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Singular Integrals: The Roles of Calderón and Zygmund Elias M. Stein
Editor’s Note : Alberto Pedro Calderón died on April 16, 1998. A memorial article appears elsewhere in this issue.
he subject matter of this essay is raries and predecessors, is contained in his fa- Alberto Calderón’s pivotal role in the mous treatise Trigonometrical Series, published creation of the modern theory of singu- in 1935. The time in which this took place may be lar integrals. In that great enterprise viewed as the concluding decade of the brilliant Calderón had the good fortune of work- century of classical harmonic analysis: the ap- ingT with Antoni Zygmund, who was at first his proximately one hundred-year span which began teacher and mentor and later his collaborator. For with Dirichlet and Riemann, continued with Can- that reason any account of that theory has to be tor and Lebesgue among others, and culminated in part the story of the efforts of both Zygmund with the achievements of Kolmgorov, M. Riesz, and Calderón. With this in mind, I shall explain the and Hardy and Littlewood. various goals that motivated them, describe some It was during that last decade that Zygmund of their shared accomplishments and later work began to turn his attention from the one-dimen- of Calderón, and discuss briefly the wide influence of their achievements. sional situation to problems in higher dimensions. At first this represented merely an incidental in- Zygmund’s Vision: 1927–1949 terest, but then later he followed it with increas- In the first period of his scientific work, from 1923 ing dedication, and eventually it was to become the to the middle 1930s, Zygmund devoted himself to main focus of his scientific work. I want now to de- what is now called “classical” harmonic analysis: scribe how this point of view developed with that is, Fourier and trigonometric series of the cir- Zygmund. cle, related power series of the unit disc, conjugate In outline, the subject of one-dimensional har- functions, Riemannian theory connected to unique- monic analysis as it existed in that period can be ness, lacunary series, etc. An account of much of understood in terms of what were then three what he did, as well as the work of his contempo- closely interrelated areas of study and which in many ways represented the central achievements Elias M. Stein is professor of mathematics at Princeton Uni- versity. His e-mail address is stein@math. of the theory: real-variable theory, complex analy- princeton.edu. sis, and the behavior of Fourier series. Zygmund’s This essay, published in the Notices by arrangement with first excursion into questions of higher dimen- the University of Chicago Press, is a modified version of sions dealt with the key issue of real-variable the- the article entitled “Calderón and Zygmund’s theory of sin- ory, the averaging of functions. The question was gular integrals”, to appear in the proceedings of the 1996 as follows. The classical theorem of Lebesgue guar- conference in honor of Calderón’s seventy-fifth birthday anteed that for almost every x held at the University of Chicago. That article also con- tains an extended list of references and further biblio- 1 (1.1) lim f (y) dy = f (x), graphical notes which have been omitted from the pre- x I I Z diam(∈I) 0 | | I sent version. →
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where I ranges over intervals and when f is an in- fact that M. Riesz proved the Lp boundedness tegrable function on the line R1. properties of the Hilbert transform In higher dimensions it is natural to ask whether p.v. ∞ dy a similar result holds when the intervals I are re- f H(f )= f (x y) 7→ π − y placed by appropriate generalizations in n. The −∞ R R p fact that this is the case when the I’s are replaced by applying a contour integral to (F) , where F is by balls (or more general sets with bounded “ec- the analytic function whose boundary limit has f centricity”) was well known at that time. What as its real part. It should be noted that the Hilbert must have piqued Zygmund’s interest in the sub- transform has a simple expression as a Fourier mul- ject was his realization (in 1927) that a paradoxi- tiplier, that is, cal set constructed by Nikodym showed that the sign(ξ) H(f ) (ξ)= f (ξ), answer is irretrievably false when the I’s are taken i to be rectangles (each containing the point in ques- where denotes the Fourier transform;b from this tion) but with arbitrary orientation. To this must b the L2 boundedness is an immediate consequence be added the counterexample found by Saks sev- via Plancherel’sb theorem. eral years later, which showed that the desired analogue of (1.1) still failed even if we now re- (ii) The theory of the Hardy spaces Hp stricted the rectangles to have a fixed orientation These arose in part as substitutes for Lp, when (e.g., with sides parallel to the axes) as long as one p 1, and were by their very nature complex- 1 ≤ allowed f to be a general function in L . function-theory constructs. (It should be noted, It was at this stage that Zygmund effectively however, that for 1
