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Ck)(x)\ ~ 1, then (1.1)
eill.Cx) dxl :s ckl\.-1 /k
a
* The
reader will note that there are several related topics not touched on in this survey. Chief among them is the subject of oscillatory integra Is arising in the solution of hyperbolic equations and their generalizations - the class of "Fourier integral operators." For an elegant introduction to that subject see [1], Chapter 4.
f
b
holds when
310 (i)
k
'2: 2
By the previous case
(ii) or k = 1, if in addition it is assumed that ¢/(x) is monotonic.
f
Proof. Let uS show (ii) first. We have
b iA
J
311
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
E. M. STEIN
e 4>dx =
a
f
b
i D(e A.4»dx = -
a
biA
f
e 4>t_D(1)dx +
iA~,eiA4>
a
Similarly a
f
a
b
'
ciA
d'i "
ek/(IW)' /k .
eM
j
ciAt_D(IJd+
e iA 4> dxl S ck/(A8i Ik .
a
]b
The boundary terms are majorized by 2/A., while
l
e-O
ciA
l d: U)d'
Clearly however
a
J
eM
b
SJ f Id~ ~,I
iA e 4> dx\ < 20 .
e-O
a
Thus b
1
=r
J
~(l)dX
J b
< 2ck iA e 4> dx - (Ao)1/k + 20
dx 4>'
a
by the montonicity of 4>'. The last expression equals ~ i\
1_
1 4> '(b)
a
1_1 ' 4>'(a)
which is dominated by 2/A. This gives the desired conclusion with c 1 = 4.
If 4>(k\c)
i
0 I and so c is one of the end-points of [a,b] , a similar
argument shows that ck/(A.0)1 Ik + 0 is an upper bound to the integral. In either situation the case k + 1 follows by taking 0 = A- 1/k +l which I
We now prove (i) by induction on k. Let us suppose that' the case k
proves (1.1) with ck+1 = 2ck+2.
is known, and assume (taking complex conjugates if necessary) that
4>(k+1)(x)~1. Let x=c be the point in [a,b] where 14>(k)(x)1 takes
COROLLARY.
its minimum value. If 4> (k)(c) = 0, then outside the interval (c~ 0, c+ 0),
conclude that
we have that 14>(k\x)1 ~ 0, (and of course 4>'(x) is monotonic when k = 1). Write
jc{8+{8,j
a
a
e-O
e+O
(1.2)
I a
Under the assumptions on 4> in Proposition 2, we can
b
e iA 4>(X)l/J(x)dx
< e0' /k
~4>(bJ\ ,
t '(')Idj. 14>
312
E. M. STEIN
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
This follows from (1.1) by integrating by parts an estimate of the form
f
x
eiA¢(X)dx! 'S
c0- l/k ,
for a < x < b .
(c) Asymptotics: We already know that the behavior of
principal branch of z-'/,-e/2 in the plane slit a long the negative half
r. b
eiA¢tjJdx is
'a
determined by those points x o ' where ¢'(x o) ~ 0, (the "critical" points of ¢). Assuming that the support of t/J is so small that it contains only one critical point of ¢, the character of the asymptotic expansion then depends on the smallest k so that ¢(k)(x o ) i 0, and is given in terms of powers of A in a way which is consistent with Proposition 2.
(1.3)
t/J
is a sufficiently small neighborhood of x
I(A)
f
c
e "'¢(x) "'(x) dx " A- 1 Ik
O
'
then
(d~r ~(A) ~ a{ UJo
REMARK.
\w\ < 1 ), then gives the desired asymptotic expansion (1.4). Step 2. Observe next that if 7/ (C:;' and e is a non-negative integer, then (1.5)
c
O(A c(N+!)/k
c
,)
as A
oo
f _~
iAx2 e
x
e
e iAx2 x e7/(x)dx
written as
__
When k is even, then aj ~ 0 for j odd.
We shall give a proof of the proposition when k ~ 2. There are three
~ fe ih2 xe7/(x)a(x/E)dx +fe iAX2 xe7/(x) (l-a(x/e)) dx .
with .!Df '" ~ d~
f
ci+l.
J
00
(1.4)
-00
The second integral can be
e iAx \ !D)N [x e7/(x) [1- a (x/E)]] dx
~x . A simple computation then shows that this term
is majorized by
steps. CNJ AN
Step 1. This is the observation that
1/2 ,
for Ix\::: 1, and a(x) ~ 0 when Ixl ~ 2, and write
The first integral is dominated by
~
!
qCX)dX! < Me l/2
To prove this let a be a COO function with the property that a(x) '" 1
J
~ a F j/k ,
in the sense that for any non-negative integers Nand r
(1.3 ')
aX!is. The pewer series expansion of (w_i)-y,-e/2 (which holds for
Suppose ¢ is real and smooth, k ~ 2, and
¢(Xo)~¢'(xo)···~¢(k-l)(xO)~O,while ¢(k\xo)iO. If tjJ(C~ and the support of
where e, is a non-negative integer; in the case e is odd the integral 2 vanishes. In fact the left-side of (1.4) is foo e-(I-iA)X xedx which by -00 x2 e a change of variables equals (l_iA)-V2-e/2 (" e- x dx. However, when -00 A > 0, (l-iA)-Y>-V2 ~ A-Y,-V2(1/A_i)-y,-e/2 , where we have fixed the
a
PROPOSITION 3.
\x\e-2N
dx
~ C~A-Nee-2N-l
lx\>e 00
eih2/e-x2dx""A-l/2-e/2
313
~Cfe)A-j
if e - 2N < - 1. Altogether then the integral in (1.5) is bounded by
314
E. M. STEIN OSCILLATORY INTEGRALS IN FOURIER ANALYSIS C N ll+1 +A -N/-2N+lj and we need only take
(With N > e;
9'
€
315
= A-I /2 , For (a) we use (1.4); for (b) we use (1.5); and for (c) we use (1.6). It is
to get the conclusion (1.5).
A similar (but simpler) argument of integration by parts also shows
then easy to see that their combination gives the desired asymptotic ex iA 2 pansion for J e x t/J(x)dx. Let us now consider the general case when k = 2. We can then write
that (1.6)
4>(x) = c(x-x O)2 + O(x-x O)3 with c 10 and set 4>(x) = c(x-x )2[IH(x)], o where e is a smooth function which is O(x-x o), and hence le(x)1 < 1
!eiAX2f(x)dX=O(A-N), every N>O
whenever f
(S,
and f
when x is sufficiently close to x O ' Moreover, 4>'(x) 10, when x I x O ' but x lies sufficiently close to x O ' Let us now fix such a neighborhood
vanishes near the origin.
of x O ' and let y = (x-x o) (1 + e(x))l /2. Then the mapping x .... y is a
Step 3. We prove the proposition first in the case 4>(x) = x 2 . To do this write
J e
where 'J; is a
iAx2
C~
t/J(x) dx =
J e
iAx
e-
x2
(e
x2
function which is 1 on the support of
eX t/J(x) =
N
~.
and of course cy2 = 4>(x). Thus
f
t/J(x))'J;(x) dx ,
e iA 4>(x) t/J(x) dx =
J e
iAcy2
'J;(y) dy
t/J. Now for
each N, write the taylor expansion
2
diffeomorphism of that neighborhood of x o to a neighborhood of y = 0,
~ bl J + xN+IRN(x) = P(x) + xN+1RN(x) .
with ;;; (C~ if the support of
t/J lies in our fixed neighborhood of x O '
The expansion (1.3) (for k=2), is then proved as a consequence of the special case treated before.
j=O
REMARKS: Substituting in the above gives three terms N
(a)
00
Ib j
j=O
J
ei..\x
2
e- X
2·
xJ dx
-00
00
(b)
J
2 ei..\x x N+ l RN (x) e- x2 'J;(x) dx
-00
(c)
!eiAx2p(x)e-X2(1_;;;(X))dX.
(1) The proof for higher k is similar and is based on the fact that
I
00
eiAX\-x\edx = Ck,e(l- iArCe+1)/k .
o (2) Each constant aj that appears in the asymptotic expansion (1.3) depends on only finitely many derivatives of 4> and
t/J at x o ' Note
e.g. that when k = 2, we have a o = vrr(-i4>'(x O))-1/2t/J(x O)' Similarly the bounds occurring in (1.3') depend on upper bounds of finitely many derivatives of 4> and
t/J in the support of t/J, the size of the support of
t/J, and a lower bound for 4> (k)(x0) .
316
E. M. STEIN
References: The reader may consult Erdelyi [8], Chapter II, where further citations of the classical literature may be found.
We can only state a weak analogue for the scaling principle, Proposi tion 2; it, however, will be useful in what follows. PROPOSITION
2. Oscillatory integrals of the first kind, n 2: 2
Only some of the above results have analogues when n? 2, but the
multi-index a,
5. Suppose
lal
we say that a phase Junction rP defined in a neighborhood of a point n in R has Xo as a critical point, if (vrP) (x ) ~ O.
X
t/I
I(tt rP(X)1 ? 1
leA)
=
J
t/I.
Then
t/I.
throughout the support of
lI
(C~(Rn), and rP is a smooth real-valued
function which has no critical points in the support of
rP is real-valued, and for some
o
o
4. Suppose
t/I ( C~'
> 0,
extension of Proposition 1 is simple. Continuing a terminology used above
PROPOSITION
317
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
(2.1 )
eiArP(X)t/I(x)dxl
Then
<: ck(rP)' A-Uk (I\l/JII
IR
L
00
+ I\vl/JIIL l )
n
eiArP(x)t/I(x)dX = O(A -N), as A ->00
for every N?: 0 .
Rn
and the constant ck(rP) is independent of A and t/I and remains bounded as long as the C k +l norm of rP remains bounded.
with k
=
lal ,
Proof. For each Xo in the support of t/I, there is a unit vector g and a small ball B(x o)' centered at x o' so that (g, vx)rP(x)? c > 0, for
Proof. Consider the real linear space of homogeneous polynomials of degree k in R n . Let d(k,n) denote its dimension. Of course
x (B(x o)' Decompose the integral JeiArP(x)t/I(x)dx as a finite sum
lxut\al=k is a basis for this space. However it is not difficult to see that there are d(k,n) unit vectors g(l),g(Z), ... ,gcd(k,n)), so that the
I
homogeneous polynomials
laa~:O)\ ?: 1 ,
where each t/lk is COO and has compact support in one of these balls. It then suffices to prove the corresponding estimate for each of these integrals. Now choose a coordinate system xl' x ' ''', x
f
g.
2
n
so that Xl
Then
eiArP(x)t/lk(x)dx =
lf J ~
j - 1,,," d(k,n), give another basis.
This means that if
n
lies along
(g (j). x)k,
!eiArP(X)t/lk(X)dX ,
e
for some
lal
= k, there is a unit vector g = g(XO) , so that
I(g, vx)krP(x O)\ iArP(x
1"
J
'" x ) \ n t/lk(X 1 , .. ·,X n )dX dxz, ... ,dx . l n
?: ak' with ak> 0 .
Moreover since we can assume that the C k + l norm of rP is bounded we can also conclude that
Icg, v:)rPCx)!
? ak/2 whenever x ( BCx O)
,
where
But the inner integral is O(A -N) by Proposition 1, and so our desired
B is the ball centered at x of fixed radius. (The radius of B can be
conclusion follows.
taken to be a small multiple of the Ck+l norm of
rP.) Next choose an
~~..
~;...~.
._~;;;:;E"-""_ii;;;;-.;CT7 ~~~~iiiii::~,~~'o""
318
319
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
E. M. STEIN
Suppose ¢> has a critical point at x 0 . If the symmetric n x n matrix
appropriate covering of Rn by such balls of fixed radius, and a corre sponding partition of unity, 1 = ~ l'/j(x) , with O'S l'/j 'S 1, ~ 1\7l'/jl ~ b k , and each l'/j supported in one of our balls. So
~
a:(xO)~
·ax·J
is invertible, then the critical point is said to be non
1
degenerate. It is an easy matter to see by the use of Taylor's expansion
f
iA
e ¢> ¢/dx
~
=
J
iA e ¢> l/Jl'/j dx =
~
J
that if x iA
e ¢> l/J j dx .
° is a non-degenerate critical point, then in fact it is an isolated
critical point.
J
PROPOSITION 6. Suppose ¢>(xO) To estimate Je iA ¢>ljJ j dx, with ~ determined as abo'{e, choose a
coordinate system so that Xl lies along ~. Then
Jf\lr . f
critical point at x 0. If l/J
/k -lIk)'t (f ax A
I \y
I}
Note. Again each of the constants aj appearing in the asymptotic ex
.
Xl x ' the above proposition 2 gives no better than a decrease of order A-I /2 , while the asymptotics of =
pansion depends on only finite many values of derivatives of ¢> and l/J
at x o. Thus e.g. a = (1Tn / 2 (_i ll j )-1I2) . l/J (x 0) , where
o J=1 ~ a 2¢>(xO) Ill' 1l2' ''', Il are the eigenvalues of the matrix ,' Similarly n k
.n
it
Let us go back for the moment to the case of one dimension. If ¢> has a critical point at x o ' and ¢>' does not vanish of infinite order at x o ' then after a smooth change of variables ¢> can be transformed to a simple canonical form ;:;, with ;{;(x) = ± x k (for X near 0). There is
each of the bounds occurring in the error terms depend only on upper bounds for finitely many derivatives of ¢> and l/J in the support of l/J, ~ 24J(xO) the size of the support of l/J, and a lower bound for det aXjaXk \ .
a
t
The proof of the proposition follows closely the same pattern as that of Proposition 3. First, let Q(x) denote the unit quadratic form given by x)=x2+x2+ ... +x2 _x2 _ ... _x 2 where O<m
1 and in a
special case corresponding to k = 2. To the asymptotics of the latter situation we now turn.
t
ax/lX
1
the proposition below Shows that the true order is A-I.
=
A.... ""
where the asymptotics hold in the same sense as (1.3), (1.3').
A final integration inthe other variables then leads to (2.1).
no analogue of this in higher dimensions, except for k
""
eiA¢>(X)l/J(X)dx'UA-n/2~al-j,as
Rn
n
REMARK. Let us note that in R2 if ¢>(x)
f
(2.2)
j
dl'/j (X ,···,X )dX n 1 1
and the support of l/J is a sufficiently
small neighborhood of x , then
For the. inner integral we invoke (1.2) giving us an estimate of the form
-1 ckak
0, and ¢> has a non-generate
O
, iA¢>(x "",xnh e 1 l/Jj(Xl' ... ,Xn)dX dx 2 ,···,dx n ·
eiA¢>l/Jjdx =
fCC;
=
(2.3)
r
j
Rn
00
eiAQ(X)e-\x\2xedx 'U A- n /2-\e\/2
~ c/m,e)A- j j=O
320
E. M. STEIN OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
with £ = (£
£
... £ ) a multi-index
£ n 'm
I'
/ = x/ ,"', xn
1£1
=
£ +£ + ... + £
'1
and
n'
2
C
; also note that if one £j is odd then (2.3) is identically
zero. To prove (2.3) write it as a product
n
II
j=I
(J
~
OO
e-+iA x 2 e - X2X£.J dx
=
(f
j=I
~
-00
lIn
321
---.!'!
AN
Ixl- 2N , allows one to conclude the proof of (2.4) in analogy with that
of (1.5).
~
OO
e _ X2Xp.J dx
A similar argument also show that whenever ~
. -1/2-£./2 (1 +: iA) J
f
S
and ~ vanishes
near the origin, then
_~
(2.5)
jeiAQ(X)t(X)dX = O(A- N), for every N > O.
n -'h-P./2 and expand the function n (l/A +: i) J , (for lilCge A ) in a power j=1 series in 1/A.
We then combine (2.3), (2.4) and (2.5) as before to obtain the
The analogue of (1.5) is the statement that
(2.4)
I
J
e
iAQ
asymptotic formula (2.2) in the special case when ¢(x) = Q(x).
(X)x£77(X)/ dx :::: AA -n/2-/£1 /2; if 77
C~(Rn) .
f
To pass to the general case one is then fortunate to be able to appeal to the change of variables guaranteed by Morse's lemma: Since ¢(xO)
= 0,
(\j¢) (X 0) = 0, and the critical point is assumed to be non-degenerate,
Rn
there exists a diffeomorphism of a small neighborhood of xO in the To prove it we consider the two-sided cones ,
r j {xllXjl' ~ i, lxi' }, 0
U r J9 = R n,
Then since J=1 .
r.J
and the smalle, cases
defined by
rr" {xllXjl' ~ kIxl'} .
we can find functions OJ"
j
=
1,. .. , n, each
x-space to the y-space, under which ¢
is transformed yi + y~ + ... + y~
y~+1 _ ... - y~. Observe that the index m is the same as that of the
form corresponding to
~
(~
iP¢(xO)l
JxjJxk \ .
homogeneous of degree 0, and COO away from the origin, so that n
1 = j=1 ~ D.(x), X = 0, with O.J supported in J .
r J..
References.
For a proof of Morse's lemma see Milnor [20], §2.
Then we can write
§3. Fourier transforms of surface-carried measl1l'es
Je
iAQ
(X)/71(X)dX =
~ Je iAQ (x)/77(X)D j(X)dX . J
In the cone
rj
one uses integration by parts via
Let S denote a smooth m-dimensional sub-manifold of R n (not necessarily closed). We let da denote the measure on S induced by the Lebesgue measure on R n , and fix a function ijJ in C~(Rn). Consider now the finite Borel measure dJl
=
ijJ(x)da on R n , which is of course
carried on S. The problem we wish to deal with is that of finding esti D.eMQ(x) = eiAQ(x) J
with
Dj(f) = ±lXj %:j .
mates at infinity of the Fourier transform of /.I.' i.e. d~(O. We shall consider two cases of this problem.
This, together with the fact IXjl
> 1 Ixi in v/2n
rj,
and I (tDj)NOjCX)j ::::
(1) Suppose first dim S
= n -1,
and S has non-zero Gaussian curvature
at each point. By this we mean the following: Let x
° be any point of
S,
322
E. M. STEIN
323
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
and consider a rotation and translation of the underlying Rn so that the point xO is moved to the origin, and the tangent plane of 5 at x O
°)
becomes the hyperplane x n O. Then near the origin (i.e. near x the surface 5 can be given as a graph x n ¢(x ,'" ,x _ ) with ¢ (C'Xi. 1 n 1
and ¢(O) 0, (\7¢)(O) '= O. Now consider the (n-1) x (n-1) matrix
=0
1° when TJ is sufficiently close to the" north pole" TJN 2° when Tf is sufficiently close to the "south pole" TiN
'=
(0,"·,0,1);
'=
(0,"',0,-1);
3° the complementary set on the unit sphere.
=0
Let us consider 1° first. The function ¢(x, TJ) has the property that
=0
. Its eigenvalues 1i g:~~O)/. k l<),k
(\7x¢) (0, TJN)
/11"'" /1 n- are called the princi1 J (2 ) ¢ 0 pal curvatures of S at xO, and their product det ~ ax.~) is the Gaussian curvature at x . J k
- -
°
THEOREM 1.
a
.
