Celebratio Mathematica

At the oc­ca­sion of the con­fer­ence in hon­or of Al­berto Calderón’s sev­enty-fifth birth­day, it is most fit­ting that we cel­eb­rate the math­em­at­ic­al achieve­ments for which he is so much ad­mired. Chief among these is his role in the cre­ation of the mod­ern the­ory of sin­gu­lar in­teg­rals. In that great en­ter­prise he had the good for­tune of work­ing with the math­em­atician who had para­mount in­flu­ence on his sci­entif­ic life: Ant­oni Zyg­mund, at first his teach­er, and later his ment­or and col­lab­or­at­or. So any ac­count of the mod­ern de­vel­op­ment of that the­ory must be in large part the story of the ef­forts of both Zyg­mund and Calderón. I will try to present this, fol­low­ing roughly the or­der of events as they un­fol­ded. My aim will be to ex­plore the goals that in­spired and mo­tiv­ated them, de­scribe some of their shared ac­com­plish­ments and later work, and dis­cuss briefly the wide in­flu­ence of their achieve­ments.

In the first peri­od his sci­entif­ic work, from 1923 to the middle 1930s, Zyg­mund de­voted him­self to what is now called “clas­sic­al” har­mon­ic ana­lys­is, that is, Four­i­er and tri­go­no­met­ric series of the circle, re­lated power series of the unit disk, con­jug­ate func­tions, Rieman­ni­an the­ory con­nec­ted to unique­ness, la­cun­ary series, etc. An ac­count of much of what he did, as well as the work of his con­tem­por­ar­ies and pre­de­cessors, is con­tained in his fam­ous treat­ise, “Tri­go­no­met­ric­al Series,” pub­lished in 1953. The time in which this took place may be viewed as the con­clud­ing dec­ade of the bril­liant cen­tury of clas­sic­al har­mon­ic ana­lys­is: the ap­prox­im­ately hun­dred-year span which began with Di­rich­let and Riemann, con­tin­ued with Can­tor and Le­besgue among oth­ers, and cul­min­ated with the achieve­ments of Kolmogorov, M. Riesz, and Hardy and Lit­tle­wood. It was dur­ing that last dec­ade that Zyg­mund began to turn his at­ten­tion from the one-di­men­sion­al situ­ation to prob­lems in high­er di­men­sions. At first this rep­res­en­ted merely an in­cid­ent­al in­terest, but then later he fol­lowed it with in­creas­ing ded­ic­a­tion, and even­tu­ally it was to be­come the main fo­cus of his sci­entif­ic work. I want now to de­scribe how this point of view de­veloped with Zyg­mund.

In out­line, the sub­ject of one-di­men­sion­al har­mon­ic ana­lys­is as it ex­is­ted in that peri­od can be un­der­stood in terms of what were then three closely in­ter­re­lated areas of study, and which in many ways rep­res­en­ted the cent­ral achieve­ments of the the­ory: real-vari­able the­ory, com­plex ana­lys­is, and the be­ha­vi­or of Four­i­er series. Zyg­mund’s first ex­cur­sion in­to ques­tions of high­er di­men­sions dealt with the key is­sue of real-vari­able the­ory — the av­er­aging of func­tions. The ques­tion was as fol­lows.

The clas­sic­al the­or­em of Le­besgue guar­an­teed that, for al­most every \( x \), \begin{equation} \label{eqonon} \lim_{\substack{x\in I\\ \operatorname{diam}(I)\to 0}}\frac{1}{|I|} \int_I f(y)\,dy = f (x), \end{equation} where \( I \) ranges over in­ter­vals, and when \( f \) is an in­teg­rable func­tion on the line \( \mathbb{R}^1 \). In high­er di­men­sions it is nat­ur­al to ask wheth­er a sim­il­ar res­ult held when the in­ter­vals \( I \) are re­placed by ap­pro­pri­ate gen­er­al­iz­a­tions in \( \mathbb{R}^n \). The fact that this is the case when the \( I \)’s are re­placed by balls (or more gen­er­al sets with bounded “ec­cent­ri­city”) was well known at that time. What must have piqued Zyg­mund’s in­terest in the sub­ject was his real­iz­a­tion (in 1927) that a para­dox­ic­al set con­struc­ted by Nikodym showed that the an­swer is ir­re­triev­ably false when the \( I \)’s are taken to be rect­angles (each con­tain­ing the point in ques­tion), but with ar­bit­rary ori­ent­a­tion. To this must be ad­ded the counter­example found by Saks sev­er­al years later, which showed that the de­sired ana­logue of \eqref{eqonon} still failed even if we now re­stric­ted the rect­angles to have a fixed ori­ent­a­tion (e.g., with sides par­al­lel to the axes), as long as one al­lowed \( f \) be a gen­er­al func­tion in \( L^1 \).

It was at this stage that Zyg­mund ef­fect­ively trans­formed the sub­ject at hand by an im­port­ant ad­vance: he proved that the wished-for con­clu­sion (when the sides are par­al­lel to the axes) held if \( f \) was as­sumed to be­long to \( L^p \), with \( p > 1 \). He ac­com­plished this by prov­ing an in­equal­ity for what is now known as the “strong” max­im­al func­tion. Shortly af­ter­ward in Jessen, Mar­cinkiewicz, and Zyg­mund [e8] this was re­fined to the re­quire­ment that \( f \) be­long to \( L(\log L )^{n-1} \) loc­ally.