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the “area integral’’ (S(u)(θ))2 = by conceiving some simple but fundamental ideas that go to the heart of the matter and then devel- (1.4) u(z) 2 dxdy is finite, ops and exploits these insights with great power. |∇ | ZZiθ In proving (1.3) (1.2) we may assume that u (e ) ⇒ is bounded inside the saw-tooth domain that iθ where (e Γ ) is a nontangential approach region arose in (iv) above: this region is the union of ap- iθ with vertex e . proach regions (eiθ), (“cones”) with vertex eΩiθ, for The Γcrucial first step in the proof was the ap- points eiθ E, and E a closed set. Calderón in- ∈ plication of the conformal map (to the unit disc) troduced the auxiliaryΓ harmonic function U, with of the famous saw-tooth domain in Figure 1.1 U the Poisson integral of χcE, and observed that This mapping allowed one to reduce the implica- all the desired facts flowed from the dominating tion (1.3) (1.2) properties of U: namely, u could be split as to the special⇒ u = u + u , where u is the Poisson integral of a case of bounded 1 2 1 bounded function (and hence has nontangential harmonic func- limits a.e.), while by the maximum principle tions in the unit u2 cU, and therefore u2 has (nontangential) disc (Fatou’s | |≤ theorem) and limits =0at a.e. point of E. also played a The second idea (used to prove the implication (1.2) (1.4)) has as its starting point the simple corresponding ⇒ role in the other identity parts of the (2.1) u2 =2 u 2 proof. |∇ | valid for any harmonic function. This can be com- It is ironic ∆ that complex bined with Green’s theorem methods with their great ∂A ∂B power and suc- (B A A B) dxdy = B A dσ, ZZ − Z ∂n − ∂n cess in the one- ∂ dimensional ∆ ∆ 2 theory actually whereΩ A = u and B is anotherΩ ingeniously chosen stood in the way auxiliary function depending on the domain of progress to only. This allowed him to show that Figure 1. Saw-tooth domain. higher dimen- y u 2 dxdy < , which is an integratedΩ |∇ | ∞ sions and ap- RR peared to block further progress. The only way revision of (1.4). Ω past, as Zygmund foresaw, required a further de- It may be noted that the above methods and the velopment of “real” methods. Achievement of this conclusions they imply make no use of complex objective was to take more than one generation and analysis and are very general in nature. It is also in some ways is not yet complete. The math- a fact that these ideas played a significant role in p ematician with whom he was to initiate the effort the later real-variable extension of the H theory. to realize much of this goal was Alberto Calderón. ------Calderón and Zygmund: 1950–1957 Starting in the year 1950, a close collaboration Zygmund spent the academic year 1948–49 in Ar- developed between Calderón and Zygmund which gentina, and there he met Calderón. Zygmund lasted almost thirty years. While their joint re- brought him back to the University of Chicago, and search dealt with a number of different subjects, soon thereafter (in 1950), under his direction, their preoccupying interest and most fundamen- Calderón obtained his doctoral thesis. The dis- tal contributions were in the area of singular in- sertation contained three parts, the first about er- tegrals. In this connection the first issue they ad- godic theory, which will not concern us here. It is dressed was—to put the matter simply—the the second and third parts that interest us, and extension to higher dimensions of the theory of the these represented breakthroughs in the problem Hilbert transform. A real-variable analysis of the of freeing oneself from complex methods and, in Hilbert transform had been carried out by Besi- particular, in extending to higher dimensions some covitch, Titchmarsh, and Marcinkiewicz, and this of the results described in (iv) above. In a general is what needed to be extended in the Rn setting. way we can say that his efforts here already typi- A reasonable candidate for consideration pre- fied the style of much of his later work: he begins sented itself. It was the operator f Tf, with 7→ 1 (2.2) T (f )(x)=p.v. K(y)f (x y) dy, Figure 1 is based on a diagram in Zygmund [1959], Vol. − Zn II, p. 200. R
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when K was homogeneous of degree n, satisfied he published (in 1956) an account of some regularity, and in addition satisfied− the Marcinkiewicz’s theorem and various generaliza- cancellation condition K(x) dσ(x)=0. tions and extensions he had since found. In it he x =1 | |R conceded that the paper of Marcinkiewicz “… Besides the Hilbert transform (which is the only seems to have escaped attention and does not find real example when n =1), higher-dimensional ex- allusion to it in the existing literature.” amples include the operators that arise as second The second point, like the first, also involves im- derivatives of the fundamental solution operator portant work of Marcinkiewicz. He had been Zyg- for the Laplacian (which can be written as 2 mund’s brilliant student and collaborator until his ∂ ( ) 1), as well as the related Riesz trans- ∂x2∂xj − death at the beginning of World War II. It is a mys- ∂ 1/2 tery why no reference was made to the paper by forms ∂x ( )− . For the Hilbert transform, n is ∆ j − Marcinkiewicz [1939b] and the multiplier theorem equal to 1 and K(x) is equal to 1 ; the Riesz trans- πx in