=0
Suppose 5 is a smooth hypersurface in Rn , with non
vanishing Gaussian curvature at each point, and let d/1 Then (3.1)
'=
'=
O. We want to see that for each Ti sufficiently close to
TfN there is a (unique) x
=
X(TJ) , so that (\1x) ¢(x(TJ)' TJ)
O. The latter
is a series of n equations and one can find the desired solution by the implicit function theorem, which requires that we check that the Jacobian determinant det ((\7 x\7x¢) (0, TJN»
=I 0;
but this of course is our assump
tion of non-vanishing curvature. Notice that if the l1-neighborhood of TJN
rf;da as above. is sufficiently small, then det
!(d/1) ~(e-)I ::; AIe-I-(n-l)/2 .
=
Proposition 6 (with xO
=0
(a2~~W~TJ») =I 0,
so we can invoke
x(TJ», as long as the support of
if;
is suffi
ciently small. This proves (3.2) when TJ is in region 1°. The proof when
Proof. It would be convenient (in applying Proposition 6 above) to change notation momentarily by taking n to be n + 1. Now by the compactness of the support of rf; (and since when the surface is given as a graph x n+ 1 '" ¢(x 1 ,''', xn) / then da matters to shOwing that
'=
J1 + I\7et> I2 dx 1 , .. ·/ dx ) we can reduce
f eiA¢(X'TJ)~(X)dxl::;
(3.2)
n
e-
'=
det
,I
)=1
XJ'Tf ' + ¢(x)TJ n+1 ' then \1x¢(x, TJ) J
'=
ever (TJi + ... + TJ;)1 /2 ~ c > 1 , since TJ is in 3° and \1¢(x)
'=
O(x) as
x .... 0; thus l\1x et>(x, TJ)I ::> c'> 0, if the support of if; is a a sufficiently small neighborhood of the origin. Hence for the TJ in region 3° we may
REMARK. We have used only a special consequence of the asymptotic
formula (2.2), namely the "remainder estimate" analogous to (1.3') when
'=
)'=1
=
(TJl"'" TJ n) + TJ n +l (\7¢) (x) . How
for every N. The proof of Theorem 1 is therefore concluded.
AA-n /2
xl TJl + x 2 TJ 2 + ... + x n TJ n + ¢(x '"'' x ) TJ +!, with 1 n n n+l A(TJl''''' TJn+1) , .I Ti~ '= 1, and A> 0; also et>(O) '= (\7et» (0)
(-t~~~'1
n
use Proposition 4 to conclude that the left-side of (3.2) is actually 0(11. -N)
Rn where ¢(x, TJ)
TJ is in 2° is the same. 50 we now come to the region 3°. Since ¢(x, TJ)
J
N '" r =
0,
=0
O. Had we used the full formula we can get an asymptotic expan
sion for d~(f); its main term is explicitly expressible in terms of the Gaussian curvature at those points x ( S, for which the normal is in the
0, and the support of ;;; is a sufficiently small neighbor
direction f or - f.
hood of the origin. (2) We shall now consider the problem in a wider setting. Here 5 will We divide consideration of the unit vector Ti into three cases.
be a smooth m-dimensional sub-manifold, with 1::; m
s: n-1,
and our
assumptions on the non-vanishing curvature will be replaced by the more
.,..
324
E. M. STEIN
_._ .. ,--- -_
..
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
f
general assumption that at each point S has at most a finite order contact with any hyperplane. We shall call such sub-manifolds of finite type.
-
..'._."--_.. '.
--
ei¢(x)'(;j;(x)dx =
.." .. _-----
325
O(I(I~l/k)
Rm
(These have some analogy with the finite-type domains in several complex variables, which are also discussed in Nagel's lectures [21].) The pre cise definitions required for our considerations are as follows. We shall assume that we are considering S in a sufficiently small neighborhood of a given point, and th-en write S as the image of mapping ¢: R m ... R n , defined in a neighborhood U of the origin in R n . (To get a smoothly embedded S we should also suppose that· the vectors aa¢ , Xl
~~ ,"', ~ u"2
u"m
are linearly independent for each x, but we shall not need that assump tion.) Now fix any point xO I:: U c R m , and any unit vector 'fJ in R n .
We shall assume that the function (¢(x) - ¢(x infinite order as x ... x 0.
0». 'fJ
does not vanish of
Put another way, for each xO
unit vector 'fJ, there is a multi-index a, with 1 <:
lal ,
I::
U and each
so that
with ¢ as described above, and the support of ;;; sufficiently small. Now we can write (= A'fJ' with 1'fJ1 = 1, and A > O. Then we know that there is an
a, with lal <: k,
so that
(~)a ¢(x)· 7/ f
0, whenever x is
in the support of ;j; (once the size of the support has been chosen small enough). Thus the conclusion (3.3) follows from (2.1) of Proposition 5. References. Theorem 1 in its more precise form alluded to in the remark goes back to Hlawka [14]. See also Herz [13], Littman [18], Randol [25], and Hormander [16]. When S is a real-analytic sub-manifold not contained in any affine hyper-plane, then it is of finite type as defined above. For such real-analytic S estimates of the type (3.3) were proved by Bjork [2].
(txt (¢(x)· 'fJ)lx=xo Ie- O. Notice that if (x','fJ') are sufficiently close to (xo''fJ), then also
4.
(~t ¢(x')'7/'l x=x' /0.
The smallest k so that for
each unit vector 'fJ then :!la, lal<:k, with be called the type of ¢ the type of ¢ in U I in U I .
at xO. Also if U
I
.aa (¢(x)'7/)1
ax a
%
will
x=x
is a compact set in U,
will be the least upper bound of the types for x O
2. Suppose S is a smooth m-dimensional manifold in Rn of finite type. Let dp. = t/Jdo, with t/J I:: C~(Rm). Then
THEOREM
Restriction theorems for the Fourier transform n The Fourier transform of a function in LP(R ), 1 < P <: 2 is most
naturally thought of as an LP' function (via the Hausdorff-Young Theorem) and so at first sight it is viewed as defined only almost-everywhere. This impression is further supported by the case p ~ 2, when clearly the Fourier transform can be completely arbitrary on any given set of zero Lebesgue measure. It is therefore a noteworthy fact that whenever n 2': 2 and S is a sub-manifold of R n (with some appropriate "curvature") then there exists a Po = p(S) , Po> 1, so that every function in LP,
1 <: p Id~(()1 < A!(I-E, for some E> 0,
(3.3)
<:
PO has a Fourier transform restricting to S (i.e. with respect to
the induced measure on S). Let us make this precise. Suppose that S is a given smooth sub-manifold in R n , with do its
and in fact we can take support of
E =
11k, where k is the type of S inside the
t/J.
Proof. By a s uitab Ie partition of unity we can reduce the prob lem to showing that
induced· Lebesgue measure. We shall say that the LP restriction property holds for S, if there exists a q = q(p), so that the inequality
326
E. M. STEIN OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
(~. II(OI'dO(f~'/,
(4.1)
"Ap,,(S.) 1I 1 11 p
Thu s (4.2) follows from Holder's inequality if we can show that
IIT(f)llp~ ~ Allf11po '
(4.3) hOlds for each f closure in S. THEOREM 3.
(S,
whenever 50 is an open subset of 5 with compact where P~ is the dual exponent to PO'
Suppose 5 is a smooth hypersurface in Rn with non-zero
To prove (4.3) we consider the function Ks (initially defined for Re(s) > 0) by
Gaussian curvature. Then the restriction property (4.1) holds for
1 ~ p ~ 2; : }, (with q
=
2 ).
. (4.4)
Proof. Suppose ljJ ~ 0 and ljJ ( C~. It will suffice to prove the
inequality
(4.2)
327
(L
li(OI2O/4
(fj
f .
e- 21Tlx ' sI: I~ _4>(~')I-l+S 7J(~ -4>
Rn
Here we have abbreviated (~l""'~n-l) by ~'; we have set ;;;(~') =
~ 1/2
ldO
s2 K (x) = _e_ s l(s/2)
< Allf Il •• L
ljJ(~')(l + 1'Vc/>(~')12)1/2 , so that ;;;(~') d~' = dt!; also 7J is a C;;'(R) function which equals 1 near the origin.
2 < Po will then follow by for Po = 2n n++ 3 , and f (S; the case 1 < _P_
Now the change of variables ~n
interpolation. * By covering the support of ljJ by sufficiently many small
~s(xn)
open sets, it will be enough to prove (4.2) when (after a suitable rotation and translation of coordinates) the surface S can be represented (in the
)1
2
dt!
dfl
=
=
e-21Ti(X"~'+xn4>(~'»;;;(~')dt = ~s(xn)K(x),
T(f) (x)f(x) dx
f e-21Tix'~dt!(~)
(, (x ) _ e S2 s n - r(s/2)
JOO
.
e
-21TlX nSI:n
I~nl
-1+s
7J(~n)d~n .
-00
Now it is well known that ~s has an analytic continuation in s .
*In fact the interpolation argument shows that we can take q so that (4.1) holds with q = (n-l) P'. which is the optimal relation between p and q.
n+l
J
with
where (Tf )(x) = (f * K)(x), with
K(x) =
~n + 4>(~') in the above integral
Rn - l
support of ljJ )as a graph: ~n =4>(~l'''''~n-l)' Now with dt! = ljJda we have that
JIf(~ Jf(~) f(~) f
->
shows that it equals
which is an entire function; also ~o
='
1; and l~s(xn)1 ~ c\xnl-Re(s),
where Ixnl ~ 1 , and the real part of s remains bounded. From these facts it follows that Ks has an analytic continuation to an entire function s (whose values are smooth functions of xl,''',x n of at most polynomial growth). One can conclude as well that
r~~~ I
------.----------------::::.-""_--
-~;;;;,t'~:--
328
c.-----------
----
:"tL
._--~-
'-_.
-.~-
'..,----
- -
.-~.~~~--
_.~,-
...--.'"
";;;;:~
E. M. STEIN
(a) Ko(x)
=
K(x) , It would of course be of interest to know what are the exponents p
(b) IK-n/2+it(x)1 ~ A, all x ( R n , all real t ,
~
(c) IKl+it(OI
:s A
and q (if any) for which the restriction holds if we are dealing with a
n
1
all ~ (R , all real t.
given sub-manifold S. This problem is highlighted by the fact quite
In fact (c) is immediate from our initial definition (4.4), and (b) follows from Theorem 1. Now consider the analytic family Ts of operators defined by Ts(f) =
f * Ks ' From (b) one has (4.5)
329
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
general sub-manifolds S (those which are of finite type in the sense described in §3) have the restriction property: THEOREM 4.
Suppose S is a smooth m-dimensional sub-manifold of R n
of finite type. Then there exists a Po
IIT_ n / 2 +it (f)11 00:S Allfll l' all real t, L
L
=
po(S) , 1 < PO' so that S has
the LP restriction property (4.1) with q = 2, and 1 ~ P:S Po' (In fact if the type of S is k, we can take Po = 2nk/(2nk-l.)
and from (c) and P1anchere1's theorem one gets COROLLARY.
(4.6)
II Tl+it(f)11
L
2
S Allfll
L
2' all real t ,
':S Allfll
L Po Theorem 3 is complete.
L Po
, with PO = 2n +32 , and the proof of n
+
Proof. As we saw above, it suffices to prove (4.3). However Tf
=
f * K,
and K(x) =d~(-x), Theorem 2 tells us that IK(x)1 ~Alxl-l/k. So according to the theorem of fractional integration, (see [26], Chapter V),
we therefore get (4.3) with
REMARKS:
(i)
hyperplane. Then S has the LP restriction property for 1 ~ p ~ PO' for some Po > 1 .
An application of a known convexity property of operators (see [28]) then shows that IITo(f)11
Suppose S is real-analytic and does not lie in any affine
For hypersurfaces with non-zero Gaussian curvature this theorem
-4 = ~ .- ~, Po Po n
where
relation among exponents is the same as Po =
a
= n - 11k, and this
2;~~1
.
Q.E.D.
is the best possible, only insofar as it is of the form (4.1) with q ;::: 2. If q is not requ ired to be 2 or greater, then it maybe
Further bibliographic remarks. The initial restriction theorem dates from
conjectured that a restriction theorem holds for such hypersurfaces
1967 but was unpublished. The sharp result for n
in the wider range 1 ~ p < 2n/(n + 1). This is known to be true
C. Fefferman and the author and can be found essentially in [9]; see also
when n = 2 (see also §7 below).
Zygmund [33]. Further results are in Thomas [30], [31], Strichartz [29],
(ii) For hypersurfaces for which only k principal curvatures are non
=
2 was observed by
Prestini [24], Christ [4], and Drury [7].
vanishing, Greenleaf [12] has shown that then the corresponding results hold with 1 Theorem 3.
~ p :s 2lk: ~), giving an extension of
(iii) In the case of dim (S) = 1 (i.e. in the case of a curve) there are
5.
Oscillatory integrals of the first kind related to singular integrals A key oscillatory integral used in the theory of Hilbert transforms
along curves is the following:
a series of results extending our knowledge of the case n = 2
alluded to above. For further details one should consult the
references cited be low.
(5.1)
p.v.
foo -00
e iPa(t)
dt t '
"
"
330
E.
M.
STEIN
where Pa(t) is a rea I polynomial in t of degree d, P (t) a
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS d
=
.
.~ aJ.t J . It
331
LEMMA 1:
)=0 00
was proved by Wainger and the author in [27], that the integral is bounded with a bound depending only on the degree d and independent of the coefficients a o,a 1 ,"',ad' The relevance of such integrals can be better understood by consulting Wainger's lectures [32]. We shall be i.nterested
(5.3)
f
(e
ip (t) iPb(t) dt a -e )t
< Ad
(1 ~ 1 _1:: II), +
10
with Ad independent of (ajl, lbjl and E> O.
here in giving an n:.dimensional generalization of this result. We formulate it as follows. Let K(x) be a homogeneous function of degree -n; suppose also that IK(x)1 S Alxl- n (i.e. K is bounded on the unit sphere); moreover, we assume the usual cancellation property: ~X'I=l K(x')da(x')=O. We let P(x) =
LEMMA 2. Let P(x)
=
~
a x a be a homogeneous polynomial of
lal=d a
degree d on Rn . Write
a x a be any real polynomial of degree d.
lalSd a
~
mp =
f
!P(x')! da(x') .
Ix'!=l
THEOREM 5:
(5.2)
p.v.
If
Then, eiP(X)K(x)dxl SAd
Rn
(5.4)
IIog
(I ~ [) Ida(x') S Bd '
!x'!=l
with the bound Ad that depends only on K and d, and not on the coefficients aa'
I
with B d independent of P. *
Nagel and Wainger observed that if K were odd, one could prove (5.2) One can if one wishes give an elementary (but complicated) proof of
from the one-dimensional form (5.1) by the method of rotations (passage to polar coordinates). To deal with the general case we need two lemmas. d
Let Pa(t) d
Pb(t)
= .~
)=1
=
)=1 .
bJ.t J .
.
.~ aJ.t J denote a real polynomial on R1 , and write also
Lemma 2. It may however be more interesting to obtain it as a conse quence of a general property of polynomials in Rn related to the class of functions of bounded mean oscillation. This property can be stated as follows. It is very well known that the function log Ixl is in B.M.O., and this is usually the first example discussed in that theory. It is surprising therefore that the follow ing natura I genera lization seems to be been overlooked.
mp
*Of course > O.
in writing (5.4) we assume that P
is not identically zero, Le.
332
E. M. STEIN
THEOREM
6. Let P(x) be any polynomial of degree
In Rn . Then
f
loglp(x)! is in 8.M.a. and in addition Illog IP(x)111 HMO :S B d '
J ""
We now pass to the proof of Lemma 1. We prove it by induction on d.
I
~
eiPa(t)
(eiat_eibt)
9f1
:S A
becomes
(1 + IIOglsll)
{
is classical. Let us now assume (5.3) for polynomials of degree d - 1,
!adl
and observe that the estimates (5.3) we wish to prove are unchanged if we
at, a j
O. Thus we may assume that b d
=
1 and ladl:S 1.
Now write
e l
/
l'
+{ ladll/d
if1a (t) dt
d
1
where PaCt) is a polynomial of degree d, with t d having coefficient one. Again
J"
ipact) iPb(t) dt (e-e )t
~
,
f
+
{Xl .) I
and that integral is estimate like
"f
J""
E'
and we treat these two integrals separately. (If
f";' , E
Q!:t I :S Cd '
1, Le. the estimate
""
replace t by
iPb(t)
e
t ... \ad l- I /d t, the integral
The proofs of Theorem 6 and Lemma 2 will be given in an appendix.
=
""
by the corollary of Proposition 2 in §1. Next, by a change of variables
where B d depends only on d, and not otherwise on P.
The case d
333
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
I
E'
> 1, we have only
e;Pa(I)"/.
J-
t l-
d'
log
I} I
while
Let us consider the second
f f
integral. It equals
J-
if1a(t) dt
e
I
:S
e;Pb(l)
~.
I
Now since b d = 1, we see that (d/dt)d Pb(t) = d!, and hence
\adl
since b d
f
I
E'"
(e
=
l d /
\adl l / d
~
= }
= }
IIOg
I~~ I)
1. Next
ipaCt)
--e
iPb(t) dt ) -"-' t
J 1
E ..
(e
iQa(t)
-e
iQb(t) dt ) -+ 0 ( t
J ~
I
dt _, t
E ...
--_.
__._-----------
.---
---------
334
--------=--=====-.....:....-=---==
E. M. STEIN
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
with d-l
J, (t
d-l
j
Qa(t) = I ajt , Qb(t) = I j=1 j=1
bjt j ,
since IPa(t)-Qa(t)l:::: \tl and !Pb(t)-Qb(t)[:::: Itl. However, by induction
Ix 1=1
E
[exP{i
I
Pi(x')t i) - exp (i
I
(eiQ,(I) _eiQb(l\
~
"A'd_l
o
t ~ IIOgl~ll)
Armed with Lemmas 1 and 2 we can now prove Theorem 5. We may
!
a
lal::::d
has no constant term, and using polar
XU
n(x')do(x') .
However by Lemma 1 the inner integral is bounded by
Ad ( 1 +
~ d
J'
log
II)
~ I
IP'(X')
+
Gathering all these terms together then proves (5.3).
assume that P(x) =
~\ )
1
hypothesis (using (5.3) for 0' = 0, and 0 =1, and d - 1 ),
f
mjl i)]
335
and so an appeal to Lemma 2 shows that
J J
E2
1
x'I=1
j exp (i IPj(X')t ) - eXP(i
I
j m/ ) dtt I (sup 111(x')I)du(x'):::: Ad
01
a d .
coordinates x = tx', t > 0, Ix'i = 1, we write P(x) = ! PJ.(x')t J , j=1
proving (5.2) and the theorem.
where P /x ') are restrictions to the unit sphere of homogeneous poly nomials of degree j. Let us also set mj = ~x'I=1 IP/x')1 du(x) , and n write K(x) = C 11(x'), with 11 bounded and ~x'I=I11(x')du(x') = O. Then to prove (5.2) it suffices to show that
f
6. Oscillatory integrals of the second kind: an example related to the Heisenberg group To motivate the interest in this example we recall the definition of the Heisenberg group Hm . The underlying space of H m is C m x R, i.e. Hm=l(z,t)l, with zfC m , tfR; the multiplication here is (z,t)(w,s)= m
eiP(x)K(x)dxl :::: Ad '
01 ::::1 x !::::E 2
(z+w,t+s+
with Ad independent of 01' 02 and P. The above integral can be
their corresponding quasi-distances. The first are the usual dilations (z ,t)
written as
J J 0
(
Ix'I=1
2
'Ip e1
, tJ.
