2,..., a,) > 0 so that C, is in&pendent of pLz,p3,..., u,,, and so that for each t > 0 at least one of the following n inequalities holds: LEMMA

I@‘(t)1 z C,F

)..., p’k’(t)l

z c, tul-k )...) I@(“)(t)l 2 Cl P-“.

Proof Let us recall (3.1) and write teai +k@(k)(t)= c,“=, ak,jujt~~ul where we have set 11,= 1, and ak,,=aj.(ajl)...(aj-k+ 1). Then if the vector w = (w, , w2,..., w,) is defined by wk = t- a~fk@(k)(t), and the vector v = (v,, v2,..., v,) is defined by v, = pjtq--u’, we have w = M(v), where M is a Vandermonde matrix whose determinant equals n;=, aj ni Cl P’ - *, we have that @’ is also monotone there. Thus we can write 1: e‘“@‘(r)dtas a corresponding sum of integrals, where for each integral we apply lemma 1. If I= a, -k is nonnegative we get the estimate

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O(A - ‘l(k + “) = ()(A - ‘lOI). If [ =

a, -k is negative we use merely the fact that in this case I@‘k’(t)( 2 C, (since /?< 1),2 and get O(2-‘lk) = O(n-‘lE). Thus if a, 2 n our sum is O(I-“‘I), while if a, < n we are satisfied with O(l-“n). This concludes the proof of the Proposition. An immediate consequence,via integration by parts, allowing a and fi to vary in [0, 11, is the following. COROLLARY. Suppose Q(t) satisfies the conditions stated after (3.1), and $I E C’ [a, p J. Then SUP Irl/(t)l + j-” IV(t)l a ICf<@

dt

with s = min( l/a,, l/n).

4. THE MAIN THEOREM

We now formulate our main result. Suppose K(x) is a Calderon-Zygmund kernel in R”, which we will assume satisfies the following conditions: (i) K is C’ away from the origin; (ii) K is homogeneous of the critical degree --n; and (iii) the mean-value of K on the unit sphere vanishes. We designate by P(x, y) a general real-valued polynomial in R” x R”, and consider the operator T of the form (Tf)(x)

= P.V. lR. e ip(~r*v)K(x- y) f ( y) dy

(4.1)

initially defined for smooth f with compact support, THEOREM 1. The above operator can be extended to be a bounded operator on Lp(R”) to itself, with 1

Preliminary to the proof of this result it will be useful to point out a general feature of “truncated” singular integrals. Let us consider an operator T of the more general form (Tf)(x)=/ 2 The restriction

Wvy)f(y)d~, !R”

p < 1 is required only when a, < n.

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INTEGRALS

where K is a distribution, which away from the diagonal agrees with a function satisfying the estimate IK&Y)I

(4.2)

GA/lx-YI”

For every E> 0, consider the truncated operator T, defined by

Fix p, 1 6 p < co. If T is bounded on Lp( W) to itself, then so is LEMMA. each T,. Moreover $ 1) 11denotes the operator norm in Lp, we have the estimate (/T,ll d C{ 1)TIJ+ A >, with A arising in (4.2), and C independent of T. Proof: Since our assumptions (and conclusions) are scale-invariant, it suffices to prove the conclusion for E= 1. We shall do this for the operator T’ = T - T, We shall see that the estimate

s Iv-hl

< 114

IT1f(x)lPdx9Cj,~,-h,
If(Y)IP&

(4.3)

holds uniformly in h E R”. Once we have achieved (4.3) we need merely to integrate with respect to h to obtain the desired result. To prove (4.3) we split f into three parts

f=f,+f~+f3.

Here f,(v)=f(~)

when ly-hl


when

+< ly-hl<$, andf,(y)=f(y) when jy-hi as. NOW with Jx-hi<+, T’f,=Tf,,since Jy-hJ
s

,‘i~h,<1,4IT’fAPdx= j,,x_,,<,,4lTfAp~~~CllfiIl;~CIlfIlp.

Next when /x-h/ < l/4, and l/2< Jy-hJ ~514, then l/4< /x--y/ <3/2, and in this range (4.2) gives trivially that

s Ix-

hl < l/4

I~‘f~IP~~~~llfiIIpP~~IlfIlpP~

Finally, when Jx - hJ < 4, and Jy - hJ> $, then Jx -yj 2 1 and so T’ f3 (x) = 0. Putting those three estimates together proves (4.3) and hence the lemma. We now turn to the proof of the theorem. We shall carry out the argument by a double induction on the degrees in x and y of the polynomial P as follows. We let k and I be strictly positive integers, and we