it. This theorem had been proved by forms are given∆ (up to a constant multiple) by Marcinkiewicz in an n-dimensional form (as a K(x)=x / x n+1,j=1,...,n. j product “consequence” of the one-dimensional All of this| | is the subject matter of their historic form). As an application the Lp inequalities for the 2 memoir “On the existence of singular integrals”, ∂ 1 3 operators ( − ) were obtained ; these he which appeared in Acta Mathematica in 1952. ∂xi ∂xj There is probably no paper in the last fifty years had proved at the behest of Schauder. ∆ which had such widespread influence in analysis. The ideas in this work are now so well known that I will only out- line its contents. It can be viewed as having three parts. First, there is the Calderón-Zygmund Bk lemma and the corresponding Calderón-Zygmund decomposition. The Qk main thrust of the former is as a sub- stitute for F. Riesz’s “rising sun” lemma, which F. Riesz used in re-proving Lebesgue’s theorem about the almost everywhere differentiability of monot- one functions and which had implicitly played a key role in the earlier treat- Figure 2. Calderón-Zygmund decomposition of a nonnegative function f at 1 n ment of the Hilbert transform. A altitude α. The cubes Q = Qk have the property that α< Q − Q fdx 2 α. | |1 ≤ schematic representation of their de- In the complement of the balls Bk, a favorable L estimateR holds. composition is given in Figure 2.2 Second, using their decomposition, they then proved the weak-type L1 and Lp, 1
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To each operator T of this kind there corre- At MIT we would meet quite often, and over time sponds its symbol a(x, ξ), defined by an easy conversational relationship developed be- tween us. I do recall that we, in the small group (2.4) a(x, ξ)=a (x)+K(x, ξ), 0 who were interested in singular integrals then, felt a certain separateness from the larger community where K(x, ξ) denotes the Fourierb transform of K(x, y) in the y variable. Thus a(x, ξ) is homoge- of analysts—not that this isolation was self-im- neous ofb degree 0 in the ξ variable (reflecting the posed, but more because our subject matter was homogeneity of K(x, y) of degree n in y), and it seen by our colleagues as somewhat arcane, rar- − depends smoothly and boundedly on x. Con- efied, and possibly not very relevant. However, versely, to each function a(x, ξ) of this kind there this did change, and a fuller acceptance eventually exists a (unique) operator (2.3) for which (2.4) came. I want to relate now how this occurred. holds. One says that a is the symbol of T and also writes T = Ta. ------The basic properties that were proved were, Starting from the calculus of singular integral first, the regularity properties operators that he had worked out with Zygmund, p p Calderón obtained a number of important appli- (2.5) Ta : L L , k → k cations to hyperbolic and elliptic equations. His p most dramatic achievement was in the unique- where Lk are the usual Sobolev spaces, involving Lp norms of the function and its partial derivatives ness of the Cauchy problem (Calderón [1958]). through order k, with 1 smooth and bounded. On can write Besides (3.1) and (3.1 0) it suffices that the co- m/2 1+ (2.8) L = Ta( ) +(Error)0 efficients a belong to C , that the characteris- − α tics are simple, and that n =3 or m 3. Under for an appropriate symbol a, where the operator 6 ≤ ∆ these hypotheses u vanishes identically in a neigh- (Error)0 refers to an operator that maps p p borhood of the origin. L L , for k m 1. It seemed clear that k → k m+1 ≥ − Calderón’s approach was to reduce matters to this symbolic− calculus should have wide applica- a key “pseudo-differential inequality” (in a termi- tions to the theory of partial differential operators nology that was used later). This inequality is com- and to other parts of analysis. This was soon to be plicated but somewhat reminiscent of a differen- borne out. tial inequality that Carleman had used in two Acceptance: 1957–1965 dimensions. The essence of it is that At this stage of my narrative I would like to share a some personal reminiscences. I had been a student ∂u 2 φ +(P + iQ)( )1/2u dt of Zygmund at the University of Chicago, and in k ∂t − Z 1956 at his suggestion I took my first teaching po- 0 (3.2) ∆ sition at MIT, where Calderón was at that time. I a 2 had met Calderón several years earlier when he c φk u dt, ≤ Z k k came to Chicago to speak about the “method of ro- 0 tations” in Zygmund’s seminar. I still remember my where u(0) = 0 implies u 0 if (3.2) holds for ≡ feelings when I saw him there; these first impres- k . →∞ sions have not changed much over the years: I was Here P and Q are singular integral operators struck by the sense of his understated elegance, of the type (2.3), with real symbols and P invert- 2 reserve, and quiet charisma. ible; we have written t = xn, and the norms are L 1134 NOTICES OF THE AMS VOLUME 45, NUMBER 9 stein.qxp 8/12/98 8:40 AM Page 1135 norms taken with respect to the variables ference of the dimen- x1,...,xn 1. The function φk is meant to behave sion of the null space k − like t− , which, when k , emphasizes the ef- and the codimension of fect taking place near t =0→∞. In fact, in (3.2) we can the range of the