This study of the ex­ten­sion of \eqref{eqonon} to \( \mathbb{R}^n \) was the first step taken by Zyg­mund. It is reas­on­able to guess that it re­in­forced his fas­cin­a­tion with what was then de­vel­op­ing as a long-term goal of his sci­entif­ic ef­forts, the ex­ten­sion of the cent­ral res­ults of har­mon­ic ana­lys­is to high­er di­men­sions. But a great obstacle stood in the way: it was the cru­cial role played by com­plex func­tion the­ory in the whole of one-di­men­sion­al Four­i­er ana­lys­is, and for this there was no ready sub­sti­tute.

In de­scrib­ing this spe­cial role of com­plex meth­ods we shall con­tent ourselves with high­light­ing some of the main points.

As is well known, the Hil­bert trans­form comes dir­ectly from the Cauchy in­teg­ral for­mula. We also re­call the fact that M. Riesz proved the \( L^p \) bounded­ness prop­er­ties of the Hil­bert trans­form \[ f\longmapsto H( f) = \frac{1}{\pi}\,\textit{p.v.}\,\int^{\infty}_{-\infty} f(x-y)\frac{dy}{y} \] by ap­ply­ing a con­tour in­teg­ral to \( (F)^p \), where \( F \) is the ana­lyt­ic func­tion whose bound­ary lim­it has \( f \) as its real part.

These arose in part as sub­sti­tutes for \( L^p \), when \( p \leq 1 \), and were by their very nature com­plex-func­tion-the­ory con­structs. (It should be noted, however, that for \( 1 < p < \infty \) they were es­sen­tially equi­val­ent with \( L^p \) by Riesz’s the­or­em.) The main tool used in their study was the Blasch­ke product of their zer­oes in the unit disk. Us­ing it, one could re­duce to ele­ments \( F \in H^p \) with no zer­oes, and from these one could pass to \( G = F^{p/2} \); the lat­ter was in \( H^2 \) and hence could be treated by more stand­ard \( (L^2) \) meth­ods.

This pro­ceeded by study­ing the dy­ad­ic de­com­pos­i­tion in fre­quency space and had many ap­plic­a­tions; among them was the Mar­cinkiewicz mul­ti­pli­er the­or­em. The the­ory ini­ti­ated and ex­ploited cer­tain ba­sic “square func­tions,” and these we ori­gin­ally stud­ied by com­plex-vari­able tech­niques closely re­lated to what were used in \( H^p \) spaces.

The main res­ult ob­tained here (Privalov 1923 [e2], Mar­cinkiewicz and Zyg­mund 1938 [e10], and Spen­cer 1943 [e13]) stated that for any har­mon­ic func­tion \( u (re^{i\theta}) \) in the unit disk, the fol­low­ing three prop­er­ties are equi­val­ent for al­most all bound­ary points \( e^{i\theta} \): \begin{align} & u \text{ has a nontangentive limit at } e^{i\theta},\label{eqontw}\\ & u \text{ is nontangentially bounded at } e^{i\theta},\label{eqonth}\\ & \text{the “area integral” }(S(u)(\theta))^2 = \textstyle \iint_{\Gamma (e^{i\theta})} |\nabla u (z)|^2 \,dx\,dy \text{ is finite}, \label{eqonfo} \end{align} where \( \Gamma (e^{i\theta}) \) is a nontan­gen­tial ap­proach re­gion with ver­tex \( e^{i\theta} \).

The cru­cial first step in the proof was the ap­plic­a­tion of the con­form­al map (to the unit disk) of the fam­ous saw­tooth do­main (which is pic­tured in Zyg­mund [e36], vol. 2, p. 200).

This map­ping al­lowed one to re­duce the im­plic­a­tion \eqref{eqonth}\( \Rightarrow \)\eqref{eqontw} to the spe­cial case of bounded har­mon­ic func­tions in the unit disk (Fatou’s the­or­em), and it also played a cor­res­pond­ing role in the oth­er parts of the proof.

It is iron­ic that com­plex meth­ods with their great power and suc­cess in the one-di­men­sion­al the­ory ac­tu­ally stood in the way of pro­gress to high­er di­men­sions, and ap­peared to block fur­ther pro­gress. The only way past, as Zyg­mund foresaw, re­quired a fur­ther de­vel­op­ment of “real” meth­ods. Achieve­ment of this ob­ject­ive was to take more than one gen­er­a­tion, and in some ways is not yet com­plete. The math­em­atician with whom he was to ini­ti­ate the ef­fort to real­ize much of this goal was Al­berto Calderón.

Zyg­mund spent a part of the aca­dem­ic year 1948–49 in Ar­gen­tina, and there he met Calderón. Zyg­mund brought him back to the Uni­versity of Chica­go, and soon there­after (in 1950), un­der his dir­ec­tion, Calderón ob­tained his doc­tor­al thes­is. The dis­ser­ta­tion con­tained three parts, the first about er­god­ic the­ory, which will not con­cern us here. However, it is the second and third parts that in­terest us, and these rep­res­en­ted break­throughs in the prob­lem of free­ing one­self from com­plex meth­ods and, in par­tic­u­lar, in ex­tend­ing to high­er di­men­sions some of the res­ults de­scribed in 1.1.4 above. In a gen­er­al way we can say that his ef­forts here already typ­i­fied the style of much of his later work: he be­gins by con­ceiv­ing some simple but fun­da­ment­al ideas that go to the heart of the mat­ter and then de­vel­ops and ex­ploits these in­sights with great power.