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assume the theorem is known for all polynomials which are sums of monomials of degree < k in x times monomials of any degree in y, together with monomials which are of degree k in x times monomials which are of degree < I in y. Our inductive step will be to add to this all the monomials which have degree k in x and degree I in y. The procedure requires that we start with the result for k = 0 and 1 arbitrary, in order to use induction on k, plus the result at any given k for I= 0, in order to use induction on 1. If k = 0, then the polynomial P depends only on y and therefore eiPCrlcan be assimilated in the functionf: The resulting operator is then an ordinary singular integral operator. Assume now we are at the stage where the theorem is proved for any polynomial that has degree at most k - 1 in x and no restriction on the degree in y, and we add a homogeneous polynomial P(x) of degree k in x and degree I= 0 in y. Then the factor eip(X) can be taken outside the integral, reducing to the case that has been already proved. We proceed now to the proof of the inductive step. We put (4.4) Ial =k IBI = 1

and also write f’(x,y)=

c

x’Q,(~,+Nxt~h

(4.4’)

(x(=k

where the inductive hypothesis applies to Z&(x, y); Here also Q,(y) = &, =laaB yB + polynomial in y of degree < 1, and R(x, y) has x-degree -Ck. Since our theorem is clearly dilation invariant, we may after resealing assume that

c IQ3l=l.

(4.5)

‘,“s’, :: We split K, as K, + K,, where K,,(x) = K(x), 1x1< 1, and K,(x) = K(x), 1x12 1 and take the corresponding splitting of T= To + T,. We deal with the local part, To, first. We shall show that

OSCILLATORY

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INTEGRALS

In fact by (4.4)

e iP(*.~)_,i(~a(x,y)+~o.8~~+~}

=

o(I~-~I),

if 1x1< 1 and JyJ< 2. Now since the kernel of ?‘, has support in Jx-y[ < 1, then when 1x1< 1 only the points 1yl 6 2 matter. Therefore if (~1< 1

+O (s

,x~y(<, Ix-Yl-“+‘lS(.Y~l~u

>

.

For the first term we can use the inductive hypothesis, and the fact that it applies by the lemma for truncated integrals. We conclude that (4.6) holds. We can similarly conclude that

holds for all h E R” (with a bound independent of h). We observe that we merely need to replace P(x, y) by P(x+ h, y + h), and then by (4.4), P(x + h, y + h) equals C ,=,= k, ,p,=,u~~x”~~ + R(x, y, h), where again the inductive hypothesis holds for R(x, y, h) (Here it is crucial that the inductive hypothesis guarantees bounds independent of h.) If we integrate (4.7) with respect to h we obtain

II~ofll, GA, II& and our treatment of the local part is complete. We turn to T,. We write K, = c,?,, tij where 11/0is supported in f< 1x1~ 1 and is bounded, while Il/j(x)=2-“j$(2-jx), j> 1, with I,$ an appropriate function of class C’ supported in + < 1x1< 1. We set T,f(x) = j eip(x,J’) ~jcveKJ4

&

Since bounds for Ti when j= 0 are trivial, we turn to T, for ja 1, and the main point will be to show that the L2-operator norms satisfy the estimate 11 T;II G C2 -jE,

for some E> 0.

(4.8)

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To do this we consider T,*T,. This operator has as its kernel Lj( y, z) given by P(x.Y))@~(~ 5&(Pkz)-

-

z) 6(X

-y)

dx.

By resealing we would obtain the same norm if we were the replace zj( y, z) by Lj( y, z) = 2”‘zj(2’y, 2’2). The result is then

A trivial estimate for Lj that follows from (4.9) is ILj(Yt

z)l G cXB~(Y-z)~

where xB2 is the characteristic function of the ball of radius 2. Now we make the changes of variables x -+ x + y in (4.9), and then write the x-inr = 1x1, lx’1 = 1, tegration in polar coordinates with x=rx’ dx = rn- ‘dr do(x). We also write, after invoking (4.4/j, P(2’(x +y), 2’2) = 1

2’““x”Q,(2’2)

+ R,(x, y, z),

where R, has x-degree strictly less than k. Similarly P(2’(x + y), 2’~) = c

2’““x”Q,(2’2)

+ R;(x, y).

/x1 =k

After these substitutions we get that L,(y,z)=s

where $=+o(x-z+y)$o(x)rnp’

‘(4.10) [.r’l= I

with

and F= Rj(x, y, z) - RJ(x, y).

OSCILLATORY

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INTEGRALS

Now the important thing is that F, when viewed as a polynomial in r, has degree strictly less than k, while E is of degree k. Thus if we use the van der Corput estimate (the corollary in Section 3 with a, = n = k) for the inner integral in (4.10) we obtain -I/k

C2-’

1

x’*(Q,(2’z)-

Q,(z’y)

as an estimate for it. Since the inner integral is obviously also bounded we can replace this by c2 -3 , f;- k x’YQol(2’z) - Qol(2’~)

~ d/k

CI

for any 0 < 6 < 1. Next carry out the integration in x’, invoking corollary 1 in Section 2. This gives us - 6Jk

IQ~(~‘z)-Q~(~‘Y)I Since &I=k,IPI=Ila,DI=l,

>

Xs2(Y-Z).

there is an It101=k and I/&l=l,

(4.11) so that

I > c > 0. However Q,, (2’2) = 2” Cla, =, LZ,,~Z~ + lower order terms in la;IoB,, z, so

and hence by Corollary 2 in Section 2, sup 1zj (z, y ) I dz < C2 -“2 piialk= C2 -jZE, i Y if we choose 6 sufficiently small to that 6/k < l/Z. Similarly SUPI z

lZj(Z,

Y)l dy < C2-j2”.