opera- k take φk(t)=(t +1/k)− . tor and is an invariant The proof of assertions like (3.2) is easier in the under deformations. special case when all the operators commute; their The problem of deter- general form is established by using the basic facts mining it was con- (2.6) and (2.7) of the calculus. nected with a number The paper by Calderón was at first not well re- of interesting issues in ceived. In fact, I learned from him that it was re- geometry and topology. jected when submitted to what was then the lead- The result of the “See- ing journal in partial differential equations, ley calculus” proved Communications of Pure and Applied Mathemat- quite useful in this con- ics. text: the proofs pro- ceeded by appropriate ------deformations, and mat- At about that time, because of the applicability ters were facilitated if Alberto Pedro Calderón, circa 1979 of singular integrals to partial differential equa- these could be carried tions, Calderón became interested in formulating out in the more flexible the facts about singular integrals in the setting of context of “general” manifolds. This required the analysis of the effect symbols instead of re- coordinate changes had on such operators. A hint stricting attention to that the problem was tractable came from the ob- the polynomial symbols servation that the class of kernels, K(y), of the type coming from differen- arising in (2.2) was invariant under linear (invert- tial operators. A con- ible) changes of variables y L(y). (The fact that temporaneous account K(L(y)) satisfied the same regularity7→ and homo- of this development geneity that K(y) did was immediate; that the can- (during the period cellation property also holds for K(L(y)) is a little 1961–64) may be found less obvious.) in the notes of the sem- R. Seeley was Calderón’s student at that time, inar on the Atiyah- and he dealt with this problem in his thesis (1959). Singer index theorem Suppose x (x) is a local diffeomorphism. Then (see Palais [1965]); for the result is7→ that modulo error terms (which are an historical survey of “smoothing” of one degree) the operator (2.3) is some of the back- transformed into another operator of the same ground, see also Seeley kind, [1967]. Antoni Zygmund, circa 1979. T 0(f )(x) a0(x)=f (x)+p.v. K0(x, y)f (x y) dy, ------· Z − ------but now With the activity surrounding the index theorem, a0(x)=a0( (x)) it suddenly seemed as if everyone was interested and in the algebra of singular integral operators. How- K0(x, y)= K0( (x),Lx(y)), ever, one further step was needed to make this a household tool for analysts: it required a change where Lx is the linear transformation given by the ∂ (x) of point of view. Even though this change of per- Jacobian matrix . On the level of symbols this ∂x spective was not major, it was significant psycho- meant that the new symbol a was determined by 0 logically and methodologically, since it allowed the old symbol according to the formula one to think more simply about certain aspects of a0(x, ξ)=a( (x),Lx0 (ξ)), the subject and because it suggested various ex- tensions. with Lx0 the transpose inverse of Lx. Hence the sym- The idea was merely to change the role of the bol is actually a function on the cotangent space definitions of the operators, from (2.3) for singu- of the manifold. lar integrals to pseudo-differential operators The result of Seeley was not only highly satis- 2πix ξ factory as to its conclusions, it was also very timely (3.3) Ta(f )(x)= a(x, ξ)f (ξ)e · dξ, in terms of events that were about to take place. Zn R b Following an intervention by Gelfand (1960), in- with symbol a. (Here f is the Fourier transform terest grew in calculating the “index” of an ellip- 2πix ξ f (ξ)= e− · f (x) dx.) tic operator on a manifold. This index is the dif- n b RR b OCTOBER 1998 NOTICES OF THE AMS 1135 stein.qxp 8/12/98 8:40 AM Page 1136 Although the two operators are identical (when erties of the operators (2.3) and their calculus a(x, ξ)=a0(x)+K(x, ξ)), the advantage lies in the when the dependence on x was e.g. of class emphasis in (3.3) on the L2 theory and Fourier C(1+ ) —this hope was not to be realized. Further transform and theb wider class of operators that can understanding about these things could be be considered, in particular, differential opera- achieved only if one were ready to look in a some- tors. The formulation (3.3) allows one to deal more what different direction. I want to relate now how systematically with the composition of such op- this came about. erators and incorporate the lower-order terms in the calculus. ------One way of doing this was to adopt a wider The first major insight arose in answer to the class of symbols of “homogeneous type”: roughly following: speaking, a(x, ξ) belongs to this class (and is of Question: Suppose MA is the operator of mul- order m) if a(x, ξ) is for large ξ asymptotically the tiplication (by the function A), sum of terms homogeneous in ξ of degrees m j, − MA : f A f. with j =0, 1, 2,... . 