Proceedings of the International Congress of Mathematicians
Vancouver, 1974
Recent Progress in Classical Fourier Analysis
Charles Fefferman*
In what sense does JÄ„ eix'*f(g) d£ converge to a given function/ on Rnl How do
properties such as the size and smoothness of/influence the behavior of its Fourier
transform/? These simple questions lie at the heart of much of classical analysis.
Their deeper study leads naturally to certain basic auxiliary operators defined on
functions on Rn ; and Fourier analysts seek to understand these operators and their
generalizations, and to apply them to various branches of analysis. In this paper I
shall describe some basic results and applications of Fourier analysis and speculate
briefly on the future. I have left out many topics of great importance, and empha-
sized merely those subjects I know something about.
Let me begin by sketching the state of the art as of about 1950. At that time, the
field was well developed only in the one-dimensional case. Since it had long been
known that the Fourier series of a continuous function on[0, 2%\ need not converge
at every point, Lebesgue measure (and in particular U) was clearly recognized as
a basic tool. The Plancherel theorem ffi\f(x)\2 dx = 2% 2 ^ \ak\2 with/(x) ~
J^ooCikeihx 8ave a complete characterization of L2 functions in terms of their
Fourier coefficients and established norm convergence of Fourier series. However,
the study oïU(p ^ 2) was known to be much harder. As an indication of the
difficulty of the problems of Intake a function f(x) ~ J^^ aheih* belonging to U
(p < 2) but not to L2, and modify its Fourier series by writing g(;c) ~ 2-
± ak&k*
with each ± sign picked independently by flipping a coin. Then with probability
one, g does not belong to U (or even to V) but is merely a distribution with
nasty singularities. Consequently, the assertion / ~ Ti-ooCt^ikx e Lp depends not
only on the sizes \ak\ of the Fourier coefficients, but also on subtle relationships
among the phases arg(aÄ).
*I could not have prepared this article without very generous help by Mrs. Yit-Sin Choo and
Dr. K. G. Choo.
© 1975, Canadian Mathematical Congress
95
96 CHARLES FEFFERMAN
Despite the difficulty of the problem, a fair amount was known by 1940 about
the relationship between the size of a function and the nature of its Fourier
series, thanks to pioneering efforts by Hardy and Littlewood, M. Riesz, Paley,
Zygmund, Marcinkiewicz and others. A result typical of the deepest work is as
follows (see [95]) :
THEOREM 1 (LITTLEWOOD-PALEY). Let {Sk}^-oobe a sequence of ± signs which
stays constant on each dyadic block. (A dyadic block is an interval of the form
[2N9 2^+1) or ( - 2^+1, - 2N]; the collection of all dyadic blocks will be denoted by A)
Then iff(x) ~ T^° ake^x belongs to U (I < p < oo), it follows that S ^ x Ä ^ * *
also belongs to ZA
Thus, although the phases arg(a^) play a decisive role in determining the size of
S-oo a^e'**, only the relationship of arg(aA) to relatively "nearby" arg(ffÄ,) really
matters.
Although the original techniques used to prove this and related theorems are
very complicated, the underlying strategy is simple. The starting point is to rewrite
Dirichlet's formula for the Nth partial sum of a Fourier series as
iti -
etNx r . .*-wu-v>/v- IAJy
*= e~w*H(eWyf(y)) - e+>'N*H(e-™yf(y))
SNf(x) = e-'"* J , *"<*->> /{x - y ) ^ ~ *m* J* erw<*-y>(x - y)
with Hf(x) s= JÄi (f(x - y)/y) dy, the integral being interpreted in the principal-value
sense. (Hf is called the Hilbert transform of/.) This is a bold step, since for
CQXJR1) (say), the integral in Dirichlet's formula converges absolutely, while that
defining the Hilbert transform does not.