/x)
~~ 11(x')du(x').
01
Since 11 has vanishing mean-value this integral may be rewritten as
->
(pz, pt), P > 0, and the metric could be defined in terms of the
usual distance. The second are the dilations (z,t)
->
(pz,p2 t ), and the
appropriate quasi-distance (from the origin) is then (lzI 4 +t 2)1/4. The latter dilations and metric are closely tied with the realization of the Heisenberg group as the boundary of the generalized upper half-space holomorphically equivalent with the unit ball in C n+ 1 . This point of view, as well as related generalizations, is elaborated in Nagel's lectures [21].
M··-:,:::,::-"··
it'!
jjiiiji
336
_._.
-- - ~
_._,
L_
__
._~.
.~.;I!'.i
.i'.~L__
In the present context the first type of dilations and corresponding
=
f. (One sees this for example in Krantz's lectures [17],
where in the formula of Henkin we have a kernel made of
pr~ucts
=
J
L(z-w)ei.\
Cm
conflicting types of dilations (and related metrics) occur in e.g. the
au
(T.\)(F)(z)
(6.3)
metric would be appropriate if one considered expressions related to ordinary potential theory in H m viewed as R2m + 1 . However the two
""""','!,,;"""l!'~
337
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
E. M. STEIN
solutions of
.;....
__
.!!:.~_~~~~~_"!!!1!!_.
of
functions each belo~ging to one of the two above homogeneities.)* Other expressions of this type occur in the explicit formulae for the solutions
(with < z ,w > the anti-symmetric form which occurs in the multiplication law for the Heisenberg group) is bounded on L 2(C m) to itself, uniformly in .\, -
00
< .\ <
DO •
*
We now change our notation and call 2m B(' , '), and F
=
=
n, L
=
K, -'\<',' >
=
f. Then the operators T,\ have the form
of the a-Neumann problem (see [1], Chapter 7). Let us now consider the simplest operator on the Heisenberg group (Tf)(x)
(6.4)
displaying simultaneously these two homogeneities. The prime example is given by
=
J
K(x~y)eiB(X'Y)f(y)dy
.
Rn
(6.1)
Tf
=
f*K We shall suppose B is a real bilinear form, but we shall not suppose
where convolution is with respect to the Heisenberg group, and the kernel K is a distribution of the form
(6.2)
K(z ,t)
that B is necessarily anti-symmetric nor that K is homogeneous of degree -n.
=
L(z )o(t) .
THEOREM
L(z) is a standard Calderon-Zygmund kernel in Cm = R 2m , i.e. L(pz)
=
p -2m L(z) , L is smooth away from the origin, and L has vanishing
mean-value on the unit sphere. Here o(t) is the Dirac delta function in the t-variable, and in an obvious sense is homogeneous o(pt)
= p -1
o(t).
smooth away from the origin, and with vanishing mean-value when /l
~2m
- 2 with
bounded on L2(R n ) to itself, for O:S /l S n; when 1 < P < tor is bounded on LP(R
n) to itself if
I~-tl S
;n'
00,
the opera
n, then T is bounded on LP(R n ) to itself for 1 < P < 00. The
fJ.
cal ones.
bound of T can then be taken to be independent of B.
Fourier transform in the t-variable. This leads to the problem of showing
=
We shall give only the highlights of the proof, leaving the details, further variants, and applications to the papers cited below. Let us con sider first the L 2 part of assertion (a) when n/2 < /l S n. Suppose T/
that the family of operators T,\ defined by
*In particular the terms Al and A
2
that appear in §6 of [17].
n.
(b) If we drop the assumption that B is non-degenerate but require that
respect to the other dilations; in both instances the degrees are the criti
We turn next to the question of proving that the operator (6.1) is bounded on L 2(H m). The most efficient way is to proceed via the
=
(a) If B is non-degenerate, then the operator T given by (6.4) is
Thus K is homogeneous at degree -2m - 1 with respect to the standard dilations, and at the same time homogeneous of degree
7. Suppose K is homogeneous of degree - fJ.' O:S /l S n ,
*For further details see Mauceri, Picardello and Ricci [19] and Geller and Stein [l0] .
---_.
338
OSCILL~ORY
E. M. STEIN
is a C;;' function, with T/(x) = 1 for Ixl ~ 1/2, and T/(x) = 0, for
J
with K replaced by K o = T/K, and T"" with K replaced by K = (l-T/)K. Observe first that since Ko(x-y) is supported where Ix-YI ~ 1,
2 ITo(f)(x)1 dx
~A
J
~A
f
If(x)\2dx.
Rn
-J
T:T ",,' The kernel L of this operator is given by
Ko(x-y) eiB(y ,y) f(y) d Y[ L(x,y)
~c
J
IX-YI-tL+1lf(y)1 dy ,
Ix-yl9
-iB(Z ,x-y) K",,(z-x) K",,(z-y) dz .
Schwarz's inequality implies
While operators of the type (6.4) are not translation invariant they do satisfy T-h TThf = eiB(h,h)eiB(X,h)T(eiB(h'·)f(.))
!L(x,y)1
(6.7)
~A
f ly-hl~2
:s A .
We next integrate by parts in the definition of L(x ,y), using the fact that (Dzle-iB(Z,x- y ) = e-iB(z,x-y), where D = i(a,\7z)/lx-YI, with z y 1 a = B- (X- ), and B denotes the matrix so that B(x,y) = (Bx,y). Ix-yl The result is IL(x,y)1 ~ ANlx-y!-N , for every N ~ 0, and hence
f(x-h). Applying this to To gives the following generali
zation of (6.5)
ITo(f )(x)1 2 dx
=Je
Now since K"" is in L 2(R n ) (here the assumption n/2 < tL is used),
and thus the L 2 theory for f -> f * Ko (which is non-trivia I only when tL = n ) proves (6.5).
Ix-hl::;1
2 If(y)1 dy.
This will be done by proving the corresponding result for the operator
~Ko(x-y) eiB(x,y) fey) dy
f
2 IT""f(x)1 dx
Rn
In fact when Ixl ~ 1,
=
J Rn
J
If(y)1 2 dy.
lyl~2
Ixl9
with Th(f )(x)
~ A2 n
We now turn to the proof of
J
(6.6)
\To(f)(x)\2dx
Rn
estimating To(f)(x) in the ball Ixl ~ 1 involves only f(y) in the ball
(6.5)
339
and an integration in h shows that as a consequence
\xl~l. We write T=To+T"", where To is defined as in (6.4),but
Iyl ~ 2. We claim
INTEGRALS IN FOURIER ANALYSIS
IL(x,y)! 'S AN(l + \x_y!)-N, N ~ O.
This shows that L is the kernel of a bounded operator on L 2 If(y)1 2 dy ,
proving the boundedness of T:'T "" and thus of T ",,' The proofs of the L 2 boundedness when 0 ~ tL:S n/2 (in part (a) of the theorem), and the L 2 boundedness when tL = n but when B is not assumed to be non degenerate, are refinements of the above argument.
=ri;;~~~~~~-_
340
....",",~
_'''_
-
-,..",~,-'--
-...--
- ,~..
..."".-
f~~"
~'-'~.-
E. M. STEIN
'8"~i':~~;.=,~';,.~;;,;';-::21~:'2k~;;e;:~_~}~kc"''';:''''';_~''---''''F~~';:'t:;'~"S".;~~}~"'t:"::"t11':!--r;t-=:.s:[:"..
. ,,,,,, ...-~.-",,:e~",~,~~c",,,",",,-= .... ;~~Z'i~~~~E~
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
Let us now describe the main idea in proving the LP inequalities
(6.9)
IlfllLP 'S
stated in (a) and (b) above. We shall need a generalization of BMO (and
of HI) which may be of interest in its own right. Suppose E = I e I is
Q
a mapping from the collection of cubes Q in R n to complex-valued
functions on Rn so that
.
To prove this we need only observe that (!fll 'S 2f~, and use the result (see [10)) for the standard It function. The point of all of this is that for operators of the form (6.4), there is a naturally associated
leQ(x)1 = XQ(x), all x ,
Apllf~IILP
341
Hk
and BMO E theory, and it is given by
choosing where X Q denotes the characteristic function of the cube Q. Let us
eQx)=e (
(6.10)
-iB(X
,C Q )
,
define on "E-atom" to be a function a so that for some cube Q (i)
a is supported in Q
(ii)
la(x)1 'S
(iii)
Ja(x) eQ(x) dx = 0 .
where c Q is the center of the cube Q. The basic step in the LP theory (besides an appropriate interpolation which goes via (6.9)), is the
1/IQI
Hk
is then given by If If = ~ Ajaj' with each a j an E atom, and ~ IAjl < 001. In a similar vein the function f~ will be defined as The space
f~(x) = sup.-l.
(6.8)
xfQ
IQI
J!f-f~1
proof that when /l = n our operator T maps L 00 to BMO E . Let us give the proof in the case (a). We may assume that I\fll 00 'S 1 , and L
suppose first that Q is a cube centered at the origin. Then we have to show that there exists a constant YQ' so that
dx ,
i~1
(6.11)
Q
f
ITf -yQI dx
'S A .
Q
where The corresponding inequality for a cube centered at another point, say
f~ ~~I =
J
c
f e Q dy ,
since we must split f into three parts to take into account the oscilla
and we take BMO E = !flf~ ( L 001. Some of the basic facts about the standard HI and BMO spaces* go
Hk
and BMO E , and sometimes these come free of charge. Po One such case is the following assertion: Suppose f (L , P'S Po
1;
< 00, and f~ ( LP. Then f (LP and *The standard situation arises of course when e
' then follows from the translation formula (6.6), (and this is the reason
for defining e Q as we do). Turning to (6.11), the argument is not exactly the same as in the standard caSe (see e.g. Coifman's lectures [5] or [9]),
Q
through for
Q
Q
= XQ' all Q.
tions of eiB(x,y). Suppose Q = Q(j' has side-Ienghts (j, then write f
=
f 1 +f 2 +f 3 , where f 1 = f in Q20' f 1 = 0 elsewhere, f 2 = f in ~Q20
n Qo-l'
f 3 = f in ~Q2(j
n cQo-l'
f 2 = 0 elsewhere, f3
=
0 elsewhere.
342
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
E. M. STEIN
We shall now state a generalization of this theorem which also in
(Note that f 2 occurs only when 8:s "j2/2 .) We have F = T(f) = F1 +F2 +F3 , where F j = T(f j ). For F1 we make the usual estimate, using the fact that T is bounded on L 2 . Next observe that IK(x_y)eiB(x,y)_ K(-y) I :S
co
cludes the oscillatory integral result given in Theorem 5 (in §S). Suppose P(x,y) is a real polynomial on Rn x Rn of total degree d. Consider the
[_1_ _l_J '
operator
+
lyln+1
!yjn-l
(Tf )(x) = p.v.
(6.12)
(Qo IF2 (x)-y Q I :S
A{O J ~ +8J ~} < Iyln+l
£0 20
!yln-l
J
eiP(x,y) K(x-y)f(y)dy
Rn
if x (Qo and y (~Q20' Thus if YQ = fK(-y)fiy)dy, we get that for x
343
where K is homogeneous of degree -n, smooth away from the origin and with vanishing mean-value.
A' .
THEOREM 8. The operator T
Qo-l
given by (6.12) is bounded on L 2(R n ) to
itself, with a bound that can be taken to depend only on K and the degree d of P, and is otherwise independent of P.
Finally
This is a recent result obtained jointly with F. Ricci. The proof is
JK(X- y ) eiB(x,y)f iy)dy
based in part on a combination of ideas used in the proof Theorems 5 and
Fix) =
7. This result has also many variants, and we now state some of these:
=f(K(X- y)- K(_y))eiB(x,y)f /y)dy +
J
(i)
One may also show that the operators (6.12) are bounded on LP,
l
K(-y)f iY) eiB(x,y) dy
(ii) Given p, with 1 < P < 00, then there is an c =c(p ,d), so that if K =
is homogeneous of degree - f1, n - c:S f1
F~ + F~ .
must assume that P(x,y) is not of the term P(x,y) = Po(x) + PI (y). (iii) One can replace K(x-y) in (6.12) by a more general" Calderon
Plancherel's formula (which we may since we have assumed that B(x,y)
Zygmund kernel" K(x,y), a distribution for which the operator when P = 0 is bounded in L 2 , and which in addition is a function (when
is non-degenerate). The result is
JIF~(x)12dx:s J 1F~(x)12dx J =A
Rn
n, T is still bounded on
LP. However now the bounds may depend on P, and in addition one
For F~ we again make the standard estimates, and for F~ we use
Q
'S
R n
!K(-y)fiy)!2dy:S Al
J ~Q
dy lyj2 n
a- 1
=Afl.
x
f- y ) which satisfies IK(x,y)!:S A\x_yl-n, !V'xK(x,y)! + lV'yK(x,y)1
:S A!x_yl-n-l . References. For the detailed proof of Theorem 7, other variants, and
Combining these estimates proves (6.11), and hence the fact that T takes
applications to the (j Neumann problem see the papers of Phong and the
L 00 to BMO .
author [22], [23].
E
U
344
.
E. M. STEIN
tl
OSCILLATORY INTEGRALS IN FO=R ANALYSIS
7. Further oscillatory integrals related to restriction theorems and
Proof. Write
B ochner-Riesz s urnmabi Ii ty
fJ 1
We have seen that if S is a hypersurface in R n with non-vanishing
F(O = ((Tf) (e-n 2
(7.2)
Gaussian curvature, then
=
o
Je~x.e-
tf;(x) du(x) =
0(1(1~(n-1 )/2),
as e-
1
eie--(y(s)+y(tnf(s)f(t)ds dt ,
0
and we shall try to apply Plancherel's theorem (more precisely, the
00
Hausdorff-Young inequality) to F. To do this break the above integral
S
into two essentially equal parts according to t whenever tf;
f
345
'#
C~, and this is a typical oscillatory integral of the first
> s or t <:: s , which
divides [OJ) x [0,1] into the union of two regions R 1 and R 2 · We then consider the mapping of R I .... R 2 given by x = yes) + y(t) , Le. Xl =
kind. We may pass to an oscillatory integral of the second kind when we replace the C~ function tf; by an L r function f, and consider the re sulting linear operator on f. The resulting operator, as is easy to
= Y2(s) + Yit). It is easy to verify on the basis of our assump 2 tions that this mapping is one-one, and its Jacobian J satisfies \JI =
observe, is in fact the dual to the restriction operator considered in §4.
Iy;(s)-y;(t)! ::: cls-tl.
s + t, x
Hence by Theorem 3 we can state that operator is in fact bounded from L 2(du) to LP'(R n), where p' is the dual exponent to 2n + 2. We shall n+3 now describe the sharper result in this setting that can be obtained for n =
(7.3)
Therefore
f e~·(y(s)+y(t))f(s)f(t)dsdt f =
R2
R1
2.
We fix a curve t .... yet) = (t, Y2(t)), 0 <:: t <:: 1, lying in R 2 , with yet) f C 2 , and having non-vanishing curvature, i.e. IY'i(t)l::: c > O. Con
with f(x t ,x 2) = f(s)f(t)IJI- l . So if we denote by F (0 the quantity appearing in (7.3) then when t
ever 1 <:: r <:: 2, and l/r' + l/r = 1, we know that
sider the transformation T, which maps function on the interval [0,1] to functions on R 2 , given by
(7.4)
f e~·y(t)f(t)dt.
11 F t II
I
(7.1)
(Tf)(O
=
eie--Xf(xt,x2)dxtdx2
r'R
L (2 )
<:: cllfll r R 2 . L()
However
o
Ilfll~r(R2)
9. Under the assumption above T is bounded from LP[O,I] to L q(R 2) , whenever 3/q + lip = 1 and 1 <:: p < 4.
THEOREM
(Note that when p .... 4, then q .... 4 in the above relation between p and q .)
= JI1f(Xl
=
,x 2 )[r dX t dX 2
f1f(S)\r If(t)jf\J\t- r dsdt
1-'
\
1 <:: c J1f(SW If(tW Is_tI - r ds dt .
346
E. M. STEIN
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
To estimate the last integral we need to invoke the theorem of fractional
where Ijt is a fixed C ~(Rn x R n ) cut-off function; is a real-valued
integration in one dimension in the form
COO phase function which we assume satisfies the assumption that its
fg(S)g(t)(s-t)-1+adsdt~Allgllu, ~-l=a,
Hessian is non-vanishing, Le. O
(7.6)
(
det
So we take g(t) '" If(tW, then Ilgll u = Ilfll~ whe~ p = ur. Then if we fix a so that -1 +a = 1-r, then 3:2 r = ~ , and p = 3 ~rr' The limitation
PROPOSITION:
0< a becomes r < 2, and with q = 2r' we obtain from (7.4) that
(7.7)
I[T,\(f)11 2
IIF II
I
1
L',R', ~ c'
1'1
(
~
1
If(tll
Pd~2/p
a2(x, t")\
I 0.
axiat"j)
n
(R )
L
iI
347
~ A,\-n/21Ifll 2 L
(R n )
.
COROLLARY: I\T,\(f)11
(7.8)
LP
'
(R n )
< AA- n /p'l\fl\ -
LP(R n )
,
with a similar estimate for Fit") which is the analogue of (7.4), but where 1:S p
taken over R 2 . Since F = F1 + F and F = (Tf)2 we obtain 2
~
2, and lip + lip'
=
1.
REMARK. The boundedness of T,\ for any fixed ,\ is trivial, but what /IT(f)IILq(R2):S AilfllLP[O,l]'
is of interest is the decrease in the norm as A ->
00.
This decrease is
consistent with the special case when (x, t") is bilinear (and non
Ii
Not e th at -3 = -3 = 3r --3 = 1 - -1 , so -3 q 2r' 2r P q
-t
" t'Ion -1 = 1 ,an d the 1"Imlta P
degenerate); when we take ,\
1 ~ r < 2 is equivalent with 1 ~ P = l~ r < 4. Theorem 9 is therefore proved.
To prove (7.7) we argue as in the proof of Theorem 7; as in the treat ment of the operator Too it suffices to show that the operator norm of I
T~T,\ is bounded by A,\-n. Now this operator has as its kernel the
function K,\(t",n) given by
Suppose we consider the family of operators T, depending on the parameter '\, ,\
> 0, def ined by
T,\(f)(t")
in that case we recover the usual
vexity theorem.
in the above argument. If we want to generalize Theorem 9 it is natural to look for a corresponding extension of the L 2 boundedness of the these lines is as follows.
00
corollary follows from the propos ition by the use of the M. Riesz con
It is clear that inequa lities for the Fourier transform playa key role
Fourier transform and the Hausdorff-Young theorem. One result along
->
(LP,LP') inequalities for the Fourier transform. Notice also that the
=
J Rn
(7.9)
K,\(t",77)
=
J Rn
eiA.(x,t")Ijt(x,t")f(x)dx,
ei,\((X,71)-(X,t"))Ijt(X,77)Ijt(x,t")dx.