This proves that 11 TTTjll d C2P2’“, and hence (4.8). however, that the norms of Tj on L’ or L” are uniformly by interpolation with (4.8) we get that the Lp norms of Tj decreasing as j + co, if 1

It is obvious, bounded. Thus are exponential and concluding

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5. FURTHERRESULTS We shall here describe briefly some extensions of Theorem 1 which can be obtained by the same methods. First, with almost no change in the proof, we can replace the Calderon-Zygmund kernel K(x - y) by a distribution kernel K(x, y) which is assumed to satisfy the following hypotheses: (i) K is a C’ function away from the diagonal for which the estimates jK(x,y)l
= !” eiP(“~-“)K(x,y)f( y) dy

(5.1)

satisfies similar conclusions to Theorem 1. Next we observe that the proof of Theorem 1 did not use the full force of the decrease at infinity of the kernel K, (as long as the phase P(x, y) was nontrivial). We can formulate this as follows: COROLLARY1. For each d > 2, there exists an CC,>0, so that whenever (i): P(x, y) is a real polynomial of total degree
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193

We shall denote by K(x) the appropriate analogue of a Calderon-Zygmund kernel for G. Namely: (i) K is C’ away from the origin: (ii) K is homogeneous of the critical degree in the sense that K(~,(x)) = f-“K(x), t>O, with a=a,+a,+ ... + a,. (iii) the mean-value of K on the unit sphere vanishes. If P(x, y) designates a real-valued polynomial on IR”x R we consider the operator T of the form (VW

= PJ. jR. eiP(xyy)K(y - ’ . x)f( y) dy.

(5.2)

COROLLARY 2. For the operator T in (5.2) the same conclusions hold as those stated in Theorem 1.

The proof is almost identical with that of Theorem 1. The following comments may help clarify the situation. Each monomial, xa, has besides it (standard) degree, (which equals ~1~+ t12+ . *. + a,), its G-degree, which equals a, . a, + a2. a2 + . . . + a,. a,. The induction is carried out respect to the G-degreesof the polynomials. Note that if the G-degree is d d, then the standard degree d does not exceed max(l/a,). d, and thus the proposition and its corollaries in Section 2 apply. Also note that when we decompose integrals on G with respect the “polar coordinates” given by the dilation structure, the resulting polynomials in the “radius” r become generalized polynomials of the type treated in Section 3. Extensions of our theorem to nilpotent groups without automorphic dilations, and to kernels carried on lower dimensional manifolds, will be treated in a subsequent paper of this series.

REFERENCES 1. R. COIFMANAND C. FEFFERMAN, Weighted norm inequalities for maximal functions and singular integrals, Sfudia Math. 51 (1979), 241-250. 2. G. DAVID AND J. L. JOURN~,A boundedness criterion for generalized Calderbn-Zygmund operators, Ann. Math. 120 (1984), 371-397. 3. D. GELLER AND E. M. STEIN,Estimates for singular convolution operators on the Heisenberg group, Mad Ann. 267 (1984), 1-15. 4. G. MAUCERI, M. A. PKARDELLO AND F. RICCI, “Twisted convolution, Hardy spaces,and Hijrmander multipliers,” Suppl. Rend. Circ. Mar. Palermo 1 (1981), 191-202. 5. D. MOLLER, Singular kernels supported by homogeneous sub-manifolds, J. Rein. Angew. Math. 356 (1985), 9&l 18. 6. D. H. hIONG AND E. M. STEIN,Singular integrals related to the Radon transform and boundary value problems, Proc. Nat. Acad. U.S.A. 80 (1983), 7697-7710. 7. D. H. PHONG AND E. M. STEIN, Hilbert integrals, singular integrals and Radon transforms I, Acra. Math. 157 (1986), 99-157. 8. F. RICCI AND E. M. STEIN, Oscillatory singular integrals and harmonic analysis on nilpotent groups, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), 1-3.

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9. P. SJijLIN, “Convolution with oscillating kernels,” Indiana Univ. Math. J. 30 (1981), 47-56. 10. E. M. STEIN, Oscillatory integrals in Fourier analysis, Ann. Math. Stud. 112 (1986). 11. E. M. STEIN AND S. WAINGNER, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Sot. 84 (1978), 1239-1295. 12. R. S. STRICHARTZ, Singular integrals supported on sub-manifolds, Studiu Math. 74 (1982), 137-151. 13. S. CHANILLO AND M. CHRIST, Weak (1.1) bounds for oscillatory singular integrals, preprint.