In prov­ing \eqref{eqonth}\( \Rightarrow \)\eqref{eqontw} we may as­sume that \( u \) is bounded in­side the saw­tooth do­main \( \Omega \) that arose in 1.1.4 above: this re­gion is the uni­on of ap­proach re­gions \( \Gamma (e^{i\theta}) \) (“cones”), with ver­tex \( e^{i\theta} \), for points \( e^{i\theta} \in E \), and \( E \) a closed set. Calderón in­tro­duced the aux­il­i­ary har­mon­ic func­tion \( U \), with \( U \) the Pois­son in­teg­ral of \( \chi_{cE} \), and ob­served that all the de­sired facts flowed from the dom­in­at­ing prop­er­ties of \( U \): namely, \( u \) could be split as \( u = u_1+u_2 \), where \( u_1 \) is the Pois­son in­teg­ral of a bounded func­tion (and hence has nontan­gen­tial lim­its a.e.), while by the max­im­um prin­ciple, \( |u_2| \leq c U \), and there­fore \( u_2 \) has (nontan­gen­tial) lim­its \( = 0 \) at a.e. point of \( E \).

The second idea (used to prove the im­plic­a­tion \eqref{eqontw}\( \Rightarrow \)\eqref{eqonfo}) has as its start­ing point the simple iden­tity \begin{equation} \label{eqonfi} \Delta u^2 = 2|\nabla u|^2 \end{equation} val­id for any har­mon­ic func­tion. This will be com­bined with Green’s the­or­em \[ \iint_{\Omega} (B\Delta A - A \Delta B)\, dx\,dy = \int_{\partial\Omega} \biggl(B\frac{\partial A}{\partial n}-A \frac{\partial B}{\partial n}\biggr)\, d\sigma, \] where \( A = u^2 \), and \( B \) is an­oth­er in­geni­ously chosen aux­il­i­ary func­tion de­pend­ing on the do­main \( \Omega \) only. This al­lowed him to show that \[ \iint_{\Omega} y|\nabla u|^2\,dx\,dy < \infty, \] which is an in­teg­rated ver­sion of \eqref{eqonfo}.

It may be noted that the above meth­ods and the con­clu­sions they im­ply make no use of com­plex ana­lys­is, and are very gen­er­al in nature. It is also a fact that these ideas played a sig­ni­fic­ant role in the later real-vari­able ex­ten­sion of the \( H^p \) the­ory.

Start­ing in the year 1950, a close col­lab­or­a­tion de­veloped between Calderón and Zyg­mund which las­ted al­most thirty years. While their joint re­search dealt with a num­ber of dif­fer­ent sub­jects, their pre­oc­cupy­ing in­terest and most fun­da­ment­al con­tri­bu­tions were in the area of sin­gu­lar in­teg­rals. In this con­nec­tion the first is­sue they ad­dressed was — to put the mat­ter simply — the ex­ten­sion to high­er di­men­sions of the the­ory of the Hil­bert trans­form. A real-vari­able ana­lys­is of the Hil­bert trans­form had been car­ried out by Be­sicov­itch, Titch­marsh, and Mar­cinkiewicz, and this is what needed to be ex­ten­ded to the \( \mathbb{R}^n \) set­ting.

A reas­on­able can­did­ate for con­sid­er­a­tion presen­ted it­self. It was the op­er­at­or \( f \mapsto T f \), with \begin{equation} \label{eqonsi} T(f)(x) = p.v. \int_{\mathbb{R}^n} K(y) f (x-y)\,dy, \end{equation} when \( K \) was ho­mo­gen­eous of de­gree \( - n \), sat­is­fied some reg­u­lar­ity, and in ad­di­tion the can­cel­la­tion con­di­tion \[ \int_{|x|=1} K(x)\,d\sigma(x) = 0.\]

Be­sides the Hil­bert trans­form (which is the only real ex­ample when \( n = 1 \)), high­er-di­men­sion­al ex­amples in­clude the op­er­at­ors that arise as second de­riv­at­ives of the fun­da­ment­al solu­tion op­er­at­or for the Lapla­cian, (which can be writ­ten as \( \smash{\tfrac{\partial^2}{\partial x_2 \partial x_j}}(\Delta)^{-1} \)), as well as the re­lated Riesz trans­forms, \( \smash{\tfrac{\partial}{\partial x_j}}(-\Delta)^{-1/2} \).

All of this is the sub­ject mat­ter of their his­tor­ic mem­oir, “On the ex­ist­ence of sin­gu­lar in­teg­rals,” which ap­peared in the Acta Math­em­at­ica in 1952. There is prob­ably no pa­per in the last fifty years which had such wide­spread in­flu­ence in ana­lys­is. The ideas in this work are now so well known that I will only out­line its con­tents. It can be viewed as hav­ing three parts.

First, there is the Calderón–Zyg­mund lemma, and the cor­res­pond­ing Calderón–Zyg­mund de­com­pos­i­tion. The main thrust of the former is as a sub­sti­tute for F. Riesz’s “rising sun” lemma, which had im­pli­citly played a key role in the earli­er treat­ment of the Hil­bert trans­form. Second, us­ing their de­com­pos­i­tion, they then proved the weak-type \( L^1 \), and \( L^p \), \( 1 < p < \infty \), es­tim­ates for the op­er­at­or \( T \) in \eqref{eqonsi}. As a pre­lim­in­ary step they dis­posed of the \( L^2 \) the­ory of \( T \) us­ing Plancher­el’s the­or­em. Third, they ap­plied these res­ults to the ex­amples men­tioned above, and in ad­di­tion they proved a.e. con­ver­gence for the sin­gu­lar in­teg­rals in ques­tion.