348
E. M. STEIN
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
Now since
we can find a = (al,···,a n ), so that the aj depend smoothly on x and III (x , ~ ,1]) ~ cl~ -1]1 on the support of K,\ (~,1]). Set Dx = (a, 'Vx)· iA Then since (Dx)~ei.\(lI>(x,1])-lI>(x,O)= e (lI>(x,1])-lI>(xl», we can
ill
, I I
det (\7[( IT, \7 ~lI»)(tO, ~ 0) ",0 .
(7.12 b )
ll(x,~,1]) = (a,'Vx )[lI>(X,1])-lI>(x,O] =(a:~ a,1]-~) + 011]-~12
These assumptions will be supposed to hold at all (to, ~ ~ in the n support of ljJ(t,O, where ljJ is a fixed function in C~(Rn-l x R ). THEOREM 10.
Under the assumptions above the operator (7.11) satisfies IIT,\(f)llq 'S AA-n/qllfll p
(7.13)
integrate by parts N times in (7.9) and obtain (7.10)
IK,\(~,1])1 'S A N(l +'\I~-1]I)-N, N ~
349
. h q= (nn+1 - 1) p, , 1p+p= 1 1,
WIt
o.
(a) when n
It follows from (7.10) with N = n + 1, that the operator TAT,\ which has
=
2, if 1 'S p < 4
(b) when n ~ 3, if 1 'S p 'S 2.
kernel K,\ has a norm bounded by A,\-n and the proposition is proved. We shall now formulate some theorems for oscillatory integrals of the form (7.11)
(T,\f)(O =
J
eiAlI>(tl)ljJ(t,~)f(t)dt, ~ (Rn ,
Rn - l
REMARKS:
(1) When lI>(t,O = tl~l +ti2+ .. ·+tn-l~n-l + (tl, .. ·,tn-l)·~n' and (V4J) (0) = 0, then the conditions (7.12) are near the origin equivalent with the non-vanishing Gaussian curvature of the graph t n = (tl, .. ·,tn_ l ). If we apply the result (7.13), letting ,\ --> 00, it is not difficult to recover Theorem 9 from part (a), and Theorem 3 from part (b).
which will generalize the restriction theorems (Theorem 9 above, as well as Theorem 3 in §4) and also give results for Bochner-Riesz summability. Notice that (7.11) are mappings from functions on R n - l to functions on R n . The basic assumptions on the real phase function lI> are as
(2) The proof of part (a) follows the same lines as the proof given for Theorem 9, once we use (7.8) as the substitute for the Hausdorff-Young theorem; further details as well as relations with Bochner-Riesz summa
follows: We consider for each fixed (to,~ ~ the associated bilinear form B(u,v) defined by B(u,v) = (v ,'Vt)(u, 'V~)(lI>)(tOl 0), with u (R n- l ,
bility may be found in the papers of Carleson and Sjolin [3] and Hormander
v ( R n . Our first assumption is that
This will also serve as a good review of many of the notions we have dis
[15]. Since part (b) has not appeared before, we will outline its proof. cussed here.
a
(7.12 )
B is of rank n - 1 .
Proof of part (b). It suffices to prove the case p = 2, since the case Thus there exists (an essentially unique),
u(
Rn ,
lui
= 1, so that the
scalar function t ... (U,'V~lI>(t,O) has a critical point at (to,~o). We
p = 1 is trivial and the rest follows by interpolation. Now the case p = 2 is equivalent by duality to the statement
shall also assume that this critical point is non-degenerate, i.e. we suppose the non-vanishing of the (n-1)x(n-1) determinant:
(7.14)
II T A(F)II
2 Rn-l
L ()
'S
AA-n/r'IIFII
r
L (
Rn
)
I
350
E. M. STEIN
with r ~ 2(n+1), where n+3
(T,t) (F) (t)
~
a In fact 'i7t 'i7~ct> already has rank n -1 by assumption (7.12 ), and
J e-iAct>(tl)!/J(t,OF(Od~,
so we need only choose ct>o(~) so that (u,'i7~)ct>o(~) 0/ 0 to increase the rank of 'i7x 'i7~ lI> to n. t t Rn-l .
Now, as in the proof of Theorem 3 in §4, we form K~ defined by
Rn
K~(~,17) ~L
We can calculate •
1(s/2)
1\
II
Rn -
f
e iA «f>(X,77}--(f>(X,O)!/J(t,77)J;(t,Olx 1-1+SlJ(x )dx, n
n
n
R
J
II
351
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
T,t(F) T,t(F) dt
with dx
l
~
dtdx n , and where lJ is a
C'(;'
function which equals 1 near
the origin. We easily verify
II
and write as
K~(~,77) ~ KA(~'77)
(7.16)
II
J
II II
since
KA(~'17)F(~)F(77)d~d17 ,
(f>(X, 0 ~ ct>(t, 0, when x ~ (t,O) .
RnxRn
Next
I:
with
K~+it is the kprnel of a bounded operator from
II
!
(~,77) ~
(7.15)
KA
J Rn -
II
(7.17) e iA (ct>(t,77)-ct>(t l »!/J(t,77)J;(t,
~)dt .
L 2 (R n ) to itself with norm S AA- n / 2 .
This follows by applying the estimate (7.7) of the proposition above and
l
using the non-degeneracy of the Hessian of lI>(x,~). Finally we claim 9
It suffices therefore to see that KA is the kernel of a bounded operator
that
n
from L\R ) to Lr(R n ), with norm not exceeding A,\-2n/r'.
~
on
n
R
x
n
R so that the following holds: we will write x t Rn , as
(t,x n ) with t~(xI,···,xn_l)tRn-l. The ~ we can construct will satisfy: (i)
~(x,~) ~ ct>(t,O + ct>o(Ox n
(ii) the determinant of the nxn matrix 'Vx'i7~(f> is non-vanishing.
IK~n/2+1 /2+i\~'17)1 SA.
(7.18)
Because of our assumptions on ct> we can construct a phase function
To see this write K~(~'17) as KA(~'17)v(A(ct>o(17)-ct>o(O» where
K' 00
vs(u)
~
s2 e 1(s/2)
f
-00
ix u e n lJ(Xn)!Xnl-I+Sdxn .
352
E. M. STEIN
Then since IV-n / 2+ 1 / 2+it(u)\
'S clul n /2-1/2,
as u
"" we see that to
Cf.
Qo
prove (7.18) it suffices to show that
i
IP(x)1 2 dx)1 /2 and
'S
A(AI71~c;i)-n/2+1/2 .
Qo
Pd ,
finite-dimensional space IK A(C;'71)1
(7.19)
353
OSCILLATORY INTEGRALS IN FOURIER ANALYSIS
IP(x)ldx are two (equivalent) norms on the so (8.1) holds for Q
=
Qo' and then for
general Q. Now it is well known (see e.g. (6]) that a function which satisfies a
·1 I
In proving this estimate for the integral KA given by (7.15) we may suppose that the integrand is supported in a sufficiently small neighbor
hood of a given point t
=
°
"reverse Holder" inequality belongs to the weight space A"", Examining the proof of this fact one obtains an r
=
r(d) , 0 < r < 00, and a constant
to' (for otherwise we can write it as the sum of
finitely many such terms). When we write lI>(t,71)
=
lI>(t,
=
Cd' so that
(\7c; 11»(t ,71) .
(71-c;) + O(71-c;)2 we see that these are two cases to consider as in the
(8.2)
proof of Theorem 1 in §3: 1° when the directions 71 - c; or c; - Tf are close to the critical direction u arising in condition (7.12 b ); or 2° in
~--.L f IP(X)ldX~' (-lIQI JIP(x)l-rdx~l/r < IQI Q
Cd
Q
for a 11 cubes Q.
the opposite case. In the first case we use stationary phase (i.e. Proposi tion 6 in §2) to obtain (7.19). In the second case, we actually get
From (8.2) and Jensen's inequality, Theorem 6 follows easily.
O(AI71-c; I)-N , for every N ~ 0 as an estimate, by Proposition 4. This
Let us now assume that P is homogeneous of degree d. Observe
completes the proof of (7.18), and shows that K~n/2+l/2+it is the kernel of a bounded operator from L1(R n ) to L ""(R n ) , with bounds uniform in A.
also that since (8.2) holds, if we normalize P by the condition that mp
=
The proof of the theorem is then concluded by applying the interpolation theorem, as in the proof of Theorem 3. 8.
'S
f,
\x\=1
IP(x)\ da(x) = 1 , we can conclude that
f
(8.3)
IP(x)rr da(x)
'S cd
.
Ixl=1
Appendix Here we shall prove Lemma 2 and Theorem 6 which were stated in §5. First let Pd denote the linear space of polynomials in Rn of degree d. We claim tha t there is a constant Ad' so that
(8.1 ) (
_1
IQ\
J
IP(x)1
Q
2
d~
1/2
< Ad
-
-.L IQI
However, when u > 0, Ilog u\
=
log+u + log+
fr 'S u + } u-r .
Therefore
(8.3) implies (5.4) whenever m p = 1, and so that result also holds in general.
J
E. M. STEIN
DEPARTMENT OF MATHEMATICS
PRINCETON UNIVERSITY
PRINCETON, NEW JERSEY 08544
IP(x)1 dx
Q REFERENCES
holds for all P (. :Pd , and all cubes Q. The space Pd is invariant under translations and dilations, and so a moment's reflection shows that to prove (8.1) for all P (. Pd , it suffices to prove it for Q = Qo' the unit cube centered at the origin. However
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···'·'·::·····.'·.•·· ..
!' .1····.·
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II
..-.,..-"-,-.-,.. ,~-,-,----_ ..
OSCILLATORY INTE:GRALS IN FOURIER ANALYSIS
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Ii \
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forms," preprint. E. Prestini, "Restriction theorems for the Fourier transform to some [24] manifolds in Rn in Harmonic analysis in Euclidean spaces," Proc. Symp. in Pure Math. 35, part 1 (1979), 101-109.
[25] B. Randol, "On the asymptotic behaviour of the Fourier transform of the indicator function of a convex set," Trans. Amer. Math. Soc.
139(1969),279-285. [26] E. M. Stein, "Singular integrals and differentiability properties of functions I" 1970, Princeton University Press. [27] E. M. Stein and S. Wainger, "The estimation of an integral arising in multiplier transformations," Studia Math. 35(1970), 101-104. [28] E. M. Stein and G. Weiss, "Introduction to Fourier analysis on Euclidean spaces," 1971, Princeton University Press.
[291 R. S. Strichartz, "Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations," Duke Math.
J.
44(1977),705-713. [30] P. A. Tomas, A restriction theorem for the Fourier transform," Bull. A.M.S. 81 (1975),477-478. , "Restriction theorems for the Fourier transform in Harmonic Analysis in Euclidean spaces," Proc. Symp. in Pure Math. 35, part 1 (1979),111-114. [32] S. Wainger, "Averages and singular integrals over lower dimensional sets," in these proceedings.
[31]
[33] A. Zygmund, "On Fourier coefficients and transforms of functions of two variables," Studia Math. 50(1974), 189-201.
AVERAGES AND SINGULAR INTEGRALS
OVER LOWER DIMENSIONAL SETS
Stephen Wainger(l)
I.
Introduction These lectures deal with work primarily due to Alex Nagel, Nestor
Riviere, Eli Stein, and myself dealing with certain averages of and singu lar integral operators on functions, f, of n variables, n ~ 2. These averages and singular integrals differ in character from the classical theory in that the integration is over a manifold of dimension less than n. Let us begin with an example of the type of problem we have in mind. The classical differentiation theorem of Lebesgue asserts for any locally integrable function f
f(x) = lim _1_ HO IQrl
J
f(x-y)dy
a.e.
Qr
(where Q r is the square, Qr = Ixt:Rnlsuplxil ~rl, and IQrl denotes the Lebesgue measure of Qr)' and
f(x) = lim _1_ r->O IBrl
J B
f(x-y)dt
r
(where B r is the ball, Br=lxllxl:;r!)·
•Supported
in part by a grant from the National Science Foundation.
357
a.e.,
~ ~......~--;r-~-;...~~~,m;F~~"'-~~~~-==-~"~"""""'"
358
-.
~
~
-....".
.
~,_-'-
•
- .• ~.~
·..."'u
"_
'.;":"'~'~~"":~i::'''''~~'''"'''~ o,~,~~~~;~n~z:;,~~,~:~,~;,~*J&:,~E.Er-:-"';;~{":":':7":"~S:"":.71i:'~"-S~~~':~h~~tl'.£:~~~..l7£'~~_!::::':.,.~~,:~;.~,,";,;,,,,~
STEPHEN WAINGER
Our first problems are the following:
MB f(x) ... f(x) r
a.e.
,6)
Does
!
lim
r ... O
faQ
I
I
I
f(x-y) da/y) "" f(x)
a.e.?
The standard approach to this type of problem involves consideration
\
\\of appropriate maximal functions. r
\
We define the maximal functions
Here aQ r denotes the boundary of Qr ~nd dar is n-1 dimensional Lebesque measure on aQ r normalized so that dar(aQr) = 1.
7)
Problem IB:
and
Does
2)
359
and
Problem IA:
1)
AVERAGES AND SINGULAR INTEGRALS
lim
r"'O
JaB
mQf(x)
=
sup MQ (Ifl)(x) ,
r>O
mBf(x)
8)
=
sup MB (jfl)(x) .
r>O
f(x-y)d/Lr
=
f(x)
r
r
a.e.? mB is called the spherical maximal function.
r
Since 1) and 2) hold for f which are continuous 1) would follow for
Here d/lr is the unit rotationally invariant mass on aB .
every f in LP, 1
r
'S
p < 00, if we could show
1) and 2) trivially hold if f is continuous, and the questions only become interesting when we consider functions in a class like L 00, L2, or L I .
MQ/(x)
=
JaQ
'S A; Ilfll~
Il x lm Bf(x»x!1 'S {p
10)
Ilfll~ .
f(x-y)dar(y) The argument showing that 9) and 10) imply 1) and 2) is the same as the
r
argument showing that Lebesgue's differentiation theorem follows from
and
4)
IlxlmQf(x»AlI
for every f in LP, and 2) would follow if we could show
In questions IA and IB, we are considering certain averages
3)
9)
MB/(x)
=
JaB
the weak type inequality for the Hardy-Littlewood maximal function given in chapter 1 of [S]. While it is not quite as well known, there are appropri f(x-y) d/l/y) .
ate estimates on maximal functions that guarantee 1) and 2) hold for all L 00 functions. In our case this means the following;
r
Let E be a measurable set and XE its characteristic function.
We are asking if
Then if
5)
MQ f(x) ... f(x) r
a.e.
(11)
Ilx\mQxE(x»AII
'S
C(A)
lEI
-
360
III
I
ti
I
t-
~j
-;
STEPHEN WAINGER
IlxlmBxE(x»AII:s C(A)IEI ,
---~-
_
-_lI:_IF
AVERAGES AND SINGULAR INTEGRALS
361
One need only consider F(x I , .. ,x n ) = f(x I )h(x 2 , .. · ,x n ) where f is as above and h is a nice function. So there are no interesting pos itive results in problem IA. However
I
i as we shall see later there are positive results for problem IB in 3 or more
'I
I
I;
I
i II
-
whe,e C(A) may depend on A hn' not on E, then 1) holds for every f in L 00. If 12)
II'II
.
then 2) holds for every f in LOO'. A discussion of this can be found in [BF]. Let us try to see if 9) or 10) could be true in Some simple cases. We consider for example the one-dimensional case. Here B = Q = lxl-r<x
I
MQ f(x) = MB f(x) =2_1 [f(xH)+ f(x-r)] .
r r
I
\ dimensions. One might ask if there is a simple geometric reason why "'-\there should be positive answers for the sphere and only negative answers \f'or the boundaries of squares. It turns out that the underlying basic reason that we have positive results for the boundary of balls and nega tive results for the boundary of squares is that spheres are round and boundaries of squares are flat. In other words an important word for us will be
I
So if we take f(x) to be log...l near x = 0, have compact support,
Ixl
and be in COO away from the origin, we would have a function f in
every LP class such that mQf(x) = mBf(x) = oo for every x. We can
also see that 11) and 12)are false in one dimension. We just take
E e = lxlo<x<el. Then IE eI . . 0 but
- -
CURVATURE. We will come back to the role of curvature in our problem in a little while, but first we shall discuss the other problems that we will consider. Problem II: Let yet) be a curve passing through the origin in R n . Is it true that lim
mQf(x) = mBf(x) ~ }
~
JOh f(x-y(t»dt
Problem III: Let vex) be a smooth vector field in R n . Does
for all x.
I
= f(x) a.e., for f in L 00 or L 2 or L I ?
We could still ask if 1) and 2) hold in some interesting class even
lim
h ....O
though 9), 19), 11), and 12) fail. However an important idea of Stein
!. h
f
h
f(x-tv(x»dt = f(x)
a.e.
o
shows that the failure [SI] of 9), 10), 11), and 12) implies that 1) and 2)
fail even in the class of locally bounded functions. The statement of the
for f in LOO or L 2 or L I ? Corresponding to problems II and III there are interesting singular
-main theorem of [SI] requires that the underlying space be compact. But if 1) or 2) were true for an LP class on R n , it would also hold for the
integrals. We let yet) be a curve and vex) be a smooth vector field as
corresponding LP class on the torus. Furthermore, the theorem of Stein
in problems II and III. We set
requires the hypothesis that 1:S p
OS
2. However due to the positive 8
nature of the averages under consideration, his ideas can be modified to show that 1) fails for at least some LOO functions. See [SW]. Thus we
13)
=
J
f(x-y(t» dt t '
-8
obtain negative results in one dimension. Similar reasoning gives the same negative conclusion for question IA in any number of dimensions.
Hi(x)
(where sometimes we wish to think of a as finite and sometimes as oo), and
' .. P ~_
,=••••""" •••),."'"'"~''.:~_ .......''''''''•••,.,..'''''''
362
'.."
14)
STEPHEN WAINGER
HVf(x) =
f
AVERAGES AND SINGULAR INTEGRALS
363
(K(x) is a function on R n ) to the one-dimensional Hilbert transform
1
f(x-tv(X)) dtt .
-1
Hf(x) =ff(X-t) dtt We call Hy the Hilbert transform along the curve y and H
the v Hilbert transform along the vector field v(x). We then have the following
(f a function on R 1 ).
two problems:
We will explain how the method of rotations can lead to problem II'. Problem II'; Can we have an estimate 15)
II H/II LP ~
II
Let K(x,y) be a function of two variables x and y which is odd, K(-x,-y) = -K(x,y)
CpllfllLP
for some p's ?
and which has a "parabolic homogeneity," that is
Problem III': Can we have an estimate
18)
16)
IIH)II LP ~ CpllfllLP
K(h,A2 y) =1. K(x,y).
,\3
We wish to consider the LP boundedness of the transformation
for some values of p?
f f 00
Tf(u,v)=
19)
The classical development of singular integrals and maximal functions suggests that problems II' and III' should be related to problems II and III.
-DO
In fact the progress on problems I, II, III, II', and III' is all interrelated.
00
f(u-x,v-y)K(x,y)dxdy.