Now the Hilbert transform also arises in complex analysis, for if F = u + iv
is a well-behaved analytic function on the upper half-plane R\, then on the bound-
ary R1, v is the Hilbert transform of w. Therefore we may hope to prove theorems
on the Hilbert transform and related operators via complex analysis (e.g., Cauchy's
theorem, Jensen's formula and Blaschke products, conformai mapping) and then
translate the results into information on Fourier series. To illustrate the "complex
method", let us prove a simple case of M. Riesz's famous theorem that the Fourier
series of an LP function on [0, 2%\ converges in norm (1 < p < oo). This comes down
to proving that the Hilbert transform is bounded on &>(Rl), and we give the argu-
ment for the easiest nontrivial case p = 4. Given a well-behaved analytic function
F ;= u + iv on R%, we have to show that j"#. v4 dx ^ C JÄ. w4 dx with C independent
of F. However, Cauchy's theorem for JF4 = w4 + 4/«3v — 6w2v2 - 4/wv3 + v4
yields JÄ, F* dx = 0 so that 0 = J*, Re(F4) dx = JÄ. (M4 - 6w2 v2 + v4) dx. Hence
JJP v 4 à ^ 6 Jjpu2v2dx£6(fjp w4dx)l/2(j*. v4dx)1/2by Cauchy-Schwarz. Dividing
both sides by (J^i vidx)in and squaring gives the desired inequality j"Äi vAdx S
36 JÄi u*dx. The general case (p ^ 4) is similar, though not so easy.1
^ e e the ingenious paper of S. Pichorides [72] for the exact norm of the Hilbert transform on Lp
and other related constants.
RECENT PROGRESS IN CLASSICAL FOURIER ANALYSIS 97
Now I can give a vague idea of the proof of the Littlewood-Paley theorem. The
idea is to relate an auxiliary operator S arising from complex analysis with an op-
erator G arising from Fourier series. Specifically, given / <~ Tkkakeikx o n IP? 2TC]
(say aQ = 0), we break up the Fourier series into dyadic blocks
/ - S ahe"* = S ( S arf**) = E/7(x)
and define G(f) as G(f)(x) = (£ / G„ |//(x)|2)1/2. The function S(f) is defined in
terms of the Poisson integral u(r, 0) off by the equation
S(f)(x) = ( Jf | Vw(r, 6)|* r dr doT
where r(x) is the Stoltz domain {(r, 6)\ \x - 0\ < 1 - r < •£} in the unit disc. £2(/)
has a natural interpretation as the area of the image of r(x) under the analytic
function u + iv whose real part is u. For our purposes, the basic facts concerning
S and G are:
(a) \\S(f)\\p~\\f\\p(l -boundedness of the Hilbert transform, since for F = w + iv analytic we
have | Vu | = |Vv| by the Cauchy-Riemann equations, and hence S(u) = S(v).
(b) || S(f) | | j ~ || G(f)\\p (1 < p < oo). Limitations of space prevent even a vague
description of the proof, but the basic tool here is the ZAboundedness of the
Hilbert transform acting on functions which take their values in a Hilbert space.
Once we know (a) and (b), the Littlewood-Paley theorem follows at once, since
evidently/= E /G^ / / and g = S/ej ±fj always have the same G-function. An
extensive discussion of the Littlewood-Paley theorem and of complex methods in
general may be found in Zygmund [95]. It must be admitted that the ingenious
complex-variable proofs of classical Fourier analysis leave the researcher in the
unhappy position of accepting the main theorems of the subject without any real
intuitive explanation of why they are true.
Now I want to speak of the profound changes which took place in classical
Fourier analysis, starting with the fundamental paper of Calderón and Zygmund
[17] in 1952.2 We shall be concerned here with efforts to generalize the basic oper-
ators, especially the Hilbert transform, from Rl to Rn. These generalizations are
anything but routine, because Blaschke products do not generalize to functions of
several complex variables, and consequently (for this and other reasons) the whole
complex method has to be abandoned and the results reproved by real-variable
techniques. Moreover, the real-variable methods and the «-variable analogues of
the Hilbert transform, ^-function, etc., play an important role in partial differential
equations, several complex variables, probability and potential theory, and will
probably continue to find further applications as time goes on.