7→ · The change in point of view described above What are the least regularity assumptions on A came into its full flowering with the papers of needed to guarantee that the commutator [T,MA] Kohn and Nirenberg (1965) and Hörmander (1965), is bounded on L2 whenever T is of order 1? (after some work by Unterberger and Bokobza 1 d In R if T happens to be dx, then [T,MA]=MA0, (1964) and Seeley (1965)). It is in this way that sin- and so the condition is exactly gular integrals were subsumed by pseudo-differ- 1 ential operators. Despite this, singular integrals, (4.3) A0 L∞(R ). ∈ with their formulation in terms of kernels, still re- In a remarkable paper Calderón [1965] showed tained their primacy when treating real-variable is- that this is also the case more generally. The key sues, issues such as Lp or L1 estimates (and even case, containing the essence of the result he proved, for some of the more intricate parts of the L2 the- arose when T = H d , with H the Hilbert trans- ory). The central role of the kernel representation dx form. Then T is actually d , its symbol is 2π ξ , of these operators became, if anything, more pro- dx | | and [T,M ] is the “commutator” C , nounced in the next twenty years. A 1 1 ∞ A(x) A(y) Calderón’s New Theory of Singular (4.4) C1(f )(x)= p.v. − f (y) dy. π (x y)2 Integrals: 1965– Z − In the years 1957–58 there appeared the funda- −∞ Calderón proved that f C1(f ) is bounded on mental work of DeGiorgi and Nash dealing with 7→ L2(R) if (4.3) holds. smoothness of solutions of partial differential There are two crucial points that I want to em- equations, with minimal assumptions of regular- phasize about the proof of this theorem. The first ity of the coefficients. One of the most striking re- is the reduction of the boundedness of the bilin- sults, for elliptic equations, was that any solution ear term (f,g) C1(f ),g to a corresponding u of the equation →h i property of a particular bilinear mapping, ∂ ∂u (F,G) B(F,G), defined for (appropriate) holo- (4.1) L(u) aij(x) =0 → ≡ ∂xi ∂xj morphic functions in the upper half-plane Xi,j z = x + iy,y > 0 by in an open ball satisfies an a priori interior regu- { } ∞ larity as long as the coefficients are uniformly el- (4.5) B(F,G)(x)=i F0(x + iy)G(x + iy) dy. liptic, i.e., Z0 This B is a primitive version of a “para-product” 2 2 (4.2) c1 ξ aij(x)ξiξj c2 ξ . (in this context, the justification for this | | ≤ ≤ | | Xi,j terminology is the observation that F(x) G(x)=B(F,G)(x)+B(G, F)(x) ). It is, in In fact, no regularity is assumed about the aij · fact, not too difficult to see that f C1(f ) is except for the boundedness implicit in (4.2), and 2 1 7→ the result is that u is Hölder continuous with an bounded on L (R ) if B satisfies the Hardy-space estimate exponent depending only on the constants c1 and c . 2 (4.6) B(F,G) H1 c F H2 G H2 . Calderón was intrigued by this result. He initially k k ≤ k k k k expected, as he told me, that one could obtain The second major point in the proof is the as- such conclusions and others by refining the cal- sertion needed to establish (4.6). It is the converse culus of singular integral operators (3.2), making part of the equivalence minimal assumptions of smoothness on a0(x) and (4.7) S(F) 1 F 1 K(x, y). While this was plausible—and indeed in his k kL ∼k kH work with Zygmund they had already derived prop- for the area integral S (which appeared in (1.4)). 1136 NOTICES OF THE AMS VOLUME 45, NUMBER 9 stein.qxp 8/12/98 8:40 AM Page 1137 The theorem of Calderón, and in particular the essentially the Hilbert transform. Also, when γ methods he used, inspired a number of significant has some regularity (e.g., γ is in C(1+ ) ), the ex- developments in analysis. The first came because pected properties of (i.e., L2,Lp boundedness, C of the enigmatic nature of the proof: a deep L2 the- etc.) are easily obtained from the Hilbert transform. orem had been established by methods (using The problem was what happened when, say, γ was complex function theory) that did not seem sus- less regular, and here the main issue that pre- ceptible to a general framework. In addition, the sented itself was the behavior of the Cauchy inte- non-translation-invariance character of the oper- gral when γ was a Lipschitz curve. ator C1 made Plancherel’s theorem of no use here. If γ is a Lipschitz graph in the plane, It seemed likely that a method of “almost-orthog- γ = x + iA(x),x R , with A L , then up to a { ∈ } 0 ∈ ∞ onal” decomposition—pioneered by Cotlar for the multiplicative constant, classical Hilbert transform—might well succeed in this case also. This led to a reexamination of Cot- (4.9) lar’s lemma (which had originally applied to the ∞ 1 (f )(x)=p.v. f (y)b(y) dy case of commuting self-adjoint operators). A gen- C x y + i(A(x) A(y)) Z − − eral formulation was obtained as follows: Suppose −∞ that on a Hilbert space T = Tj. Then where b =1+iA0. The formal expansion 2 P (4.8) T sup Tj Tj∗+k + Tj∗Tj+k . 