-00
We now introduce parabolic polar coordinates into 19)
We have presented our problems as variants of Lebesgue's Theorem on
e
the differentiation of the integral. These particular variants arose from
x = rcos
other considerations.
y =r 2sine
Riviere was led to problem II' from the consideration of a problem of and find
singular integrals, namely from trying to generalize the method of Rota tions of Calderon and Zygmund. Calderon and Zygmund developed the
Tf
=
20)
Tf(u,v) =
o
K*f
f(u-rcos
e, v-r 2sin e)
0
K(rcos e, r 2 sin e)r 2N(e) dr de
where K is a kernel having "standard homogeneity" that is 17)
J f
2TT
00
method of rotations to reduce the study of operators
where r 2 N(e) is the Jacobian factor in the change of variables. N(e) is K(Ax) = A-nK(x)
A>O
smooth and N(e+TT) = N(e). By 18 we see that
\'
.~'
I
364
STEPHEN WAINGER
21) Tf(u,v)
=
f
277
<Xl
N(O)K(cosO,sinO)dO
o
}f(u-rcosO,v-rsinO)dr
277 N(O)K(c OS(O+77) ,sin(O+77)) dO
= -
J
I
f(x-yO(r)) d: '
with YO(r) = (rcos O,r 2 sin 0) . Now we prove IITfll
J} ~
N(O)K(cos O,sin O)dO
o
LP
:s cl\fliLP
f(UHCOS O,v+rsin O)dr by showing
0
IIHOfIILP:S cpllfllLP .
since N(O+77)
=
This is a problem of the type II'.
N(O) .
Stein was led to consider problem II by his study of Poisson integrals
Finally
on symmetric spaces. We are not going to launch into a discussion of
I I
J
277
21A) Tf(u,v)
I
=
J
symmetric spaces, but instead we consider an example. Let
<Xl
N(O)K(cost,sinO)dO
o
tf(u-rcosO;v-r 2sinO)dr.
_~
Tf(u,v)
=}
f
J
=h 1e K(~e 'e~) .
<Xl
N(O)K(cos O,sin 0) dO
}f(u-rcosO,v-r 2sinO)dr.
-~
If K(cos O,sin 0) is in L 1 of [0,277], we can apply Minkowski's inequality 77 IITfl1 LP
:s c
J o
drds
f,l
277
o
where
1 fYf(X-r,y-s) I~~ 2)(1+ s2) M,f(x'Y)~;2J. ~ 1
~.
Now adding 21) and 21A) we find that
I
J
}f(u-rcosO,v-rsin(O))dr
0
277
Tf(u,v)
~
J ~
since K is odd. Thus
I
=
365
-~
o
I
Hef
0
f
= -
J
AVERAGES AND SINGULAR INTEGRALS
2
If K(x,y) were dominated by a decreasing, radial, L 1 function, the classical theory would imply lim M f(x,y) = f(x,y) e~O
a.e.
e
dO IIHefllLP see [SWE]. However K(x,y)
1
=
(1+x 2) (1+y2)
,
"__
..
~_~~
~.
,,~_"_,_._
..
~
="""''''''''''=~.
366
-
~
•
......iW'"'"
~''''04UiiC·
.
wsiBi:li
!l:~U;"lJ~~
-''''''''''''''
_
AVERAGES AND SINGULAR INTEGRALS
STEPHEN WAINGER
so the smallest radial majorant of K is 1 which is not integrable. . 1+x 2+y2
We now consider the intersection of two of these balls of the same size
In effect K has too much of its mass along the coordinate axis. The
/
extreme case of this phenomena would be to have a kernel with all of its mass on the coordinate axis. In other examples, kernels have too much of their mass along curves, and the extreme case of difficulties arising in problems of Poisson Inte
367
Clearly one of these "balls" is not contained in a fixed multiple of the
grals on symmetric spaces lead to Problem II. Appropriate positive results to problem III would have implications for !
I
Ii
the boundary behavior of functions holomorphic in pseudoconvex domains in C n . The natural balls in these problems are long, thin and twisting.
other (uniform in E). We can also see the difficulty of using the Calderon-Zygmund theory to study Problem II'. Suppose yet) is the parabola (t,t 2) in R
2
and
The idealized situation is that of a vector field. In the case of a strictly
I I
I~
III III
II!
pseudo convex domain, the balls satisfy the standard properties that en
Tf(x,y)
sure that the usual covering arguments apply. See [SBC]. For progress
=
2 ff(X-t,y-t )d: .
in the case of pseudoconvex domains see [NSW] and [NSWB].
Now that we have seen some of the roots of our problems, let us con
Then we may formally write
sider why these problems don't fit into the framework of the standard theory of Maximal functions and singular integrals as presented for Tf(x,y) =
example in [S].
II
f(x-t,y-st 2 ) o(1;S) ds dt
In the s tanda rd treatment of avera ges over Euclidean balls an important geometric property of the Euclidean balls is used. If two balls Bland B 2 of the same radius, r, intersect, then B;, the ball having the same
=
center as B 2 but having radius 3r contains B 1 . To see how badly this property fails for our problems let us suppose we were considering averages
II
~ 0 (1- t~) dv dt .
f(x-t,y-v)
Or h
~EJo
2
J
t +E
t
22)
f(x-r,y-s)drds
2
where 22A)
over slightly thickened parabolas or balls BE,h = l(r,s)IO~r~h, t2~s~t2
Tf=Kd,
+E
I.
1
K(x,y) = 1. 0 1 -x3
\
2) x2
'
368
AVERAGES AND SINGULAR INTEGRALS
STEPHEN WAINGER
The Calderon-Zygmund theory deals with convolution operators with
11
kernels K(x,y), but in their theory K(x+h,y+k)- K(x,y) should be much
~(~,r,) = ll(ei~Xei1]Y) =
less than K(x,y) if hand k are much smaller than x or y. However for our K if (x,y) is a point on the curve y
=
x~xis,
is supported on a straight line, namely the
x 2 and (x+h,y+k) is not
369
and
f ei~tdt 1
o
on the curve, no cancellation in the difference K(r+h,y+k)- K(x,y) can occur, no matter how small hand k are. The Calderon-Zygmund Theory is based on 4) tools
which is independent of 1] and hence cannot decay at infinity along the 1]-axis. Now let us consider a measure supported on a parabola,
a) The Fourier transform (The Fourier transform is even used in the 00
L 1 theory) 23)
b) Interpolation
v(¢)
=
J
2
e- t ¢(t,t 2 )dt.
-00
c) Covering lemmas d) Calderon-Zygmund decomposition.
Then
Perhaps the natura 1 attack on our problems would be to find appropriate
i/(~,1])
covering lemmas and suitable variants of the Calderon-Zygmund decom
=
v(e~xei1]Y)
pcsition. Some progress in finding covering lemmas for related problems
00
J
was made by Stromberg (Str] and (STRO] and Cordoba [CORI], (COR2], Cordoba and Fefferman (CFI], [CF2], (CF3], and Fefferman (FE£].
2 '1:
.
2
e-t elC, t e l1]t dt.
-00
Our approach will however be different. We shall try to use the Fourier transform or other orthogonality methods and interpolation to reduce our problems on averages and singular integrals to the more standard averages
This integral may be computed exactly by completing the square, and it is easy to see that ~2
and singular integrals. In retrospect we see that some of these ideas Iv(g,1])1
occurred in (SPL], (CS], and in [KS].
:s
e- 7f C ~
We have said earlier that curvature and Fourier Transform would be important for us. Actually they go together. If one has a nice measure on
Thus V(~,1]) tends to zero at infinity.
a curved surface, the Fourier transform of that measure decays at infinity
Another example is afforded by rotationally invariant Lebesgue n measure on the n-l dimensional sphere Ixl = 1 in R . If we denote n this measure by dll, we have for g ( R
even though the measure is singular. Let us consider some examples. Define, for a test funct ion ¢,
J 1
Il(¢)
=
o
¢(t,O)dt.
dll(g)
=
C
n]n-2 (I~l) I~I- (n~2) . 2
,' .//
I I
__
~i"WT·'rC·--4~~Ww.."$.~~,~~:"J:'::;~"".k~,",t<"'";Jr21
/
I
370
STEPHEN WAINGER
AVERAGES AND SINGULAR INTEGRALS
~~'''~'''''''''_'
371
/
11/
See [SWE]. Thus
1 d IAl1(e,T/) 1< - C (1 -+ IT/I) 1/2
25)
'" s C (l + lel)- (n-l\ -2-J. n
I
If.l
Id l1(e)!
Now if 1111 < 8'
Of course we want to have a tool to estimate the Fourier transform of
Ih'(t)1
=
le+2T/tf > e/8.
measures in general, not in just a few specific cases. This tool is
a lemma
Let h(t) be a real function. For some j,
VAN DER COR PUT'S LEMMA.
assume Ih(j)(t)l?
11.
So by Van Der Corput's lemma with j
of Van Der Corput.
S t S b.
in an interval a
1 ,
C IAdl1(e ,T/)1<- IZ1
26)
If j = 1, assume also that
h' (t) is monotone, then
if
J b
lei> 81T/1
C·
exp(ih(t))dt -<~J 1I.1 /j .
For the proof of Van Der Corput's lemma for j
=
1 and 2 see [Z]. The
1~(e'T/)1 S (1+e 2C+1T/1)'
27)
proof for higher j is similar.
I<
for some 0 > O.
Let us consider the measure
J
Stein pointed out in retrospect that we can already see from an
2
d/l(¢)
.
Putting 25) and 26) together we have
a
24)
=
=
¢(t,t2 )dt.
estimate like 27) that d/l has interesting properties from the point of harmonic analysis - namely even though dl1 is singular,
1
Tf
Then A dl1(e ,T/)
J
e i eteil1t2 dt .
1
This integral cannot be eva luated explicitly, but we wish to see that Van Der Corput's lemma may be applied. We take h(t) = et+T/t 2 . First we use the fact that hH(t) = T/. Thus by Van Der Corput's lemma with j = 2, we see
dl1
*f
maps LP into L 2 continuously for some p < 2. For
2
=
=
'''''~--''::
372
-- f
-_.-
STEPHEN WAINGER
J(Tf)2
=
Idlt(~'77)12If(~'77)12 d~d77
J
~1t(e'77)1\1+~2+/771)20
_....
_::~_
.=----:;;_.__
373
into 13) or recognize the fact that Hf
~
If(~,77)12
(1+e 2 +1771)2B
=D *f
where D is a distribution
If(~'77)12 (1 +e 2 + 1771 20 ~
. . ( f·If(~ ,77)1
<
n.
f(x) =Je-if x f(~ )d~
A
<
- _._-"
~
J
=
.---==
AVERAGES AND SINGULAR INTEGRALS
JI~(~'77)1 2 d~ d77 A
=
--....:.__ . .:..-
2q)1/q(
1
J(1+<'+I.!)28Q
D¢ =J¢(y(t»dt .
)l/q, , So
A
Hf
~~
=
Df ,
The second integral is bounded if q' is sufficiently large which means for some q> 1. But then the first integral is bounded for f 1 +-1- = 1. p 2q
II.
f
LP where
The Hilbert transform along curves
and D may be computed by evaluating D on an exponential. Thus to prove that Hy is bounded on L 2 one needs to show that my is bounded. The first result of this type was obtained b~ Fabes [F]. Fabes showed H is bounded on L 2 in 2-dimensions for the curve y
The first progress in our series of problems was made on the Hilbert
yet) = (t, \tlasgnt) ,
a> O.
transform along curves. The Hilbert transform along a curve can be So Fabes' proof consisted in showing that the integral
thought of as a multiplier transformation ~
A
28)
m(~,77)=
-00
where
is uniformly bounded in ~ and 77. To this end Fabes employed the
00
29)
J eXP(it~+ilt\a(sgnt)77)d: 00
Hi(~) = my(Of(O
my(O =
J
e~·y(t) dt
T'
method of steepest descents. The method of steepest descents is a method of obtaining very precise asymptotic information for large A about
-00
integrals of the form To see that 29) is true we may either substitute the formula
J
exp(iAh(t»dt
374
I
///
/~
I
AVERAGES AND SINGULAR INTEGRALS
STEPHEN WAINGER
/ by contour integration. However to employ the method one has to have
may assume ~ n = ± 1 in 30). Then by using Van Der Corput's lemma
very precise information on where the real part of h(z) is positive and
with j
negative in the complex plane. Thus already to employ the method of steepest descents for the curve (t,t 2 ,t 3), one would ha ve to understand
=
n, we find
f
31)
the zero set of
t
exp
(i~1s + ... + i~ n_l sn 1 ± isn)1 'S C(n) .
1
Real Part l~lz+~2z2+~3z31
An integration by parts together with 31) shows that
uniformly in ~1' ~2 and ~3' So it is hard to imagine using the method of steepest descents, and for the curve t, t 2 , t 3 , t 4 , t S it would seem close
f
32)
to impossible. Fabes' result was very important in that it gave the first
However a better method would have to be found - a method that
J
f
eXP[i(~lt+"'+~ntan-l)]dtt.
t
t
< C(n) .
n exp (i~ 1t + .,. + i~ n-l t n- 1 ± it ) dtt
- 00
J
exp (i~lt + ... + i~n-l tn-I) dtt I <
n t dtt'S C(n) .
e
e<\ tl<1
Hence we have reduced the proof 30) for n to proving 30) for n - 1 .
The proof was by way of the Van Der Corput lemma but was unnecessarily
Also the case of n = 1 is easy. So we are done by induction.
complicated because at that time we only knew the lemma for j = 1,2. Let us see how Van Der Corputs'lemma works in the case y(t) (t,t 2 , ... ,t n ). We then have to show that
1
e
00
f
n n c t - 1 -+ it ) (i~ 1t + ... +i"n-l
Now
needed less precise information about h(t). The next step was to a a an 1 show that if y(t) = (t,t 1 ,t 2 ,···,t -), 1 < a 1 < a < ... < a n_ 1 2 a· a· a· (here t J can mean either It\ J or It I J sgnt ) then Hy was bounded on L 2 (R n ) (SWA]. Here we had to prove the boundedness of the integral
I
exp
1'S1 t\'SR
clue that problems such as II and II' could have positive answers.
30)
375
We now turn to the LP theory. We wish to emphasize how curvature
=
and Fourier Transform are joining together to help us. So we shall com pare the case of the parabola (t,t 2 ) to the straight line (t,t). In the case of the parabola we are studying
exp(i~1 t+ .. · +i~ntn) ~tl 'S C(n) .
Hpf
34)
e'Sltl'SR
where D
where C(n) does not depend on e,Rll'· .. l n' We shall prove 30) by induction on n. By changing variables, replacing t by __t _ we I~ \l/n n
k'
d it,
p
=
Dp
*f
is a distribution. For a test function
1>
376
STEPHEN WAINGER
35)
Dp ¢
=
We shall now show that 6p(~,1)
f¢(t,t 2 ) dtt .
is continuous near I~\
f
00
00
HLf
DL
=
*f
41)
lim 7]->0
,
t
L
T'
> O.
Of course
J
¢(t,t)
r
d: . 42)
J
lim 7]->0
I\(~,7]) = Dp(ei~Xei7]Y) =
1
I
00
ei~t dt
ei~tei7]t2 dt
T'
=
f
ei~t dt
t
-00
1 1) 1/3
00
and
-00
I 1) 1~3
and
0L(~,7]) = DL(ei~Xei7]Y)
39)
f
ei~t dt
-00
Assume for simplicity that 7]
D¢ =
38)
e~tei7]t2 dt
-00
where
37)
f
43)
ei~t(eiT/t2 -l)dtt
1
1)1/3
00
J
ei~tei7]t dt
T'
1
-00
We can calculate
DL
f
1)1/3
explicitly, and we find
<::21)
tdt~1)1/3->0
as 1)->0,
o
0L(~,7]) = c sgn (~+ n) ,
40)
In view of 42) and 43) we can show 41) by showing Notice that
DL
is discontinuous along the line ~ = -7]' We shall
show in contrast that Dp(~,7]) is continuous away from the origin. It is very easy to see that Dp(~,1)
is COO away from the line 7]
plex integration. If for example 1)
=
44)
0 by com
> 0, we think of t as a complex l.
~ E,
for some positive
E,
f
1)-1/3
t2
variable and integrate along the line lmt = Ret. Then the factor e 7] 2 decays as fast as e- c 1)l t I , and one can easily justify differentiation under the integral sign as long as 1)
=
1. What we
must show is that lim Op( ~,T/) exists for ~ near ± 1. We shall show 7]->0
In the case of a straight line we are studying
36)
377
AVERAGES AND SINGULAR INTEGRALS
II
'· I
l~;
i
ei~tei1)t2 d t -> 0 . t
·1 /~78
_.
STEPHEN WAINGER
"
'
..
~~._---_.---_._-379
AVERAGES AND SINGULAR INTEGRALS
j-//
-----/1
We shall prove 44) by using Van Der Dorput's lemma. By using Van Der Corput's lemma with j
2, we see
=
and hence the continuity of Dp(~'''') away from the origin. If one is a
I
f
I
.
.t:
little more careful in the above argument, one can prove that Dp(~'''')
2
e1sSe1"'S ds
< -.£.- .
satisfies a Lipschitz condition away from the origin. One may then use
- V1J
Riviere's [R] version of Hermander's multiplier theorem [H] to obtain some LP results for p I- 2. In fact if y(t) = (t,t 2) one obtains
."-2/3
I I
45) and 46) together with similar estimates for negative t prove 44)
(Let h(s) = ~s + ."s2 , then h"(s) = 2.".) So an integration by parts
shows
47)
IIH/II LP
~ CpllfllLP'
i < P < 4.
00
J
45)
I
;t:
2
For quite a while we tried to prove the range of p, :!. < P < 4, in • 3 47) was optimal - with no success. Also, there was a suspicion that the
e"Stei."t dt t
." -2/3.-.
use of the Hbrmander Riviere theorem lost something. In the Hbrmander
f
00
~
I
.,,2/3 + __1_ .,,1/2
argument, one wishes to estimate an expression of the form
dt t2
.,,-2/3 ]
~ C .,,2/3/.,,1/2
Note that if t
< C."l /6 .
=
.J" ."
1~+2."tl > I~I - 2.,,1/3.
K.
One does this by using Schwartz's
inequality and Plancherel's theorem.
]
-2/3
J
=
ei~sei."s2 ds
Ix lal K(x+h)- K(x)ldx
-1/3
is bounded if ~ is close to minus one. Hence an integration by parts
46)
IK(x+h)- K(x)ldx
where K is a kernel, in terms of
So
r
J
R~lx19R
< .,,-2/3 Ih'(.t)1
shows
=
~(
J
~
Ixt2a dx
R~lxl~2R
1/2
.(J
1
Je~tei."t2 1
.,,1/3
J ~~ 00
dtt
+ ." 1 /
3
~ C ." 1 / 3 .
~~. R a- -2
l 1"1
J=a
Jll-ei~'hI21.t K(OI2d~ at: J S
1
.,,1/3
1
Ix!2a l (x+h)-K(X) 12d
where the sum is over all a'th derivatives of K.
"E.- _'::...
~;-I=T-'-'=~=="~~~=='="~O~~
380
IllllJI
_..-
._~-"---=:'._
•...... - ..."="
_.-
.._"-'=-.