The operators. Let us begin with the Laplace equation au — f in Rn (n > 2)
aIn retrospect we can see many of the ideas anticipated in the work of Titchmarsh, Besicovitch,
and Marcinkiewicz. (See [95].)
98 CHARLES FEFFERMAN
which one solves with the standard Newtonian potential
(1) f(y)dy w W
-
c
» J*" \x-y\»~* '
If/belongs to some function space (LP, Lip(a), C(Rn), etc) does it follow that the
second derivatives of w all belong to the same function space? Differentiating the
right-hand side of (1) (carefully) under the integral sign, we obtain for the second
derivatives of w the formula
<2> w" <*> = I § f = '*•><*>+ J* fê^f-w * •
where Qß is homogeneous of degree zero, and smooth away from the origin. Note
that the integral in (2) diverges absolutely, but at least for "nice" functions /we
may define that integral as
lim J M^ZJlf(y)dy,
e-o+ I*4I>* \*-y\n
and the limit exists by virtue of the essential cancellation j^ «-« Qjk(y) dy = 0. In
general, a singular integral operator is defined on functions on Rn by
(3) Tf(x) = lim J F?~*imdy,
£-0 \x-y\>* \x ~ y\
where Q is reasonably smooth and homogeneous of degree zero, and Js.-i Q(y)dy
= 0. For example, if we set Q(y) = sgn(j>) on R1, then (3) becomes Tf(x) —
IR1 (/OO *?K/(* ~ J0)> i-e-j Tis the Hilbert transform. Thus regularity properties of
solutions to the Laplace equation come down to boundedness on various function
spaces of a few specific singular integral operators ; that is, certain w-variable gener-
alizations of the Hilbert transform,
More generally, the theory of singular integral operators plays an essential role
in a host of problems of partial differential equations. To see why, start with a pure
wth order differential operator
and write
where Rj = (d/dxj) (— A)~l/2. Now Rj is called the jfth Riesz transform, and is given
as a singular integral operator by the formula
x*m =
CJ*- , *' ~ yi f(y) dy-"
\x- y\n+l
(Note that in one dimension, the single Riesz transform is just the Hilbert trans-
3See Horvâth [52] and Stein [85].
RECENT PROGRESS IN CLASSICAL FOURIER ANALYSIS 99
form.) Therefore, L factors as L = T(— A)m/2> where Tis a variable-coefficient
singular integral operator, i.e., an operator of the form
(4) TAX) = c(x)f(x) + j * ^x-y)i\x-y\lf(y) dy,
\x-y\
with c(-)e C°°(R»), Qe C(Rn x 5*"1), and J^ -i Q (x, œ) dœ *= 0 for all x. In other
words, modulo the factor (— A)m/2 a partial differential operator is merely a special
type of singular integral operator.
As a substitute for the Fourier transform, we associate to the operator T of (4)
its symbol a( T) defined by
(5) a{x, 0 ~ e(x) + J* Qforc/M) e*" do>.
hi"
Clearly, a(x, £) is homogeneous of degree zero in £ and smooth on Rn x (Rn\0).
In the special case T = (H\a\=m ^a(x)(d/dx)a)(- A)~m/2 the symbol is just a(x, £)
= H\a\=maa W(?f)a/|£|w. Moreover,
(6) Every smooth homogeneous a(x, £) on R2n arises as the symbol of a unique
singular integral operator, which we denote by a(x, D).
(7) The class of all symbols forms an algebra of functions. The mapping a(x, £)
-> a(x, D) is an approximate homomorphism from functions to operators. That is,
a\(x,D) oa2(x, D) = (o\'(T2)(x, D) + a "negligible" error.