It should not de­tract from one’s great ad­mir­a­tion of this work to note two his­tor­ic­al an­om­alies con­tained in it. The first is the fact that there is no men­tion of Mar­cinkiewicz’s in­ter­pol­a­tion the­or­em, or to the pa­per in which it ap­peared (Mar­cinkiewicz 1939 [e12]), even though its ideas play a sig­ni­fic­ant role. In the Calderón–Zyg­mund pa­per, the spe­cial case that is needed is in ef­fect re­proved. The ex­plan­a­tion for this omis­sion is that Zyg­mund had simply for­got­ten about the ex­ist­ence of Mar­cinkiewicz’s note. To make amends he pub­lished (in 1956) an ac­count of Mar­cinkiewicz’s the­or­em and vari­ous gen­er­al­iz­a­tions and ex­ten­sions he had since found. In it he con­ceded that the pa­per of Mar­cinkiewicz  “… seems to have es­caped at­ten­tion and does not find al­lu­sion to it in the ex­ist­ing lit­er­at­ure.”

The second point, like the first, also in­volves some very im­port­ant work of Mar­cinkiewicz. He had been Zyg­mund’s bril­liant stu­dent and col­lab­or­at­or un­til his death at the be­gin­ning of World War II. It is a mys­tery why no ref­er­ence was made to the pa­per Mar­cinkiewicz [e12] and the mul­ti­pli­er the­or­em in it. This the­or­em had been proved by Mar­cinkiewicz in an \( n \)-di­men­sion­al form (as a product “con­sequence” of the one-di­men­sion­al form). As an ap­plic­a­tion, the \( L^p \) in­equal­it­ies for the op­er­at­ors \( \smash{\tfrac{\partial^2}{\partial x_i \partial x_j}}(\Delta^{-1}) \) were ob­tained; 1 these he had proved at the be­hest of Schaud­er.

As has already been in­dic­ated, the \( n \)-di­men­sion­al sin­gu­lar in­teg­rals had its main mo­tiv­a­tion in the the­ory of par­tial dif­fer­en­tial equa­tions. In their fur­ther work, Calderón and Zyg­mund pur­sued this con­nec­tion, fol­low­ing the trail that had been ex­plored earli­er by Giraud, Tricomi, and Mih­lin. Start­ing from those ideas (in par­tic­u­lar the no­tion of “sym­bol”) they de­veloped their ver­sion of the sym­bol­ic cal­cu­lus of “vari­able-coef­fi­cient” ana­logues of the sin­gu­lar in­teg­ral op­er­at­ors. To de­scribe these res­ults one con­siders an ex­ten­sion of the class of op­er­at­ors arising in \eqref{eqonsi}, namely of the form, \begin{equation} \label{eqonse} T ( f ) ( x ) = a_0(x)f (x) + p.v. \int_{\mathbb{R}^n} K (x, y ) f (x - y)\,dy, \end{equation} where \( K (x, y) \) is for each \( x \) a sin­gu­lar in­teg­ral ker­nel of the type \eqref{eqonsi} in \( y \), which de­pends smoothly and boundedly on \( x \); also \( a_0(x) \) is a smooth and bounded func­tion.

To each op­er­at­or \( T \) of this kind there cor­res­ponds its sym­bol \( a(x,\xi) \), defined by \begin{equation} \label{eqonei} a(x, \xi) = a_0 (x) + \widehat{K} ( x, \xi), \end{equation} where \( \widehat{K} ( x, \xi) \) de­notes the Four­i­er trans­form of \( K (x, y) \) in the \( y \)-vari­able. Thus \( a(x, \xi) \) is ho­mo­gen­eous of de­gree 0 in the \( \xi \) vari­able (re­flect­ing the ho­mo­gen­eity of \( K (x, y) \) of de­gree \( -n \) in \( y \)); and it de­pends smoothly and boundedly on \( x \). Con­versely to each func­tion \( a(x, \xi) \) of this kind there ex­ists a (unique) op­er­at­or \eqref{eqonse} for which \eqref{eqonei} holds. One says that \( a \) is the sym­bol of \( T \) and also writes \( T = T_a \).

The ba­sic prop­er­ties that were proved were, first, the reg­u­lar­ity prop­er­ties \begin{equation} \label{eqonni} T_a: L^p_k \rightarrow L^p_k, \end{equation} where \( L^p_k \) are the usu­al So­bolev spaces, with \( 1 < p < \infty \).