1 k k ≤ j {k k k k} Xk x y + i(A(x) A(y)) − − Despite the success in proving (4.8) this alone (4.10) k 1 ∞ A(x) A(y) was not enough to re-prove Calderón’s theorem. = ( i)k − x y · − x y As understood later, the missing element was a cer- − kX=0 − tain cancellation property. Nevertheless, the gen- eral form of Cotlar’s lemma, (4.8), quickly led to a then makes clear that the fate of Cauchy integral number of highly useful applications, such as sin- is inextricably bound up with that of the C gular integrals on nilpotent groups (intertwining commutator C1 and its higher analogues Ck given operators), pseudo-differential operators, etc. by Calderón’s theorem also gave added impetus to k p.v. ∞ A(x) A(y) f (y) the further evolution of the real-variable Hp the- C f (x)= − dy. k π x y x y ory. This came about because the equivalence (4.7) Z − − −∞ and its generalizations allowed one to show that The further study of this problem was begun by the usual singular integrals (2.2) were also bounded Coifman and Meyer in the context of the commu- 1 p on the Hardy space H (and in fact on all H , tators Ck, but the first breakthrough for the Cauchy 0 OCTOBER 1998 NOTICES OF THE AMS 1137 stein.qxp 8/12/98 8:40 AM Page 1138 The first was a complete analysis of the L2 the- The decisive application of the Cauchy integral ory of “Calderón-Zygmund operators”. By this ter- to the potential theory of the Laplacian in a Lip- minology is meant operators of the form schitz domain was in the study of the bounded- ness of the double-layer potential (and the normal (4.11) T (f )(x)= K(x, y)f (y) dy, derivative of the single-layer potential). These are Zn R n 1 dimensional operators, and they can be re- initially defined for test functions f , with the − ∈S alized by applying the “method of rotations” to the kernel K a distribution. It is assumed that away one-dimensional operator (4.9). One should men- from the diagonal, K agrees with a function that tion that another significant aspect of Laplacians satisfies familiar estimates such as on Lipschitz domains was the understanding n K(x, y) A x y − , brought to light by Dahlberg of the nature of har- (4.12) | |≤ | − | n 1 monic measure and its relation to Ap weights. x,y K(x, y) A x y − − . |∇ |≤ | − | These two strands, initially independent, have been The main question that arises (and is suggested by linked together, and with the aid of further ideas the commutators Ck) is, what are the additional a rich theory has developed, owing to the added conditions that guarantee that T is a bounded op- contributions of Jerison, Kenig, and others. erator on L2(Rn) to itself? The answer, found by Finally, we return to the point where much of David and Journé (1984), is highly satisfying: a this began—the divergence-form equation (4.1). certain “weak boundedness” property, namely, Here the analysis growing out of the Cauchy inte- (Tf,g) Ar n wherever f and g are suitably nor- gral also had its effect. Here I will mention only the | |≤ malized bump functions, supported in a ball of ra- usefulness of multilinear analysis in the study of dius r; also, that both T (1) and T ∗(1) belong to the case of “radially-independent” coefficients, BMO. (Here BMO denotes the space of functions of also in the work on the Kato problem: the deter- bounded mean oscillation. This space first arose mination of the domain of √L in the case where in partial differential equations as a useful sub- the coefficients can be complex-valued. stitute for the space L∞ and later played a key role in Hp theory.) These conditions are easily seen to Some Perspectives on Singular Integrals: also be necessary. Past, Present, and Future The argument giving the sufficiency proceeded The modern theory of singular integrals, developed in decomposing the operator into a sum and nurtured by Calderón and Zygmund, has T = T1 + T2, where for T1 the additional cancella- proved to be a very fruitful part of analysis. Beyond tion condition T1(1) = T1∗(1) = 0 held. As a conse- the achievements described above, a number of quence the method of almost-orthogonal decom- other directions have been cultivated with great position, (4.8), could be successfully applied to success, with work being vigorously pursued up to 2 T1. The operator T2 (for which L boundedness was this time. In addition, here several interesting open proved differently) was of para-product type, cho- questions present themselves. I want to allude sen so as to guarantee the needed cancellation briefly to three of these directions and mention property. some of the problems that arise. The conditions of the David-Journé theorem, 1. Method of the Calderón-Zygmund lemma. As while applying in principle to the Cauchy integral, is well known this method consists of decompos- are not easily verified in that case. However, a re- ing an integrable function into its “good” and “bad” finement (the “T (b) theorem”), with b =1+iA0, parts, the latter being supported on a disjoint was found by David, Journé, and Semmes, and this union of cubes and having mean value zero on each does the job needed. cube. Together with an L2 bound and estimates of A second area that was substantially influenced the type (4.12), this leads ultimately to the weak- by the work of the Cauchy integral was that of sec- type (1,1) results, etc. ond-order elliptic equations in the context of min- It was recognized quite early that this method imal regularity. Side by side with the consideration allowed substantial extension. The generalizations of the divergence-form operator L in (4.1) (where that were undertaken were not so much pursued the emphasis is on the minimal smoothness of for their own sake, but rather were motivated in the coefficients), one was led to study also the po- each case by the interest of the applications. tential theory