(8) The adjoint of a(x, D) is given approximately by the complex-conjugate sym-
bol: (o(x,D))* p= a(x,D) + a "negligible" error.
By virtue of (6)—(8) we may construct useful operators merely by making ele-
mentary manipulations with symbols. For instance, an elliptic singular integral
operator a(x, D) (i.e., an operator with nonvanishing symbol) evidently has an
approximate inverse—we simply take (\ja)(x, D)—and the standard interior
regularity results on elliptic partial differential equations follow easily from these
observations.
So far we have described the theory as it first appeared in the pioneering work of
Calderón [12] on uniqueness of solutions to Cauchy problems. (Calderón used
singular integrals to diagonalize a matrix of differential operators. See also earlier
work of Giraud [43] and Mihlin [66].) Nowadays it is more common to work with
the closely related theory of pseudodifferential operators, invented by Kohn and
Nirenberg [60] and developed by Seeley [75], Hörmander [48], [49], Calderón and
Vaillancourt [16] and others. To arrive at the notion of pseudodifferential oper-
ators4 one uses (5) and the Fourier inversion formula in (4) to obtain
(9) r/(*) = J* «*•**(*, 0 / ( 8 j(x, £) in (x, £)-space to define approximate projec-
tion operators j(x, D), By microlocalizing, we hope to split up one hard problem
into many easy ones, and then patch the easy results together. In patching together,
one has to use a calculus of pseudodifferential operators with "exotic" symbols
Q satisfying merely
|(9/9*)*(3/30*71 S Caß\C IM'2~m
instead of the usual estimates (10). We shall say more about exotic symbols later on.
Now let us return to the tangential Cauchy-Riemann equations on the sphere
$2»-i c Cn, and this time suppose n > 2. A linear fractional transformation maps
the sphere to the hypersurface H = {(zl, z») e Cn~l x C1 |Re(z") = \z' |2}, which
has the structure of a nilpotent Lie group under the multiplication law (zf, zn)-
(w\ wn) — (zf + w', zn + wn + 2z' ' w'). By analogy with the Rn theory sketched above,
one expects that very sharp results on existence and regularity of solutions of the
tangential Cauchy-Riemann equations on H can be proved by using "singular
integrals" of the form Tf(x) = ]# K(xy~l)f(y) dy, where K has appropriate pro-
perties of cancellation and homogeneity with respect to the natural "dilations"
ö°(zf, zn) = (dz', ö2zn) on H. Moreover, once the results are known for H, one can
build a "variable-coefficient" theory of "singular integrals" on (say) the boundary
of a strongly pseudoconvex domain in Cn, by osculating the domain with biholo-
morphic images of H. Thus, a natural analogue of singular integrals provides a
powerful machine to study the tangential Cauchy-Riemann equations. (Note that
we cannot use the pseudodifferential operators viewpoint here, because the non-
abelian Fourier transform on H is [so far] too cumbersome even to deal with the
constant-coefficient case.) The ideas explained here come from Folland and Stein
[41], although singular integrals on nilpotent Lie groups have already appeared in
Knapp and Stein [59] in connection with irreducibility of the principal series. See
102 CHARLES FEFFERMAN
also Folland and Kohn [40] for the initial work of Kohn on tangential Cauchy-
Riemann equations, as well as Folland [39] and Stein [87].5
I have attempted to show by a few examples how w-dimensional analogues of the
Hilbert transform enter naturally into various branches of analysis. Let us now
review some techniques which have been used to study such operators, and then see
what insights we can gain into the Fourier transform in Rn.
The techniques. The first step in analyzing operators that generalize the Hilbert
transform is to prove L2-boundedness. Fortunately, this is often an easy conse-
quence of the Plancherel theorem, as in the case of a constant-coefficient singular
integral operator
Q(x - y) Tfw = i*^-y\;W)4y
where one has (f/)(£) = a(£)f(£) with a e L°°. The S-function falls into this category
—it is not hard to sho
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