Also the ba­sic facts of sym­bol­ic ma­nip­u­la­tions \begin{align} T_{a_1}\cdot T_{a_2}&= T_{a_1\cdot\mkern1mu a_2} + \textit{Error} \label{eqononze}\\ (\smash{T_a})^{\star}&= T_{\bar{a}} + \textit{Error} \label{eqononon} \end{align} where the \( \textit{Error} \) op­er­at­ors are smooth­ing of or­der 1, in the sense that \[ \textit{Error}: L^p_k \rightarrow L^p_{k+1} .\]

A con­sequence of the sym­bol­ic cal­cu­lus is the fac­tor­iz­ab­il­ity of any lin­ear par­tial dif­fer­en­tial op­er­at­or \( L \) of or­der \( m \), \[ L=\sum_{|\alpha|\leq\mkern1mu m} a_{\alpha} (x) \biggl(\frac{\partial}{\partial x}\biggr)^{\mkern-3mu \alpha}, \] where the coef­fi­cients \( a_{\alpha} \) are as­sumed to be smooth and bounded. One can write \begin{equation} \label{eqonontw} L=T_a (-\Delta)^{m/2} + (\textit{Error})^{\prime} \end{equation} for an ap­pro­pri­ate sym­bol \( a \), where the op­er­at­or \( (\textit{Error})^{\prime} \) refers to an op­er­at­or that maps \( L^p_k \rightarrow L^p_{k-m+1} \) for \( k\geq m - 1 \). It seemed clear that this sym­bol­ic cal­cu­lus should have wide ap­plic­a­tions to the the­ory of par­tial dif­fer­en­tial op­er­at­ors and to oth­er parts of ana­lys­is. This was soon to be borne out.

At this stage of my nar­rat­ive I would like to share some per­son­al re­min­is­cences. I had been a stu­dent of Zyg­mund at Uni­versity of Chica­go, and in 1956 at his sug­ges­tion I took my first teach­ing po­s­i­tion at MIT, where Calderón was at that time. I had met Calderón sev­er­al years earli­er when he came to Chica­go to speak about the “meth­od of ro­ta­tions” in Zyg­mund’s sem­in­ar. I still re­mem­ber my feel­ings when I saw him there; these first im­pres­sions have not changed much over the years: I was struck by the sense of his un­der­stated el­eg­ance, his re­serve, and quiet cha­risma.

At MIT we would meet quite of­ten and over time an easy con­ver­sa­tion­al re­la­tion­ship de­veloped between us. I do re­call that we, in the small group who were in­ter­ested in sin­gu­lar in­teg­rals then, felt a cer­tain sep­ar­ate­ness from the lar­ger com­munity of ana­lysts — not that this isol­a­tion was self-im­posed, but more be­cause our sub­ject mat­ter was seen by our col­leagues as some­what ar­cane, rar­efied, and pos­sibly not very rel­ev­ant. However, this did change, and a fuller ac­cept­ance even­tu­ally came. I want to re­late now how this oc­curred.

Start­ing from the cal­cu­lus of sin­gu­lar in­teg­ral op­er­at­ors that he had worked out with Zyg­mund, Calderón ob­tained a num­ber of im­port­ant ap­plic­a­tions to hy­per­bol­ic and el­lipt­ic equa­tions. His most dra­mat­ic achieve­ment was in the unique­ness of the Cauchy prob­lem (Calderón [7]). There he suc­ceeded in a broad and de­cis­ive ex­ten­sion of the res­ults of Holmgren (for the case of ana­lyt­ic coef­fi­cients), and Car­le­man (in the case of two di­men­sions). Calderón’s the­or­em can be for­mu­lated as fol­lows.

Sup­pose \( u \) is a func­tion which in the neigh­bor­hood of the ori­gin in \( \mathbb{R}^n \) sat­is­fies the equa­tion of \( m \)-th or­der: \begin{equation} \label{eqononth} \frac{\partial^m u}{\partial x^m_n} = \sum_{\alpha} a_{\alpha}(x) \frac{\partial^{\alpha}}{\partial x^{\alpha}}, \end{equation} where the sum­ma­tion is taken over all in­dices \( \alpha ( \alpha_1, \ldots, \alpha_n) \) with \( |\alpha| \leq n \), and \( \alpha_n < m \). We also as­sume that \( u \) sat­is­fies the null ini­tial Cauchy con­di­tions \begin{equation} \label{eqononfo} \frac{\partial^j u (x)}{\partial x^j_n}\biggr|_{x_n = 0}=0, \quad j=0,\ldots, m-1. \end{equation}

Be­sides \eqref{eqononth} and \eqref{eqononfo}, it suf­fices that the coef­fi­cients \( a_{\alpha} \) be­long to \( C^{1+\epsilon} \), that the char­ac­ter­ist­ics are simple, and \( n \neq 3 \), or \( m \leq 3 \). Un­der these hy­po­theses \( u \) van­ishes identic­ally in a neigh­bor­hood of the ori­gin.

Calderón’s ap­proach was to re­duce mat­ters to a key “pseudo-dif­fer­en­tial in­equal­ity” (in a ter­min­o­logy that was used later). This in­equal­ity is com­plic­ated, but some­what re­min­is­cent of a dif­fer­en­tial in­equal­ity that Car­le­man had used in two di­men­sions. The es­sence of it is that \begin{equation} \label{eqononfi} \int^a_0 \phi_k\biggl\|\frac{\partial u}{\partial t}+ (P+i Q)(-\Delta)^{1/2} u \biggr\|^2 \,dt \leq c \int^a_0 \phi_k \|u\|^2\, dt, \end{equation} where \( u(0) = 0 \) im­plies \( u\equiv 0 \), if \eqref{eqononfi} holds for \( k\rightarrow \infty \).

Here \( P \) and \( Q \) are sin­gu­lar in­teg­ral op­er­at­ors of the type \eqref{eqonse}, with real sym­bols and \( P \) is in­vert­ible; we have writ­ten \( t = x_n \), and the norms are \( L^2 \) norms taken with re­spect to the vari­ables \( x_1, \ldots \), \( x_{n-1} \). The func­tions \( \phi_k \) are meant to be­have like \( t^{-k} \), which when \( k\rightarrow \infty \) em­phas­izes the ef­fect tak­ing place near \( t = 0 \). In fact, in \eqref{eqononth} we can take \( \phi_k(t) = (t + 1/k)^{-k} \).