of the Laplacian (where the empha- Roughly, in order of appearance, here were some sis was now on the minimal smoothness of the of the main instances: boundary). In the latter setting a natural assump- (i) The heat equation and other parabolic equa- tion to make was that the boundary is Lipschitz- tions. This began with the work of F. Jones (1964) ian. In fact, by an appropriate Lipschitz mapping for the heat equation, with the Calderón-Zygmund of domains, the situation of the Laplacian in a Lip- cubes replaced by rectangles whose dimensions re- schitz domain could be realized as a special case flected the homogeneity of the heat operator. The of the divergence-form operator (4.1) where the do- theory was extended by Fabes, Rivière, and Sa- main was smooth, say, a half-space. dosky to encompass more general singular inte- 1138 NOTICES OF THE AMS VOLUME 45, NUMBER 9 stein.qxp 8/12/98 8:40 AM Page 1139 grals respecting “nonisotropic” homogeneity in In much the same way the general maximal op- Euclidean spaces. erator (ii) Symmetric spaces and semisimple Lie groups. 1 To be succinct, the crucial point was the extension (5.2) M(f )(x) = sup (y)f (x y) dy r n − to the setting of nilpotent Lie groups with dilations r>0 Z y r (“homogeneous groups”), motivated by problems | |≤ Ω arises from the one-dimensional Hardy-Littlewood connected with Poisson integrals on symmetric maximal function. spaces, and construction of intertwining opera- This method worked very well for Lp estimates tors. for p>1, but not for L1 (since the weak-type L1 (iii) Several complex variables and subelliptic “norm” is not subadditive). The question of what equations. Here we return again to the source of happens for L1 was left unresolved by Calderón singular integrals, complex analysis, but now in the and Zygmund. It is now to a large extent answered: setting of several variables. An important conclu- we know that both (5.1) and (5.2) are indeed of sion obtained was that for a broad class of domains n weak-type (1,1) if is in L( log L). This is the in C the Cauchy-Szegö projection is a singular in- achievement of a number of mathematicians, in tegral susceptible to the above methods. This was particular Christ andΩ Rubio de Francia. realized first for strongly pseudo-convex domains, When the method of rotations is combined with next weakly pseudo-convex domains of finite type the singular integrals for the heat equation (as in 2 in C , and more recently convex domains of finite 1(i) above), one arrives at the “Hilbert transform n type in C . on the parabola”. Consideration of the Poisson in- Connected with this is the application of the tegral on symmetric spaces leads one also to in- above ideas to the ∂-Neumann problem, and its quire about some analogous maximal functions as- boundary analogue for certain domains in Cn, as sociated to homogeneous curves. The initial major well as the study solving operators for subelliptic breakthroughs in this area of research were ob- problems, such as Kohn’s Laplacian, Hörmander’s tained by Nagel, Rivière, and Wainger. The subject sum of squares, etc. These matters also involved has since developed into a rich and varied theory, using ideas originating in the study of nilpotent beginning with its translation invariant setting on groups as in (ii). Rn (and its reliance on the Fourier transform), and The three kinds of extensions mentioned above then prompted by several complex variables, to a are prime examples of what one may call “one-pa- more general context connected with oscillatory in- rameter” analysis. This terminology refers to the tegrals and nilpotent Lie groups, where it was fact that the cubes (or their containing balls) which rechristened as the theory of “singular Radon occur in the standard Rn setup have been replaced transforms”. by a suitable one-parameter family of generalized A common unresolved enigma remains about “balls” associated to each point. While the general these two areas that have sprung out of the method one-parameter method clearly has wide applica- of rotations. This is a question that has intrigued bility, it is not sufficient to resolve the following workers in the field and whose solution, if posi- important question: tive, would be of great interest. Problem. Describe the nature of the singular Problem 1 integral operators that are given by Cauchy-Szegö (a) Is there an L theory for (5.1) and (5.2) if 4 projection, as well as those that arise in connec- is merely integrable? (b) Are the singular Radon transforms, and theirΩ tion with the solving operators for the ∂ and ∂b complexes for general smooth finite-type pseudo- corresponding maximal functions, of weak-type (1,1)? convex domains in Cn. 3. Product theory and multiparameter analysis. Some speculation about what may be involved To oversimplify matters, one can say that “prod- in resolving this question can be found below. uct theory” is that part of harmonic analysis in n 2. The method of rotations. The method of ro- R which is invariant with respect to the n-fold dila- tations is both simple in its conception and far- tions: x =(x1,x2,...,xn) (δ1x1,δ2x2,...,δnxn), reaching in its consequences. The initial idea was → δ > 0. Another way of putting it is that its initial to take the