...
'··'L'::~'::':':::·;;;:~~·=':""_'::';;"-.:::::..o~"....:::':'::",::.::,,:~:::..;:.·.~:.::.:;;;::.·.:C:'" ~:~":::::""-""_·-
···~;:··.:.::_,,,",,,,===C';:::=-'-~:::'-::'·_=":"'::":--===·=""':"=":'-";;'
381
A VERAGES AND SINGULAR INTEGRALS
STEPHEN WAINGER
Now .there was the feeling that a use of Schwartz inequality like that
Then by a duality argument T would be bounded in all LP, 1 < P <
above lost too much, and that more careful estimates for ] for particular
It turns out that one can show by a messy calculation that T z is of
kernels might lead to better results.
Calderon-Zygmund type if Re z > O. Let us try to understand why the kernel for Tz' Re z > 0, might be
To get an idea of what to do we calculated D(tf,1]) very precisely by the method of steepest descents. We found
~ D p (tf,1])
48)
I 1/2 '1:2 sgntf + C _1]_ e 1S /1]rjI . tf 1
=
(.!L) tf2
00.
a little better than the kernel for To' T is essentially H y y and so the kernel K o of To is essentially
KO(x,y)
=-.l3 0 It x
\
=
(t,t 2 ),
_1-) x2
+ Better terms,
where rjI is a C;;' function on R 1 which is one near the origin. Hence,
from 22) and 22A). If we introduce" parabolic polar coordinates"
y = r 2sin () x = rcos (), we see
the crux of the matter was to study the transformation Tf given by Tf (tf,1])
49)
K o(x,y)
~
/'\.
I:
e _ . 1]_ e 1] rjI (1])
I 11/2 ) __
tf
i--
sin()o =1. cos 2 ()0
tf2
We might expect if Re z This suggests introducing an analytic family of operators in the sense of
= E
> 0 K z to be
A
T zf(tf,1])
=
m (tf,1]) z
=
11]1 1 /2)Z
(
_I_Itf-
II
theorem we would know that To
.
singularity is better than
=
the analogous situation for the standard polar coordinates with () 0
-1/2. So if we could
bounded in each LP if Re z > O. Hence by Stein's interpolation
!1 ~1,l li
1
Suppose =
III
-II
1
a o(()-()o) singularity. To see the situation more clearly, let us examine m(tf,v).
prove that the kernel Kz corresponding to T z for Re z positive satis fied a condition of the Calderon-Zygmund type, we could prove T z was
1
1
[()_()Oll-E
Tzf is clearly bounded on L 2 if Re z 1:
better than K o ' So we
== ~ I()_() Oll-E
Now we would like to explain why a
III! 52)
K z (x,y)
~ mz (tf,1]) f(tf, 1]) ,
where
E
might expect
[SI] as follows:
iI
o(()-() 0)
where
m( S,1] -
51)
r3
m(tf,1])f(tf,1]) ,
=
where
50)
1
= -
T was bounded in all LP, p S 2.
53)
Ko(x,y)
=
where x = rcos () and y = rsin () , and
o(()) r2
=
O.
382
54)
K/x,y)
=
383
AVERAGES AND SINGULAR INTEGRALS
STEPHEN WAINGER
J
7T/4
1 r 2 18jl-E
IKo(r, 8) -
Ko(r', 8')\ d8
=
1 .
o (The factor
II
Ii
1.. r2
for ordinary polar coordinates plays the same role as 1
r3
for parabolic polar coordinates.) We are trying to see whether
I
Let us consider now what happens with KE. We should expect no help from the difference K(x,y)-K(x,y-1) when y:,:>O, if y-1<::O. But this can only happen if y
II
55)
I
x
J
IK(x,y)-K(x-h,y-k)1
{,;('
°
For either K = K o or K = KE . Let us take h = and k = 1 and con sider first K = K o ' Let us look at the contribution from y's which are
I
I
J J I~ -o;~
I I
-
Wi
r2
_1_d8rdr
8 1- E
o DO
'V
SJ
~
I I
l+E
It is not difficult to complete the argument and to show 55).
After a laborious calculation one could prove that the kernel Kz corresponding to T z of 51) was for Re z > 0 an operator of Calderon y-1
Zygmund type. This proved that Hy was bounded in LP 1 y =
So the left-hand side of 55) is at least f~ ~dr =
_1_ dr .
r
5
o~l I,
1..
5
~y
III"
f J
)!d8rdr.
r>C nearo
If y is very close to 0
KE(r, 8)d8 rdr
l/r
DO
<
I
< l/r. But over this set KE(x,y)
o
5
very close to O. We have HI
or 8
is integrable at infinity
2 +y2> c(h 2 +k 2 )
'II II
S1
00.
So 55) can't
00
if
(t,t 2 ). However it would be extremely difficult to carryover this
proof to a three dimensional curve. For example it would be hard to derive an ana logue of 48) for the curve (t, t 2 ,t 3 ). Essentially one
hold. Let us put the matter a little differently. If we consider the 8's
needed a way to define a suitable analytic family T z without using the
with () ~
asymptotic formula 48). Recall that Hi
°
where the difference K o(r,8)- K o(r',8')
56) offers no cancellation, we find there is only one bad 8, 8 = 0. But still
=
Dp
Dp(t',1]) = Jei~teint2
*f
dt t .
where
IiFi"1II
,~i
i
1lI!.
E
384
-
,.
•
-
......"
-,,-
-<"~~---'-'-'--' -------,-,,
AVERAGES AND SINGULAR INTEGRALS
STEPHEN WAINGER
I /
j /,I
,
=-
I ii
"-'z
57)
Iii
f
So one might be tempted to define H/
=
"-'z Dp
*f
,
t
e ifs e i 7) S 2 ds
1
J t
where 58)
eifseis2 dsl
=
~~(f,7)
I
=feiftei7)t2
(1+t 2 l/ 2 dt .
III
:s C .
1
So an integration by parts shows
II
It turns out that 58) is not a good idea for a very important reason. By
i'~
Ii
changing variables in formula 56) we see that DpCAf, 1-.2 7)
Iii
!
I'II
f"" (1+t 4)-Z /4 e ift e i7)t 2 dttl :s C(z)
> O. Note that also the function mz (f,7) defined in 52) also has this type of homogeneity, namely mz (l-.f,1-.27) = mz (f,7), for I-. > O. h~mobeneity is
Now experience has shown that to be tossed away lightly. However
Dp
59)
HZf y
=
('7)=1).
does not have this homogeneity . Now
J 1
DZ x f p
where
(1+t4)-Z/4eiftei7)t2 dtt
-1
~Z Dp (f,7)
60)
1
a powerful friend not
This situation can be remedied by defining
)
..
D p(f,7), for
any I-.
III
.,1,1
=
J
""
=
-J 1
2 (1 + 7)2 t 4)-Z /4 eift i7)t dt
e
t'
III
-1
Note that for I-. > 0
II
2 e ift e i 7)t dtt
Ii
~Z
2
< C(z)
-
~z
Dp(l-.f, I-. 7) ~, Dp(f ,7) .
61)
J t2~ 1
:1
ItI < C(z).
-1
Let us see how formula 61) can help us. We would like to show 62) if Re Z
16;(f,7)1
But we already know that
J
:s C(z)
> - ~. By formula 61) we may assume 7) = ±1, let us say 7)
Then by Van Der Carput's lemma with j = 2 , we See that
1
=
1.
-1
e ift e i7)t
2
~t I < C .
385
-----
,.~--':'-'.'~==-~~~~~i.."1l.;..:;:-'--
386
'''';;;.:;~.:>
._'....,~
._''l
."'~~':"~~;'~"S:.::~:;:;:iZ~~~~~~..l't.:"=.1'1-~~~~.~~~::='~T1"=~(".~:~ ~k__ ,
STEPHEN WAINGER
One may finally deal with treated 1:S t
<
00.
A VERAGES AND SINGULAR INTEGRALS
< t:S -1 in the same manner that one
00
yet)
Hence 62) is proved.
Let us calculate the kernel, Kz , corresponding to H~ if z Then KE(x,y)
=JJ
ei(x e-
iny
=
E> O.
6~((,ry)d(dry
at
f
f
dtt
J
00
e -i((X-i)dx =
However, there is a natural generalization of these curves. All of these curves satisfy an equation of the form y'(t)
-00
~ yet) t
values of A are positive. For example if yet)
eiry(t2_y)(I+ry2t4r-E/4eiryt2a(x_t)dt A
-00
J
=
where A is a real nxn matrix such that the real parts of the eigen
00
d/
-00
e -iryy (1 + ry2 t 4)-E/4 e iryt2
-00
J J
00
a a a (t 1 ,t 2 ,'" ,t n)
=
00
-00
.
387
< a 2 ,"', < an' See [NRW].
63) 00
_ ..1 .. _")1'",_.f:'~~
=
=
(t,t 2)
(~ ~).
00
=~
2 e iry (X - Y\1 + ry2 x 4)-E/4dry
-00
.t7 lry
=-l3 x
J
e
y
_ )
T.\. 1 dry (1 +ry2)E/ 4
-00
2 y2 __ ) -_ - 13 P E/2 (X--2 x x
III I
II I
I!I I .III
I
where PE/ 2 is a modified Poisson kernel. P / decays exponentially E2 fast at 00 and p/ 2 (u)'V C as u-->O. See [SWE]. E lu!1-E/2
'III
II
I
=
exp (A logA) .
Then H y f = Dy d
64) where
65)
i\(TAO = i\(O Moreover there is a distance PA (x) defined on R n such that
Thus Kix,y) has a singularity near the curve (t,t 2) of the form _1_ _ which is just the improvement over the a-function that we rJOII-E/2 seek. A modification of these ideas worked for curves
III i
real part is called a homogeneous curve. A will generate a group of transformations
2
00
A curve satisfying 63), where all the eigenvalues of A have positive
peTAx)
=
Ap(x) .
In the case of the cure (t, t 2 ) we may, as we said before take
388
A
389
AVERAGES AND SINGULAR INTEGRALS
STEPHEN WAINGER
=(~ ~).
We shall show that if 7J(u) is a curve in R
n
satisfying 68) where the
eigenvalues of A have positive real part, then either 1J(u) lies in a proper subspace of R n or for every I; J 0 and u there is a j I
Then
T=G
1
A~) ,
Sj Sn
such that
~ I; . 7J(u) /0 0
69)
.
du J
and From 68) we see that
p(x,y) ~ (x 4+y2)1/4 .
j d +1 7J (u) du j + 1
It turns out that in the case of a general homogeneous cure, we can
~ Aj1J'(u)
.
obta in a satisfactory analytic family of operators by defining By the Cayley-Hamilton theorem, we can find numbers aj' 0 <: j
H~f
66) where
B~(O p~~ (0
67)
=
=
f
D~ d
such that
n
2
ajAj
~
Sn
I
o.
j~O
Itl- Z exp (il;'y(t» dtt
. So n
p
.
d+ k~ aj-.-7J(U)
* is the distance function corresponding to A *, the adjoint of A.
A
j~O
For a detailed description of the argument see [SW]. Here we shall
J
1
du J+
1
~ 0,
just make a comment. If some of the eigenvalues of A have non-zero imaginary part, yet) can be an infinite spiral. For example the curve
and n
.
~
J k aj -d . 7J , (u) . St: j=O duJ
yet) ~ (tacos (f3logt), t a sin({3logt»
=
0 .
is an example. So one could believe it might be rather messy to prove integrals involving exp (i/;·y(t»
to be bounded. It might be difficult to
show that at each t some derivative of 1;. yet) would be non-zero. However, if one makes a change of variables t
eU
In other words 7J'(u)' /; satisfies an nth order constant coefficient differential equation. So if for some u and /;
we would be led to consideration of integrals involving /;. 1J(u) where 1J(u) ~ y(e u). If y'(t)
68) ,'1'1
I,
I I I \1
=
=
,
LJ n'(u) . I; 7J'(u) . I;
7J'(u) = A1J(u) .
0
j
=
1,2,.·· ,n-1
du
~y(t), 7J(u) satisfies t
=
=
0 for all u. Thus 7J(u)· /; is a constant. But 1J(u)' 1;, 0
Ii lllr-r~--·--·--_·
1II1 I11
11
J
since the eigenvalues of A have positive real part. Hence 7](u) is in the subspace of R n orthogonal to ~.
2 began with the study of maximal functions along the curve (t,t ). Thus
00
we wish to consider averages
We shall conclude this section with the statement of some theorems
;
,I JI
1
I
A VERAGES AND SINGULAR INTEGRALS
as u ., -
j'"
that follow from the reasoning discussed above.
I1I
Mhf(x ,y)
70)
'I
'\ 1 I, I
391
STEPHEN WAINGER
.
:I, III, 1
390
THEOREM
J h
~k
1. Let y(t) satisfy
2
f(x-t,y-t ) dt .
o
1'1,11 1
11
!
y'(t) =
,'
III
~ y(t) .
After much frustration it was decided to take Fourier Transforms and try to see if anything could be learned. It is easy to see that
Suppose the span in R n of y(t) for positive t and the span in R n I
A
I
Mhf(~,7])
of y(t) for negative t agree. Then
,II' IIHil1 LP :S Cp,y Ilfll LP' \
'I
l
where
We say that a curve y(t) in R n is well curved if
I
1,1,11:
, I
djy(t) dyj(t) t=o'
11II
mh(~,7]) =
f eit~eit27]dt. h
k
o
j = 1,2""
n span R . It turns out that well curved curves can be approximated by homogeneous curves. We can then prove THEOREM
~
mh(~,7])f(~,7])
=
Now mh(~,7]) cannot be evaluated explicitly. If one hopes to gain some insight by staring at a formula, one should have a formula that is as explicit as possible. Now a similar situation arose in the path integral approach to Quantum Mechanics. See [FH]. Feynman and Hibbs wished
2. Let y(t) be weII curved then,
to have an explicit expression for the probability amplitude that a particle
r
f -1
lies in a sphere of radius t. In essence, they had to consider an integral
< - A p,y IIf/1 L p'
f(x-y(t» dt t
l
in R n of the form
J
LP
exp Q(x)dx
A general theorem in L 2 for curves which are approximately
IxISF.
homogeneous was obtained by Weinberg [We].
where Q(x) was a quadratic function of x. Instead they considered
III. Maximal functions and g-functions We turn now to a discussion of maximal functions. We are especially interested in how the Fourier transforms and g-functions may be used as a tool to relate our maximal functions to more classical ones. The story
1
_E
J-
Ix l2
- 2 +Q(x)
e
R
E
dx
ri
III
I
II:
I I '
1I
I
II,
392
STEPHEN WAINGER
which could be calculated explicitly. This suggests to consider instead of 70)
II
, II
I
71)
"h d(x,y)
J
- i
393
~1]
l/h could be dealt with by classical argu ments, while
-00
{ exp
Then
/~
II1
Iii
~h(t,1]) = v(ht,h 21]) exp
~ k ~ ex p (- ~:) f(x t,y-t') dt .
ill 1
AVERAGES AND SINGULAR INTEGRALS
72)
~
l/h*f(t'1]) = l/(ht,h
2
~
1])f(t,1])
~ t:) f(t ,1])} = g(t,1])
where g is another L 2 function having the same norm as f. So one could hope that
where l/ is the measure considered in 23). In particular \!sup l/h * fll 2 = l\sup ~h * gil 2
74)
I ;:;(ht,h 21])
=
L
Joo e-t2eihtteih21]t2 dt ,
~
Cilgil L 2 = Cilf!1 L 2'
-00
I
II
L
and this integral can be computed explicitly by completing the square. We find 73)
;:;(h,h 2)
(nice smoothly decaying function) . exp i h4t21] . 1+h 41]2
. h4t21]. fl' IS a re ectlOn 1+h 41]2
1---
The proof of 74) was a hint on how to proceed. However, it depended (as had happened before) on very special computations. What was needed
=
One might guess that the appearance of the oscillatory factor exp
Roughly speaking this works out. See [NRWM].
0
f h f h . . 1 t e act t at l/h IS a smgu ar measure.
On the other hand we see that if h 2 1] is large 1:1
4 2
exp i h t 1] 1+h 41]2
'V
exp i t_
was a way to compare averages like l/h to more classical averages by using only the decay of
vh
and not so much the explicit expression of
vh as was used above . Stein [Ssp] and [SH] succeeded in doing this by introducting appropri ate g-functions. Stein's first argument with g-functions dealt with the averages MB f of equation 4). Recall r
2
,
75)
1]
MB/(x) =
f B
f(x-y)dj.(r(y) r
which is independent of h. Thus one might try to write (from 72)
0(f,~) -~(ht,h'~)
ex p ( - ;
f~')) « J ')r(f,~~ exp
One might now hope that if one defines a measure
;)h
where B
is the ball of radius r centered at the origin, and dj.(r is the r
unit rotationa lly invariant measure on B r . We set, as before,
76) by the formula
mf(x)
=
sup 1MB f(x)1 ..
r>O
r
=.o=..~~-"-"="".=""_.___
_ ..
_"""",=,"-""..-==."",--=""",",-==-"====,,,,,,,;,,,-,,,,,,,,,,,,="-=-,,,,,,,,,=---=,~",,,,,,,,,,,~,,,,~,_-=,,,,,,,-==",,--,,-,,,,,,,,,-, ".='''''"_.."",..-''.--,...."=,,..~,,·O'=.,,=,='=i''''=,_·~_=o,~= _~
__ ..__ ..__.-,_._---,--_ ..
~-1 I
394
A VERAGES AND SINGULAR INTEGRALS
STEPHEN WAINGER
Thus
Stein used g-functions to prove THEOREM
3: M f(x) r
77)
II
Ilmfil
I~I
S r~
S Cp,n Ilf\I LP
LP
f
r
J r
sn-l M f(x)ds + In s
r
o =
if p > ~.I and n > 3 . n-
I
sn dd Msf(x)ds s
0
I(r) + lI(r) .
Now I(r) is dominated by the Hardy-Littlewood Maximal function and Simple exampl,es of the form
J r
f(x) =
t/J(x) n-l Ixl I1l oga
II(r) <::
.l
r~
sn-l/2 s 1/2M s f(x)ds
o
Ixl ~1/2
r
I
~ ,~ ~
where t/J = I near the origin and has compact support show that p > --!!...I n is necessary in order that 77) hold. The situation for n = 2, p > 2 is
I'I
II! IIII
8,n-1
(
dj
g(f )(x)
unknown at this time.
II1I
I
78)
4. We define
=
g(f)(x)=
"I tt J tl
So if we assume 29), we have
f
Ilg(f )1\ 2 ~ C(n) Ilf\\ 2' L
I
L
Ig(f)(x)['dx"
Rn
L
f
_. I""
tJ
0Mtf(OI2 d~ dt
1 dt
Rn
o
dd snMsf(x)ds s
n
I f ""
II
J r
=n
III1I
"I
o
III I
R
-
r
~ Mt f(x)\2 dxdt
1
0
o
I~l'Ii.
f- f I
and let us see how 77) would follow. Now
rnMrf(x) =
L
We turn now to the proof of 79).
Assume that we could prove
79)
Iisup M/(x)\1 2 <:: Cllf\\ 2'
}1/2 Mt f(x)1 2 dt
o
I I
S Cg(f )(x) .