The proof of as­ser­tions like \eqref{eqononfi} is easi­er in the spe­cial case when all the op­er­at­ors com­mute; their gen­er­al form is es­tab­lished by us­ing the ba­sic facts \eqref{eqononze} and \eqref{eqononon} of the cal­cu­lus.

The pa­per of Calderón was, at first, not well re­ceived. In fact, I learned from him that it was re­jec­ted when sub­mit­ted to what was then the lead­ing journ­al in par­tial dif­fer­en­tial equa­tions, Com­ment­ar­ies of [i.e., Com­mu­nic­a­tions on] Pure and Ap­plied Math­em­at­ics.

At about that time, be­cause of the ap­plic­ab­il­ity of sin­gu­lar in­teg­rals to par­tial dif­fer­en­tial equa­tions, Calderón be­came in­ter­ested in for­mu­lat­ing the facts about sin­gu­lar in­teg­rals in the set­ting of man­i­folds. This re­quired the ana­lys­is of the ef­fect co­ordin­ate changes had on such op­er­at­ors. A hint that the prob­lem was tract­able came from the ob­ser­va­tion that the class of ker­nels, \( K ( y ) \), of the type arising in \eqref{eqonsi}, was in­vari­ant un­der lin­ear (in­vert­ible) changes of vari­ables \( y\mapsto L(y) \). (The fact that \( K ( L ( y ) ) \) sat­is­fied the same reg­u­lar­ity and ho­mo­gen­eity that \( K ( y ) \) did, was im­me­di­ate; that the can­cel­la­tion prop­erty also holds for \( K (L(y)) \) is a little less ob­vi­ous.)

R. See­ley was Calderón’s stu­dent at that time, and he dealt with this prob­lem in his thes­is (see See­ley [e18]). Sup­pose \( x\mapsto \psi( x ) \) is a loc­al dif­feo­morph­ism, then the res­ult was that mod­ulo er­ror terms (which are “smooth­ing” of one de­gree) the op­er­at­or \eqref{eqonse} is trans­formed in­to an­oth­er op­er­at­or of the same kind, \[ T^{\prime} ( f)(x) \cdot a^{\prime}_0(x)= f (x) + p.v. \int K^{\prime} (x, y)f (x- y)\,dy, \] but now \( a^{\prime}_0 ( x ) = a_0(\psi(x)) \), and \( K^{\prime} (x, y) = K^{\prime}(\psi(x), L_x(y)) \), where \( L_x \) is the lin­ear trans­form­a­tion giv­en by the Jac­obi­an mat­rix \( \partial\psi(x)/\partial x \). On the level of sym­bols this meant that the new sym­bol \( a^{\prime} \) was de­term­ined by the old sym­bol ac­cord­ing to the for­mula \[ a^{\prime} (x, \xi) = a(\psi(x), L^{\prime}_x(\xi)), \] with \( L^{\prime}_x \) the trans­pose-in­verse of \( L_x \). Hence the sym­bol is ac­tu­ally a func­tion on the co­tan­gent space of the man­i­fold.

The res­ult of See­ley was not only highly sat­is­fact­ory as to its con­clu­sions, but it was also very timely in terms of events that were about to take place. Fol­low­ing an in­ter­ven­tion by Gel­fand [e19], in­terest grew in cal­cu­lat­ing the “in­dex” of an el­lipt­ic op­er­at­or on a man­i­fold. This in­dex is the dif­fer­ence of the di­men­sion of the null-space and the codi­men­sion of the range of the op­er­at­or, and is an in­vari­ant un­der de­form­a­tions. The prob­lem of de­term­in­ing it was con­nec­ted with a num­ber of in­ter­est­ing is­sues in geo­metry and to­po­logy. The res­ult of the “See­ley cal­cu­lus” proved quite use­ful in this con­text: the proofs pro­ceeded by ap­pro­pri­ate de­form­a­tions and mat­ters were fa­cil­it­ated if these could be car­ried out in the more flex­ible con­text of “gen­er­al” sym­bols, in­stead of re­strict­ing at­ten­tion to the poly­no­mi­al sym­bols com­ing from dif­fer­en­tial op­er­at­ors. A con­tem­por­an­eous ac­count of this de­vel­op­ment (dur­ing the peri­od 1961–64), may be found in the notes of the sem­in­ar on the Atiyah–Sing­er in­dex the­or­em (see [e23]); for an his­tor­ic­al sur­vey of some of the back­ground, see also [e26].

With the activ­ity sur­round­ing the in­dex the­or­em, it sud­denly seemed as if every­one was in­ter­ested in the al­gebra of sin­gu­lar in­teg­ral op­er­at­ors. However, one fur­ther step was needed to make this a house­hold tool for ana­lysts: it re­quired a change of point of view. Even though this change of per­spect­ive was not ma­jor, it was sig­ni­fic­ant psy­cho­lo­gic­ally and meth­od­o­lo­gic­ally, since it al­lowed one to think more simply about cer­tain as­pects of the sub­ject and be­cause it sug­ges­ted vari­ous ex­ten­sions.