one-dimensional Hilbert transform, in- j concern is with operators that are essentially prod- duce it on a fixed (subgroup) R1 of Rn, rotate this 1 ucts of operators acting on each variable sepa- R , and integrate in all directions, obtaining in rately and then more generally with operators (and this way the singular integral (2.2) with odd ker- associated function spaces) that retain some of nel, which can be written as these characteristics. Related to this is the multi- (y) parameter theory, standing partway between the (5.1) T (f )(x)=p.v. f (x y) dy y n − one-parameter theory discussed above and prod- Zn | | Ω R Ω uct theory: here the emphasis is on operators where is homogeneous of degree 0, integrable in the unit sphere, and odd. 4For (5.1) we assume also that is odd. Ω Ω OCTOBER 1998 NOTICES OF THE AMS 1139 stein.qxp 8/12/98 8:40 AM Page 1140 which are “invariant” (or compatible with) speci- [1965] ———, Commutators of singular integral opera- fied subgroups of the group of n-parameter dila- tors, Proc. Nat. Acad. Sci. 53, 1092–1099. tions. [1977] ———, Cauchy integrals on Lipschitz curves and re- n lated operators, Proc. Nat. Acad. Sci. U.S.A. 74, The product theory of R began with Zygmund’s study of the strong maximal function, continued 1324–1327. [1952] A. P. CALDERóN and A. ZYGMUND, On the existence of with Marcinkiewicz’s proof of his multiplier theo- certain singular integrals, Acta Math. 88, 85–139. rem, and has since branched out in a variety of di- [1956a] ———, On singular integrals, Amer. J. Math. 78, rections where much interesting work has been 289–309. done. Among the things achieved are an appro- [1956b] ———, Algebras of certain singular integrals, Amer. priate Hp and BMO theory and the many proper- J. Math. 78, 310–320. ties of product (and multiparameter) singular in- [1957] ———, Singular integral operators and differential tegrals that have came to light. This is due to the equations, Amer. J. Math. 79, 901-921. work of S. Y. Chang, R. Fefferman, and Journé, to [1985] S. Y. A. CHANG and R. FEFFERMAN, Some recent de- p mention only a few of the names. velopments in Fourier analysis and H theory on Finally, I want to come to an extension of the product domains, Bull. Amer. Math. Soc. (N.S.) 12, 1–43. product theory (more precisely, the induced “mul- [1994] C. E. KENIG, Harmonic analysis techniques for sec- tiparameter analysis”) in a direction that has par- ond order elliptic boundary value problems, CBMS, ticularly interested me recently. Here the point is Regional Conference Series in Mathematics, vol. 83, that the underlying space is no longer Euclidean Amer. Math. Soc., Providence, RI. n R , but rather a nilpotent group or another ap- [1939a] J. MARCINKIEWICZ, Sur l’interpolation d’opérations, propriate generalization. On the basis of recent but C. R. Acad. Sci. Paris 208, 1272–1273. limited experience, I would hazard the guess that [1939b] ———, Sur les multiplicateurs des séries de Fourier, multiparameter analysis in this setting could well Studia Math. 8, 78–91. turn out to be of great interest in questions related [1990] Y. MEYER, Ondelettes et Opérateurs, vols. I and II, to several complex variables. A first vague hint that Hermann, Paris. [1991] Y. MEYER and R. R. COIFMAN, Ondelettes et Opéra- this may be so came with the realization that cer- teurs, vol. III, Hermann, Paris. tain boundary operators arising from the ∂-Neu- [1965] R. PALAIS, Seminar on the Atiyah-Singer index the- mann problem (in the model case corresponding orem, Ann. of Math. Study, vol. 57, Princeton Univ. to the Heisenberg group) are excellent examples Press, Princeton, NJ. of multiple-parameter singular integrals (in the [1967] R. T. SEELEY, Elliptic singular integrals, Proc. Sym- work of Müller, Ricci, and the author (1995)). A sec- pos. Pure Math., vol. 10, Amer. Math. Soc., Provi- ond indication is the description of Cauchy-Szegö dence, RI, pp. 308–315. [1970a] E. M. STEIN, Singular integrals and differentiabil- projections and solving operators for ∂b for a wide class of quadratic surfaces of higher codimension ity properties of functions, Princeton Univ. Press, in n in terms of appropriate quotients of prod- Princeton, NJ. C [1970b] , Some problems in harmonic analysis sug- ucts of Heisenberg groups (in yet unpublished ——— gested by symmetric spaces and semisimple groups, joint work with Nagel and Ricci). And even more Proc. Internat. Congr. Math., Nice I, Gauthier–Vil- suggestive are recent calculations (made jointly lars, Paris, pp. 173–189. with A. Nagel) for such operators in a number of [1982] ———, The development of square functions in the pseudo-convex domains of finite type. All this work of A. 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[1958] ———, Uniqueness in the Cauchy problem for par- tial differential equations, Amer. J. Math. 80, 16–36. [1962] ———, Existence and uniqueness theorems for sys- tems of partial differential equations, Proc. Sympos. Univ. Maryland, 1961, Gordon and Breach, New York, pp. 147–195, [1963] ———, Boundary value problems for elliptic equa- tions (Joint Soviet-American Symposium on Partial Differential Equations, Novosibirsk, 1963), Acad. Sci. USSR Siberian Branch, Moscow, pp. 303–304. 1140 NOTICES OF THE AMS VOLUME 45, NUMBER 9