I would like to present here Stein's original argument which proved
77) for p = 2 and n
II
395
I
J r
sn-l Msf(x)ds +
sn
0
d~ Msf(x)ds .
=
t
o
R
Itt n
0(oI2d~dt.
-= . !iii..
A
-;;'
396
•
._,,,-
.~'"._..
:-owaiii~_~"'';'''~ .,,~,"._.'_'~'-"'''' ~;-...o:;;;;;.",~'.~'~".'--~~'~'."-'-"-"-'-"';,~.o;m'"-".'
~_".~.~"
..~
~~-
~'.,,;;;;::
397
AVERAGES AND SINGULAR INTEGRALS
STEPHEN WAINGER
:11
I
But Mtf(t)
=
C(t) m(tltl),
f
""
where m(r)
1 n-2 I n_ 2 (r) .
=
r
-
£m(tltll dt
l/itl
-2
2
tI
""
:s Itl Here
J n-2
is the usual Bessel function. We sha 11 need to know
2
~
1
-
=
n-2 r 2 r l12
C _1_ , n-l r 2
Jlg(f)(x)1 2dA :S
:s
t
lm(tltl) If(t)1
1 Ixl log
f dt
f(x)
n
tld~m(tltl)2dt
We can disprove 77) for p
=
2, n
=
MB
f(x)
m(tltlf dt :s
c.
Because of rotational symmetry it suffices to prove 80) for points (a ,0) with a small. In that case
~ is bounded,
1IIti
J
MIx If(x)"
l/itl
dt < Itl 2
t 2
J
de
17
-17
Ia' (1-0 os )'+0 'sin' 01 ' /2 log a' [(1"
tdt:sC. E
o 'V
Next, since Im'(t)1 < ~, - n-l
= ""
Ixl
for all small x.
o
o
2 by showing
0
""
J
away from O.
o
80)
First since
Ixj
=
Thus to obtain 79) we need to show
t
x very near 0
1...
in C""
""
,i(t) ,2
J I~
,
Rn
J f R
SC
Let us be more precise about the counterexample in 2-dimensions. We take
J"" J I~ o
1
(tltl)n-l dt
if n>4.
is bounded. See [SWE]. Thus
I
t
I/ltl
I dm(r) I < C I dr
and
J
2
f -E
de 1 = lei In TOT
"".
O~)'
+sin' 0]
,-
i,
---
-~~--
398
STEPHEN WAINGER
A similar argument shows 81)
Then, if we set
sup 1MB f(x)1 = 1~r<S2
00
r
k 2
It {
gf(x) =
86)
.
a.e. for an appropriate f.
1MB
J=O
L 2 (R2). We set
J1/2
. f(x)-M B . f(x)1 2 J ~1 1+1+ 2
We wish now to prove a theorem indicating how bad 81) fails in
82)
399
AVERAGES AND SINGULAR INTEGRALS
k
2
k
we see mJl(x ,y) =
sup 1MB. f(x)y I ,
2 k<j<2 k + 1 ~
- -
MB -k f - MB -k+1 1+j(x)2 1+j*(x)2
1
f! ::;
gf(x) .
2k
Thus to prove 86) and hence 85) it suffices to prove where f is in L 2 (R2). We shall show
\\g(f )1\ 2 ~
87) 83)
IImJlII 2 L
L
~ CkllfllL2 .
cl\fl\ L 2'
We shall prove 87) by using the Fourier Transform.
To prove 83), it suffices to show that for each function j(x) taking the values 0,1,2"",2 k ,
84)
11
f
MB . -k f(x)l\ < Ckllfll 2' 1+J(X)2 L
2
Igf(x)1 2 dx =
k
~ lIMB J=O
. f(x)- MB
1+ -.!..2
2
with C independent of the function j(x). We show 85) by induction on k. That is given a j(x) taking the values 0,1,2, .. ·,2 k , we shall define
k
=It j=O
a function j*(x) so that j*(x) takes values in the set 0,1,2, ... ,2 k- 1
k
2
-~
\M B
.
J 1+2
k
2 '-1 f(x)1 dx 1+!.. k
~
f(O - MB
f(~»2 d~
'
J-1 1+ 2
k
and 2
85)
11
MB
. _kf-MB. -k+1 fll 2::; C\lfll 2' 1+J(x)2 1+J*(x)2 L L
=
k
flf(~)12 ~ 1m ((1+~)1~1) -m(~+~)I~l) 12d~,
85) provides the inductive step to prove 86). If j(x) is given define where m(r) = Jo(r). j(x) 2
So to prove 87) it suffices to shOlN
if j(x) is even
j*(X) =
k 2
j(x)-l
-2
if j(x) is odd .
88)
~ IJ o ((l + ~k)\~) - Jo((l+~)I~~
2 1
::; C.
400
STEPHEN WAINGER
But 88) follows because for s positive and r positive
93)
!/Jh(x,y)
=
!Jo(r» ~ C/..;r
89)
401
AVERAGES AND SINGULAR INTEGRALS
:3 !/J(~,
:2) ,
and
and
90)
IJo(r+s)-Jo(r)! ~
C"Jr.
~ 94)
g(f)(x,y)
<
(1/2
(~
00
Idl'j, .f(x'Y)-"h .f(x,Yll' d;
j
To prove 88) we use 89) if I~I ~ 2 k and 90) if I~I ~ 2 k . Finally we will show how Stein [SH] proved the maximal function
Let us first assume
along the parabola (t,t 2) is bounded by using g-functions, We start with the measure d/l defined in 24)
J
95)
2
d/l(¢)
=
IIg(f)l\ 2:SCllfll 2' L L Note that
¢(t,t 2 )dt.
1
sup e>O
We set
J
~
f
cJ/l h (¢)
=
t
1 2 /
e
¢(ht,ht 2)dt,
< sup
- '>0
1
Then
1
,1/' { {
Id/lh *f(x,y)-!/Jh *f(X,y)12dh)
t1/2
e
d/l h
* f(x,y) =
f
<
2
{
~ Id.h .f(x'Y)-"h .f(x,Yll' d; j
f(x-ht,y-ht 2 )dt,
1
~
or
92)
Id/lh *f(x,y)-!/Jh *f(x,y)ldh
o
2
91)
e
g(f) (x,y) ,
So
d/lh
J
2h
* f(x,y) = k
f(x-t,y-t 2 )dt.
h
We choose a function !/J(x,y) (C';(R 2 ) with ~(O) = 1. We set
96)
sup t>o
I~
f
e
d/l h *f(x,y)dh\
o
~ g(f)(x,y) + sup \!/Jh *f(x,y)\ .
h>O
402
STEPHEN WAINGER
A classica 1 argument Csee [R]) shows
JICgf)(X,y)!2dXdY
=
IIsup ljIh *fll p:S Cpllfll p'
h>O
L
403
AVERAGES AND SINGULAR INTEGRALS
JOO ~ Iid/l h *f-ljIh *f\2dxdy
o
L
Thus by 95), we see
supl~ E>O
If
f
= E
< Cllfll L 2'
d/lh*fCx,y)dhl
dhfA A 2 2 11 !d/lh ce-,71)-ljIh ce-,71)1 !fce-,1/)1 de-d1/
o
~1
.
~
IIL2
0
f
OO
-J-
f~O,
I£«,"JI'
J-'¥? ~h«'") -~h«,")I'
o
}J
d( d" .
0
E
d/l h * fCx,y)dh
So to prove 96) it suffices to prove
o _1
- E
JE h1 f2h o
> -
E
o
f
J
97)
J
h
J d~ I~Che-,h21/)-~Che-,h21/)12
98)
E/2
.
.
o
e
Thus by replacing h by ,.\h ,.\ > 0 we may write the expression in fCx-t,y-t 2 )dt.
98) as
J-'¥? 1;;;;(),h(,A'h'"J-~(Ah(.A2h'")12
So from %) we infer
.
o
tJ
e
e>O
:s C
00
1 dh
o
sup
A
The integral on the left side of 97) is E
fCx-t,y-t 2 )
dh A
11 !d/lh Ce-,1/)-ljIh Ce-,1/)!2
o
fCx-t,y-t 2 )dtdh
h
!.JE
~~
00
o
By choosing ,.\ so that ,.\2e- 2 + ,.\41/2 2 fCx-y,y-t )dtl!
:s Allfll L2
2' L
mate 99) when e- 2 + 1/2
=
=
1 , we see that it suffices to esti
1. In this case we see
I ~ d/lChe-,h271)_~Che-,h21/)12 :s J ~ 1
It remains to prove 96).
1
C
99)
o A
since d/lCO) = ljICO) = 1 .
0
dh
:s C
404
STEPHEN WAINGER
AVERAGES AND SINGULAR INTEGRALS
Then from 27)
IV. Vector field problems
1~1~(h(,h271)1 'S Ch- O
We turn now to problems III and III'. To study problems III and III' it is convenient to make a change of variables. It is possible to make a
for some 0 > 0. So
change of variables so that the integral curves of the vector field vex)
J ~ 1~(h(,h271)1 J 00
100)
405
become lines parallel to the x-axis. Under this change of variables the
00
'S
1
dh h1+28 'S C .
vectors vex) transform into curves y which vary from point to point. So we are led to studying problems IV and IV'. We facilitate the statement
1
of these problems with 3 definitions.
C
~
Also r/J((,Tj) 'S
N
(1 +( 2 + 71 2 )N
If y(x,t) is for each x a smooth curve in t with y(x,O) = 0, let
for any N, so
J 1
102)
00
J
101 )
dh
~
H f(x) = y
11 1r/J(h(, h2 71)1 'S C .
f(x-y(t,x» dtt ,
-1
1
let
Now we obtain 98) and hence 96) by combining 100), 101) and 102). In this section we have emphasized L 2 methods. LP results for
103)
p > 1 , can be obtained by combining the L 2 estimates presented here
M~f(x)
kJ f(x-y(t,x»dt , h
=
o
with the techniques of section 2. Altogether one can prove the following theorems:
and
THEOREM 4. If yet) satisfies y' = ~ yet) where all the eigenvalues of
104)
mf(x) = y
sup
l>h>o
IM hf(x)1 Y
'
A have positive real part,
kJ If(x-y(t» Idtll Lp 'S cI/fll LP ,
We are now ready for the statement of problem IV and IV',
h
II
sup O
1 < p 'S
00 •
Problem IV: When do we have
o
I\H/(x)"LP THEOREM 5. Let yet) be a curve in Rn . If the vectors y'(O),y"(0),y(3\0)'" span R n ,
!.J
11 sup O
o
'S CpllfllLP
Problem IV': When do we have
II m/II LP 'S CpllfllLP
h
If(x-y(t»I dtll
'S LP
CpllfllLP
l
?
?
The change of variables described above preserves tangency and curvature conditions. So for example
~
Ul
I
t=o
will be parallel to the x
====._--
..~. ---
-.-....
406
.-
.._.
-- .
..."-
,."".. ._--_._. -
."
_-
--
.
STEPHEN WAINGER
AVERAGES AND SINGULAR INTEGRALS
axis, and if the curvature of the integral curves of v(x) never vanishes
by work of Kolmogorov and Silveristov [Z] on the partial sums of Fourier
2Y
a
Series and an approach to the Poisson Integral by Paley [p]. The idea is
will not be zero. It turns out that one can prove the following
j
at 2
407
to consider H . H* and Mh(x). (Mh(x))* where . denotes composition y y y y of operators and * signifies Hilbert space adjoint. In order to gain some insight into this method we shall just discuss it in the case y(t) = (t,t 2 )
t=O
theorems: THEOREM 6. Let y(t,x) = (t,r(t,x)) be a smooth curve in R 2 satisfying
where, of course, we already know the results by a different method.
at'(t,x) I at t=O
=
0
and
2
aatr It=o
*
Let us consider K = H . H;. The kernel of K will have support on y l(u,v)\u=t-s,v=t 2 -s 2 1. Hence one might hope that K would have a much smoother kernel than Hy ' (Of course if K were bounded on L 2 so would H ') One might hope then, that K would be a Calderon-Zygmund y operator. Let us see if this could be the case. One can see that
0 .
Then 105)
IIHyfl1 2
2 ::; Cllfll 2
L (R )
2 '
L (R )
~::
and
,1'
H;f(x,y) = !f(X+t,y+y(t)) 106)
11 m fll 2
2 ::; Cllfll 2
Y L (R )
2 .
L (R )
and hence that
A consequence of Theorem 6 for vector fields will then be
If
THEOREM 7. If v(x) is a smooth vector field in R 2 such that the
HyH;f(x,y) =
integral curves of v have nowhere vanishing curvature, IIHvfl1 2 2 ::; Ilfll 2 2 L (R ) L (R )
lim
~
J
2 2 f(x+t-s,y+t -s )
If
HyH;f(x,y) =
h
f(x-tv(x))dt = f(x)
d\~s .
Let u=s-t, v=s2_t 2 , and we find
and
h....O
~t ,
f(x-u,y-v) k(u,v)du dv
a.e.
o
where
for f in L 2 (R 2 ). These theorems are announced in NSWV. Here we shall give some discussion of the ideas. Because y(t,x) depends on x, the Fourier Transform no longer seems like a good tool to study H and my' We y must find a different way to employ orthogonality. Here we're motivated
C
k(u,v) = lui
(u-~) (u+~)'
Now k(u,v) does have its support spread out. However the singularities of k across the curves v = ± u 2 are not locally integrable away from the
r-'~I
-----
~-_.~-_._-~~~-~~~~---===="'---~~~~~.:===--
408
:-~~-:-'==--~--=_._._.'-----_._---:--=:---:"'._-=
STEPHEN WAINGER AVERAGES AND SINGULAR INTEGRALS
origin. Hence HyH; cannot be a Calderon-Zygmund operator. Let us
IIPh(xlll~2 S CIIP h(x/(x)II L2 '
try to see what goes wrong on the level of the Fourier Transform. Recall Hyf
and hence
Qp * f
=
409
IIP h (xl II 2 S C . L
where 00
=
Qp(t",77)
J
Since the function hex) is arbitrary we have e(it"t+i77t2 )dt
T'
Iisup Ph *fll 2
-00
h
We know from 48) that
1/2
Qp(t",77)
ie
I 1 sgn + C _77_ e Tf t"
=
operator Rh(x,y)f(x,y) =
f
=
L
h(x~y)
J
hex ,y)
f(x-t,y)dt .
o (We know IIR h (x,Yl(x,y)II
much nicer than Qp(t"'77) or Qp(t",77) - sgnt". Now sgnt" corresponds to the operator Lf
S Cllfll 2'
The fact that we had to modify Hy suggests that we should not expect 107) to hold, but we might expect a variant to hold - perhaps involving an
+ better terms. The multiplier for Hy ' H; would be essentially IQ I2 . Notice that p 2 IQp l doesn't look any nicer than Qp' However IQp(t",77)-Sgnt"1 2 is
L
L2
:; CI 1f(x,y)II
L2
.)
We might hope to prove dt f(x-t,y) T'
L is known to be bounded in L 2 . This suggests that we try to consider
108)
Mh(x)(Mh(x))*f(x) S C(Rh(x)Mh(xl(x) + Mh(x)Rh(x)f(x) + Bh(x)f(x))
where Bh(x) is some bounded operator. Even 107) is not quite right. We refer the reader to NSWV for the correct technical modification of 107).
M = (L-H y ) (L *-H;) .
If M is bounded in L
This concludes our discussion of the vector field problem.
2
so will be L - H ' Hence so w ill be H ' This y y actually works and is the basic idea in the proof of Theorem 4. We turn now to the idea of the proof of Theorem 5. Paley showed that
V. Recent developments The positive results of Theorem 2 and Theorem 5 assume that the curve yet) has some curvature at the origin. There have been a number
107)
ph(x) . (Ph(x))*f(x) S C[Ph(xl(x)+ ph(xl(x)]
where Ph is the Poisson Kernel. It follows from 107) that
of papers trying to understand what happens when this curvature condition is dropped. See [C], [CNVWW], [NVWW] 1), 2), 2), and [NE]. Let us note that we don't have positive results for all COO curves.
'-'.' -"- ...-"'..
410
.-....-
.-..,.,,,.._--.-..... -..... -...----·c .--·'0- __
STEPHEN WAINGER
Suppose for example y(t) is odd and y(t)
r(t)
=
f
=
(t,r(t»
where
s~P~
0 for t > 1
._.-----~--~-~-----
411
f
h
If(x-t,y)\ dt +
f
s~p ~
h
f(x-t,y-t+l)dt.
o
o
[SW] for an example to show that the maximal function along a COO curve
A Hi(~'77)
has no non-trivial positive results.
~
=
my(~'77)f(~'77)
The above authors have been investigating the behavior of the Hilbert Transform and maximal functions related to convex curves. Let y(t)
where
(t, r(t»
my(~'77) =
J
ei~tei77r(t) dt
J
THEOREM 8. Let y(t)
1
sin
~t dtt +
o
J
(t, ['(t»
sin
(~t + 77t -77) ~t
II Hyfl\ 2 ~ Cllfl\ 2 L
= -77. In fact it is easy to see
intervals ai ~ t ~ b i ~ 1 and assume that the linear extensiog.of y on [ai,b i ] does not pass through the origin. Then if the ratios a~ are
1f"(Ct)/ ~21f"(t)1 for some C
f
IIHyfl1
2
h(Ct)
*)
It is somewhat more difficult to produce counterexamples for the
For example in the case of the two dimensional Curve
described above, it is easy to see that
L
~ Cllfll
L
2
if and only if
cannot exist in the L 2 sense for every f in L 2 •
LP, because it is dominated by
> o.
f(x-y(t» dt t
I t'~E
t >0
If y(t) is odd
1
lim
L
if and only if
the following: Suppose y(t) is odd and linear on a sequence of
E ->0
with r(t) convex and increasing for
t> O. If r(t) is even
oo
which is easily seen to be unbounded if ~
my.
=
1
unbounded,
=
curve y at y(t). We then have
t
-oo
=
a plane curve with r(t) convex for positive t. Let h(t)
=
tr'(t) - r(t). -h(t) represents the y-intercept of the line tangent to the
oo
(t,r(t»
.-.~.--
Both of these operators are bounded by the classical theory. We refer to
Then
operator
~_.
AVERAGES AND SINGULAR INTEGRALS
for 0 ~ t ~ 1
{ t-1
__._. ._....,_.--.--_.. - __
~_~
my
is bounded in every
for some
~
2h(t) ,
t> 0
C > O.
There are generalizations of Theorem 8 to higher dimensions, and an investigation of the LP theory has begun. We know that
....... .:::.__
--
412
.-
_,_.
ii!ii
n
::--
_.
".
_
••
*
i!!~~_~===~
for large A. Of course we can make a change of variables t =
holds for some C > 0, however the maximal functions can be
bounded on L 2 (and in fact in LP for any p curves even if
*
413
AVERAGES AND SINGULAR INTEGRALS
STEPHEN WAINGER
Ilmyfll L 2::; Cllfll L 2 if
.__
and observe
> 1 ) for some convex
fails.
r
I(A) =~
A-2)
Recently Phong and Stein [PS] introduced a general problem of which our problems are special cases. They consider at each point P in R n a submanifold Mp of dimension say r and an Zygmund kernel K(P,Q). Then they consider
x"fA
"fA
where
dimensional Calderon
A-3)
B=
f
00
2
e- t dt .