The idea was merely to change the role of the defin­i­tions of the op­er­at­ors, from \eqref{eqonse} for sin­gu­lar in­teg­rals to pseudo-dif­fer­en­tial op­er­at­ors \begin{equation} \label{eqononsi} T_a(f)(x) = \int_{\mathbb{R}^n} a(x,\xi) \hat{f} (\xi) e^{2 \pi i x \cdot \xi} \, d\xi, \end{equation} with sym­bol \( a \). (Here \( \hat{f} \) is the Four­i­er trans­form, \[ \hat{f}(\xi) = \int_{\mathbb{R}^n} e^{-2\pi i x \cdot \xi} f (x) \, dx .) \]

Al­though the two op­er­at­ors are identic­al (when \( a ( x, \xi) = a_0(x)+\hat{K}(x, \xi) \)), the ad­vant­age lies in the em­phas­is in \eqref{eqononsi} on the \( L^2 \) the­ory and Four­i­er trans­form, and the wider class of op­er­at­ors that can be con­sidered, in par­tic­u­lar, in­clud­ing dif­fer­en­tial op­er­at­ors. The for­mu­la­tion \eqref{eqononsi} al­lows one to deal more sys­tem­at­ic­ally with the com­pos­i­tion of such op­er­at­ors and in­cor­por­ate the lower-or­der terms in the cal­cu­lus.

To do this, one might ad­opt a wider class of sym­bols of “ho­mo­gen­eous-type”: roughly speak­ing, \( a(x, \xi) \) be­longs to this class (and is of or­der \( m \)) if \( a(x, \xi) \) is for large \( \xi \), asymp­tot­ic­ally the sum of terms ho­mo­gen­eous in \( \xi \) of de­grees \( m - j \), with \( j = 0,1,2,\dots \).

The change in point of view de­scribed above came in­to its full flower­ing with the pa­pers of Kohn and Niren­berg [e25] and Hörmander [e22], (after some work by Un­ter­ber­ger and Bokobza [e21] and See­ley [e24]). It is in this way that sin­gu­lar in­teg­rals were sub­sumed by pseudo-dif­fer­en­tial op­er­at­ors. Des­pite this, sin­gu­lar in­teg­rals, with their for­mu­la­tion in terms of ker­nels, still re­tained their primacy when treat­ing real-vari­able is­sues, is­sues such as \( L^p \) or \( L^1 \) es­tim­ates (and even for some of the more in­tric­ate parts of the \( L^2 \) the­ory). The cent­ral role of the ker­nel rep­res­ent­a­tion of these op­er­at­ors be­came, if any­thing, more pro­nounced in the next twenty years.

In the years 1957–58 there ap­peared the fun­da­ment­al work of De­Giorgi and Nash, deal­ing with smooth­ness of solu­tions of par­tial dif­fer­en­tial equa­tions, with min­im­al as­sump­tions of reg­u­lar­ity of the coef­fi­cients. One of the most strik­ing res­ults — for el­lipt­ic equa­tions — was that any solu­tion \( u \) of the equa­tion \begin{equation} \label{eqononse} L(u) \equiv \sum_{i,j} \frac{\partial}{\partial x_i} \biggl(a_{ij}(x) \frac{\partial u}{\partial x_{\mkern-2mu j}}\biggr) = 0 \end{equation} in an open ball sat­is­fies an a pri­ori in­teri­or reg­u­lar­ity as long as the coef­fi­cients are uni­formly el­lipt­ic, i.e., \begin{equation} \label{eqononei} c_1|\xi|^2 \leq \sum_{i,j} a_{ij}(x) \xi_i \xi_j \leq c_2 |\xi|^2. \end{equation} In fact, no reg­u­lar­ity is as­sumed about the \( a_{ij} \) ex­cept for the bounded­ness im­pli­cit in \eqref{eqononei}, and the res­ult is that \( u \) is Hölder con­tinu­ous with an ex­po­nent de­pend­ing only on the con­stants \( c_1 \) and \( c_2 \).

Calderón was in­trigued by this res­ult. He ini­tially ex­pec­ted, as he told me, that one could ob­tain such con­clu­sions and oth­ers by re­fin­ing the cal­cu­lus of sin­gu­lar in­teg­ral op­er­at­ors \eqref{eqononfi}, mak­ing min­im­al as­sump­tions of smooth­ness on \( a_0(x) \) and \( K ( x, y ) \). While this was plaus­ible — and in­deed in his work with Zyg­mund they had already de­rived prop­er­ties of the op­er­at­ors of such op­er­at­ors \eqref{eqonse} and their cal­cu­lus when the de­pend­ence on \( x \) was, for ex­ample, of class \( C^{1+\epsilon} \) — this hope was not to be real­ized. Fur­ther un­der­stand­ing about these things could be achieved only if one were ready to look in a some­what dif­fer­ent dir­ec­tion. I want to re­late now how this came about.

The first ma­jor in­sight arose in an­swer to the fol­low­ing:

Ques­tion: Sup­pose \( M_A \) is the op­er­at­or of mul­ti­plic­a­tion (by the func­tion \( A \)), \[ M_A: f \mapsto A \cdot F. \] What are the least reg­u­lar­ity as­sump­tions on \( A \) needed to guar­an­tee that the com­mut­at­or \( [T, M_A] \) is bounded on \( L^2 \), whenev­er \( T \) is of or­der 1?