-00
Tf(P) 1f(Q) K(P,Q)dm(Q) The point we wish to make here is that if A is large most of the where dm(Q) is a measure on Mp ' They show that if n k = n-l
contribution to the integral I(A) comes from a small neighborhood of the
> 3 and
origin. In fact
II Tf I LP ::; CIIfllLP
f
if Mp satisfies a kind of generalized curvature condition. See [PS]. Various authors have also considered multiple parameter problems
e-
At2
J
dt <
\t\>_l
1
e- At2 dt+
which are essentially multiple Hilbert transforms on surfaces and multi <e
parameter maximal functions on surfaces. See [NW2], [V], [STRl], and [CSS].
->
_A 1 /5
+e
~A/2
0 very fast as A ->
method of steepest descent. The interested reader can find a more
example we might consider
Let us first consider the behavior of the integral 00
J
•
can expand the integrand in a power series in that little interval. For
detailed description in [B].
=
00
interval - _1_ < t < _1_. Now if we perturb the integrand in I(A), we >(. /5 - - A2/5
Here we shall try to give an explanation of the main ideas of the
I(A)
- 2 dt
Thus the main contribution to the integral I(A) comes from the small
Appendix 1. An introduction to the method of steepest descents.
A-I)
e- i
t2
t>l
<
A2/5 _t9
A2/5
I
A
](A)
e-,\x2 dx x '
=
I -00
-00
I
00
e -Ah(t) dt
414
STEPliEN WAINGER
where h(t) ~ t for t> O.
2
for large t, h(t) = t 2 + <::l(t 3 ) for small t and h(t) > 0
J
f
g(z)eAh(z)dz
dt
Itl~A2/5
g(z) and h(z) are holomorphic, we would try to choose a contour on which Re h(z) had only a finite number of maxima, and argue that the
is exponentially small as before. Now
e-Ah(t)dt
main contribution to the integral should come from a small neighborhood
J
e-At2 .
=
It1
More, generally, if we were concerned with the asymptotic behavior for large values of A of an integral of the form
Then one can easily see that
f
(1+~)cAt3)
of the largest maximum or perhaps an endpoint and we then expand g and h in a Taylor series at such points. At such a maximum, ~, h'(')
Itl
=
O.
Thus the main contribution to our integral should come from a point where hOe')
f
e-
At2
dt +
tk 1 /5
Itl~A-2/5
=
415
A VERAGES AND SINGULAR INTEGRALS
~ + a e-A1 /5
VA
(
f
00
)
0 or an endpoint. This is also reflected in Van Der Corput's
lemma. A variant of this principle for non-analytic functions is called the e-.\t2 dt
principle of stationary phase and is discussed in Professor Stein's lectures in these proceedings.
-00
+
=
tl(
Appendix 2. The method of stationary phase and quantum mechanics
1 ) . A1/2+1 /5
The method of stationary phase lends itself to a formulation of quantum mechanics that is very appealing to at least some mathematicians. The
In the method of steepest descents we try to choose a contour of
principle of stationary phase asserts that the integral
integration so that on the new contour the situation would be essentially that of
J. Thus for example,
J
J b
if we had
1=
eiAf(t)dt
00
K=
a
eiAt2+AP(t)dt with f(t) real gets most of its contributions for large A, near a, b, or a
-00
zero of £'(t). and pet) were very negative at infinity and a(t 3 ) near t try to write t
=
0 we would
= a + iT and integrate on the line a = T for /tl < 1/A2/5,
and we would expand eAP(t) in a power series in this small interval.
If we had no endpoints for example if f were periodic with period b - a or if the interval of integration was from
-OQ
to
OQ
and f oscil
lated very rapidly for large t, we would expect the main contribution to the integral to come from small neighborhoods of a zero of f'.
f
~
* · __
...
w_
._~
-'.-
.' ••-
._-
416
'C!'IIft __ n
---
w.wmwrwii'·-··
.- -
..
-_.-
--'---ilii-
-'ijijjji;'--_.
•
,
2'1llIiUIIailQj~
STEPHEN WAINGER
AVERAGES AND SINGULAR INTEGRALS
Let us now turn to quantum mechanics. In particular let us consider a
417
where the integration is an integral over all paths x(t) such that
particle moving from a point xa to x b as time evolves from time t to a time t b . According to classical physics, the particle will follow a path
x(ta ) = x a and x(tb ) = xb' Leaving aside the question of how such an integral can be precisely defined, let us try to guess what paths contribute
for which the classical action is stationary. If x(t) is any path, and our
the most to the integral F. Since h is very small, the principle of
particle has mass m and is moving under the influence of a potential
stationary phase would indicate that the main contribution to the integral
V(x,t), the classical Lagrangian is defined by
F should come from a small neighborhood of a path of which some kind
L
=
of a derivative of S was zero. Thus we might expect the main contribu
[.x(t)]2 - V(x(t),t) .
2"m
tion to the integral F to come from paths which are very close (on a scale of h ) to the path
The action along a path x is defined by
x
defined by
*.
There have been many papers in the mathematical and physical litera t
S(x)
O~
J t
ture dealing with the problem of making sense out of the definition F.
b
See for example [CS] and references cited there. However, the original
L(x(t))dt.
definition in [FH] serves the purpose of making many formal calculations.
a
The path on which the classical particle moves will be a path if(t) such that x(t a )
=
We may imagine dividing the t interval into 2 j subintervals, Ij , of equal length
x a ' x(t b ) = xb' and such that
Ik
d :TI'" S(x +ox)1 = 0 . uo 0=0
One of the most important principles of quantum mechanics asserts that
=
[[tb-t a ]
~ + t a , [tb-ta ] 2·hl
2J
xk = X
classical mechanics. In our example it means the only paths that are im
*.
This principle is expressed in
terms of a small number h. The principle says that the only classical paths that should be important are paths which differ from
x
terms of probability amplitudes of events. In our case the probability amplitude of passing from xa at time ta to xb at time t is given by b an integral F
=
i S(x) h
e
:Dx
k
=
~a + 0[tb-t a ])
1,2,· .. ,2 L l. We then take
only on an
h scale of measuring. Now the Feynman path integral formulation is in
J
+ taJ
We consider only paths which are linear on I k . Such paths are determined by
motion on a classical scale must be essentially described by the laws of
portant are paths near the path if of
n1
F
=
.r
lim A· J
,
j->oo
1 2 e hi.S(X ,···,x L
1 ) dx
1
dx "",dXk 2
j
R
2 -l
where S(xl'""x 2 L 1 ) is the action along the polygonal path determined by xl',,·,x 2 L 1 , and A j is a normaliZing factor. For more details see [FH]. We would like to make one final remark about the book [FH]. It's
418
STEPHEN WAINGER
great. It explains quantum mechanics in terms of mechanics and does not
A VERAGES AND SINGULAR INTEGRALS
[CFl]
A. Cordoba and R. Fefferman, "A geometric proof of the strong maximal theorem," Annals of Math., Vol. 102, 1975, pp. 95-100.
[CF2]
_ _ _ _ , "On differentiation of integrals," Proc. Nat. Acad. ScL, U.S.A., Vol. 74, 1977, pp. 2211-2213.
[CF3]
_ _ _ _ , "On the equivalence between the bounded ness of certain classes of maximal and multiplier operators in Fourier Analysis," Proc. Nat. Acad. ScL, U.S.A., Vol. 74, 1977, pp. 423-425.
use notions of atomic physics as many of the standard books do. Thus it is accessible to many more mathematicians than standard quantum mechanics texts. The book also elucidates the differences between the nature of physicists and mathematicians. If you don't want to know h to 3 significant figures in ergs/sec, whatever they are, you probably would rather be a mathematician than a physicist. Finally, many mathematicians .could probably learn a great deal to improve themselves as mathematicians by reading the book.
419
[CNVWW] A. Cordoba, A. Nagel, J. Vance, S. Wainger, and D. Weinberg, "LP bounds for Hilbert Transforms along convex curves," preprint. [F]
E. B. Fabes, "Singular integrals and partial differential equa tions of parabolic type," Studia Math., Vol. 28, 1966, pp. 81-131.
[FEF]
R. Fefferman, "Covering lemmas, maximal functions, and multi plier operators in Fourier Analys is," Proc. Symp. in Pure Math., Vol. XXXV, Part 1, pp. 51-60.
[FH]
N. DeBruijn, Asymptotic Methods in Analysis, North Holland
Publishing Co., Amsterdam, 1958.
R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill, New York.
[H]
H. Busemann and W. Feller, "Zur Differentiation des
Lebesguesche Integrale," Fund. Math, Vol. 22, 1934,
pp. 226-256.
L. Hormander, "Estimates for translation invariant operators in LP spaces," Acta Math., Vol. 104, 1960, pp. 93-139.
[KS]
[CS]
R. Cameron and D. Storvick, A simple definition of the Feynrnan integral with applications, Amer. Math. Soc., Providence, 1983.
R. Kunze and E. Stein, "Uniformly bounded representations and harmonic analysis of the 2 x 2 unimodular group," Amer. of Math., Vol. 82, 1960, pp. 1-62.
[NRW]
[CSS]
H. Carlsson, P. Sjogren, and J. Stromberg, "Multipara meter
maximal functions along dilation-invariant hypersurfaces" to
appear in Trans. of the A.M.S.
A. Nagel, N. Riviere, and S. Wainger, "On Hilbert transforms along curves, II, "Amer. J. Math., Vol. 98,1976, pp. 395-403.
[NRWM]
[CW]
H. Carlsson and S. Wainger, "Maximal functions related to con vex polygonal lines, " to appear.
A. Nagel, N. Riviere and S. Wainger, "A maximal function associated to the curve (t,t 2 )," Proc. Nat. Acad. of Sci., U.S.A., Vol. 73, 1976, pp. 1416-1417.
[NSWB]
A. Nagel, E. Stein, and S. Wainger, "Balls and metrics defined by vector fields I; Basic Properties," to appear in Acta Mathematica.
[NSW]
A. Nagel, E. Stein, and S. Wainger, "Boundary behavior of functions holomorphic in domains of finite type," Proc. Nat. Acad. Sci. U.S.A., Vol. 78, 1981, pp. 6595-6599. A. Nagel, E. Stein, S. Wainger, "Hilbert transforms and maximal functions related to variable curves," Proc. of Symposia in Pure Math., Vol. XXXV, part 1,1979, pp. 95-98.
STEPHEN WAINGER
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF WISCONSIN
MADISON, WISCONSIN
REFERENCES
[B] [BF]
[C]
M. Christ, preprint.
[CS]
J. L. Clerc and E. M. Stein, "LP multipliers for non-compact
symmetric spaces," Proc. Nat. Acad. ScL, U.S.A., Vol. 71,
1974, pp. 3911-3912.
[COR1]
A. Cordoba, "The Kekeya maximal function and the spherical
summation multipliers," Amer. J. of Math., Vol. 99, 1977,
p. 1-22.
[NSWV]
[COR2]
"Maximal functions, covering lemmas and Fourier
multipliers," Proc. Symp. in Pure Math., Vol. XXXV, Part I,
1979, pp. 29-50.
[NVWW1] A. Nagel, J. Vance, S. Wainger, and D. Weinberg, "Hilbert transforms for convex curves," Duke Math. J., Vol. 50, 1983, pp. 735-744.
J.
._'••i1i.-
.._ .. _...._-_ ..j
~.ii'lliiii·;ijii'''-
420
Si\IlJllli"'-'--
t''UU'.UII''W
--;iii if
v
m
"i""W'~'~;rn;w~;:;tr'f~~~~j;
•
<;;R;;WW_¥!I'llnJW"-,JI;]II._\n;
....- - - - . - . - - - - - " ' ' ' ' ' ' ' ' ' ' ' '.......
AVERAGES AND SINGULAR INTEGRALS
STEPHEN WAINGER
[NVWW2] A. Nagel, J. Vance, S. Wainger, and D. Weinberg, "The Hilbert transform for convex curves in R n ," to appear in Amer. ]. of Math. [NVWW3]
i-jBiil;~_t
421
[SWE]
E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton.
[STRl]
R. Strichartz, "Singular integrals supported on submanifolds," Studia Math., Vol. 74, 1982, pp. 137-15l.
, "Maximal functions for convex curves," Preprint.
[NW]
A. Nagel and S. Wainger, "Hilbert transforms associated with plane curves," Trans. Amer. Math. Soc., Vol. 223, 1976, pp. 235-252.
[STR]
J. Stromberg, "Weak estimates on maximal functions with rectangles in certain directions," Ark. Mat., Vol. 15, 1977, pp. 229-240.
[NW2]
_____ , "L 2 boundedness of Hilbert transforms along surfaces and convolution operators homogeneous with respect to a multi-parameter group," Amer. J. of Math., Vol. 99, 1977, pp. 761785.
[STRO]
J. Stromberg, "Maximal functions associated to rectangles
[NE]
W. Nestlerode, "Singular integrals and maximal functions associated with highly monotone curves," Trans. Amer. Math. Soc., Vol. 267, 1981, pp. 435-444.
[p]
R. Paley, "A proof ofa theorem on averages," Proc. Lond. Math. Soc., Vol. 31, 1930, pp. 289-300.
[R]
N. Riviere, "Singular integrals and multiplier operators," Ark. Mat., Vol. 9, 1971, pp. 243-278.
[PS]
D. Phong and E. Stein, "Singular integrals related to the Radon transform and boundary value problems," Proc. Nat. Acad. Sci., U.S.A., Vol. 80, 1983, pp. 7697-770l.
[S]
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
[SBC]
_ _ _ _ , Boundary Behaviour of Holomorphic Functions of Several Complex Variables, Princeton University Press, Princeton, 1972.
[SH]
_ _ _ _ , "Maximal functions: Homogeneous curves," Proc. Nat. Acad. Sci., U.S.A., Vol. 73, 1976, pp. 2176-2177.
[Ssp]
- -_ _ , "Maximal functions: Spherical means," Proc. Nat. Acad. Sci., U.S.A., Vol. 73, 1976, pp. 2174-2175.
[SPL]
_ _ _ _ , Topics in Harmonic Analysis related to the LittIewood Paley Theory, Princeton University Press, Princeton, 1970.
[SI]
_ _ _ _ , "Interpolation of linear operators," Trans. Amer. Math. Soc., Vol. 88, 1958, pp. 359-376.
[SWA]
E. Stein and S. Wainger, "The estimation of an integral arising in multiplier transformations," Studia Math., Vol. 35, 1970, pp. 101-104.
[SW]
_ _ _ _, "Problems in harmonic analysis related to curvature," Bulletin of the A.M.S., Vol. 84, 1978, pp. 1239-1295.
with uniformly distributed directions," Ann. of Math., Vol. 107, 1978, pp. 399-402. [V]
J. Vance, "LPboundedness of the multiple Hilbert transform along a surface," Pacific J. of Math., Vol. 108, 1983, pp. 221-24l.
[WE]
D. Weinberg, "The Hilbert transform and maximal function for approximately homogeneous curves," Trans. Amer. Math. Soc., Vol. 267, 1981, pp. 295-306.
[Z]
A. Zygmund, Trigonometric Series, Vols. I & II, Cambridge University Press, London, 1959.
_,.._..
._. _.
,••
__
_,.__.
.u_
_,.•._'._••••_
._..__ ._.
__
~_
_.__
..
,
,.._, _ .__
_
.'._._ ._
~.
-
_",,_.• ,
.,._..__.__.._.,_ _
.u .•. _'w_.".. _."W""",
_ ••• _,,
• _.,
'" ••.
_'".,.~,
•.•·••••.
u"._
"._
~
__
_ ,.•.•..,_._ ."".
INDEX convex curves, 411
approach regions
admiss ible, 245
non-isotropic, 261
non-tangential, 244
curvature, 321, 361
and Fourier transform, 321, 325,
268, 375
A p classes, 73,353 area integral, 92
atomic decomposition, 114, 156,
159, 340
DeGiori-Nash regularity theory,
144, 158
Dirichlet problem, 132, 243
domain of holomorphy, 211
Bergman kernel, 230
duality of Hl and BMO, 114, 340
B.M.O. (Bounded mean oscillation),
9, 94, 331, 340
BMO(R~ x R~), 101
electrostatics, 163
Bochner-Martinelli formula, 196
Bochner-Riesz summability, 344
Calderon-Zygmund decomposition,
48
Campbell-Hausdorff formula, 267
exponential mapping, 267, 275
Fatou's theorem, 245
finite type, 269, 283, 324
Fornaess imbedding theorem, 226
functional calculus, 27
canonical coordinates, 275
g-functions, 393
Carleson measure, 16, 94
Cauchy-Fantappie formula, 195
Cauchy integral, 5, 8, 186
on a Lipschitz curve, 143
Cauchy-Riemann
~quation,
201
Hardy space, 144
HP spaces, 89
HP(R~ x R~), 101
harmonic measure, 141
convergenge of averages over
spheres, 358, 361
a long curves, 361
along vector fields, 361, 406
Hartogs extension phenomenon, 207
covering lemmas, 60
Henkin integral formula 219, 336
heat operator, 254
423
424
INDEX
Heisenoerg group, 257, 335
Hilbert transforms along curves, 361,372 along vector fields, 362,406
non-commuting vector fields, 267
Library of Congress Cataloging-in-Publication Data Beijing lectures in harmonic analysis.
oscillatory integrals (first kind), 308
(second kind), 335, 344, 349
hydrostatics, 177
hypoelliptic differential opera
tors, 281
Poisson integral, 187,243
bi-Poisson integral, 102
Kohn (canonical) solution, 209
Laplacian, 266
Rellich-type formulas, 150, 166
(Annals of mathematics studies; no. 112)
Bibliography: p.
Includes index.
1. Harmonic analysis. I. Stein, Elias M..
1931- . II. Series.
515' .2433 86-91452
QA403.B34 1986 ISBN 0-691-08418-1
ISBN 0-691-08419-X (pbk.)
res triction theorems, 325, 344
Korn-type inequalities, 145
Korteweg-de Vries equation, 25
Laplace equation, 132,242
Leray form, 194
Levi-pseudoconvex, 214, 257
polynomial, 227
Lipschitz domain, 133, 145
Littlewood-Paley-Stein theory,
48,53
Sobolev estimates, 145
space of holomorphy, 42
space of homogeneous type, 251
stationary phase and quantum
mechanics, 415
Stein-Weiss spaces, 96, 160
steepest descents, 412
Stokes theorem, 189
local singular function, 210
surfaces (non-zero curvature), 321
Lu Qi-Keng conjecture, 235
systems of elliptic equations, 133,
163
maximal functions, 48, 49, 245
strong, 60
spherical, 359, 393
on curves, 391, 400
on vector fields, 406
Szego kernel, 193,265,296,297
Transference theorem, 39
method of layer potentials, 133, 143 Van der Corput lemmas, 309, 370
Mobius transformation, 186
multilinear Fourier analysis, 18
multipara meter differentiation
theory, 57
multipliers, 72
weight norm inequalities, 72
Zygmund conjecture, 67
Elias M. Stein is Professor of Mathematics at Princeton University