In \( \mathbb{R}^1 \), if \( T \) hap­pens to be \( \tfrac{d}{dx} \) then \( [T, M_A] = M_{A^{\prime}} \), and so the con­di­tion is ex­actly \begin{equation} \label{eqononni} A^{\prime}\in L^{\infty} (\mathbb{R}^1). \end{equation}

In a re­mark­able pa­per, Calderón [1965] [10] showed that this is also the case more gen­er­ally. The key case, con­tain­ing the es­sence of the res­ult he proved, arose when \( T = H \tfrac{d}{dx} \) with \( H \) the Hil­bert trans­form. Then \( T \) is ac­tu­ally \( \bigl|\tfrac{d}{dx}\bigr| \), its sym­bol is \( 2\pi|\xi| \), and \( [T, M_A] \) is the “com­mut­at­or” \( C_1 \), \begin{equation} \label{eqontwze} C_1 (f)(x) = \frac{1}{\pi}\, \textit{p.v.} \int^{\infty}_{-\infty} \frac{A(x)-A(y)}{(x-y)^2} f(y) \, dy. \end{equation} Calderón proved that \( f \mapsto C_1(f) \) is bounded on \( L^2(\mathbb{R}) \) if \eqref{eqononni} held.

There are two cru­cial points that I want to em­phas­ize about the proof of this the­or­em. The first is the re­duc­tion of the bounded­ness of the bi­lin­ear term \( ( f, g) \rightarrow \langle C_1 ( f), g\rangle \) to a cor­res­pond­ing prop­erty of a par­tic­u­lar bi­lin­ear map­ping, \( (F, G)\rightarrow B (F, G) \), defined for (ap­pro­pri­ate) holo­morph­ic func­tions in the up­per half-plane \( \{z = x + iy\mid y > 0 \} \) by \begin{equation} \label{eqontwon} B(F,G)(x) = i \int^{\infty}_{0} F^{\prime} ( x + iy)G(x + iy)\,dy. \end{equation}

This \( B \) is a prim­it­ive ver­sion of a “para-product” (in this con­text, the jus­ti­fic­a­tion for this ter­min­o­logy is the fact that \[ F(x) \cdot G(x) = B (F, G)(x) + B(G, F)(x)) .\] It is, in fact, not too dif­fi­cult to see that \( f\mapsto C_1(f) \) is bounded on \( L^2(\mathbb{R}^1) \) if \( B \) sat­is­fies the Hardy-space es­tim­ate \begin{equation} \label{eqontwtw} \|B(F,G) \|_{H^1}\leq c\, \|F\|_{H^2}\|\,G \|_{H^2}. \end{equation}

The second ma­jor point in the proof is the as­ser­tion need to es­tab­lish \eqref{eqontwtw}. It is the con­verse part of the equi­val­ence \begin{equation} \label{eqontwth} \|S(F)\|_{L^1}\sim \| F\|_{H^1}, \end{equation} for the area in­teg­ral \( S \) (which ap­peared in \eqref{eqonfo}). The the­or­em of Calderón, and in par­tic­u­lar the meth­ods he used, in­spired a num­ber of sig­ni­fic­ant de­vel­op­ments in ana­lys­is. The first came be­cause of the en­ig­mat­ic nature of the proof: a deep \( L^2 \) the­or­em had been es­tab­lished by meth­ods (us­ing com­plex func­tion the­ory) that did not seem sus­cept­ible of a gen­er­al frame­work. In ad­di­tion, the non­trans­la­tion-in­vari­ance char­ac­ter of the op­er­at­or \( C_1 \) made Plancher­el’s the­or­em of no use here. It seemed likely that a meth­od of “al­most-or­tho­gon­al” de­com­pos­i­tion — pi­on­eered by Cot­lar for the clas­sic­al Hil­bert trans­form — might well suc­ceed in this case also. This lead to a reex­am­in­a­tion of Cot­lar’s lemma (which had ori­gin­ally ap­plied to the case of com­mut­ing self-ad­joint op­er­at­ors). A gen­er­al for­mu­la­tion was ob­tained as fol­lows: Sup­pose that on a Hil­bert space, \( T = \sum T_j \), then \begin{equation} \label{eqontwfo} \|T\|^2\leq\sum_k \sup_j \bigl\{\|T_jT^{\star}_{\smash{j+k}}\|+\|T^{\star}_j T_{j+k}\|\bigr\}. \end{equation}

Des­pite the suc­cess in prov­ing \eqref{eqontwfo}, this alone was not enough to re­prove Calderón’s the­or­em. As un­der­stood later, the miss­ing ele­ment was a cer­tain can­cel­la­tion prop­erty. Nev­er­the­less, the gen­er­al form of Cot­lar’s lemma, \eqref{eqontwfo}, quickly led to a num­ber of highly use­ful ap­plic­a­tions, such as sin­gu­lar in­teg­rals on nil­po­tent group (in­ter­twin­ing op­er­at­ors), pseudo-dif­fer­en­tial op­er­at­ors, etc.

Calderón’s the­or­em also gave ad­ded im­petus to the fur­ther evol­u­tion of the real-vari­able \( H^p \) the­ory. This came about be­cause the equi­val­ence \eqref{eqontwth} and its gen­er­al­iz­a­tions al­lowed one to show that the usu­al sin­gu­lar in­teg­rals \eqref{eqonsi} were also bounded on the Hardy space \( H^1 \) (and in fact on all \( H^p \), \( 0 < p < \infty \)). Taken to­geth­er with earli­er de­vel­op­ments and some later ideas, the real-vari­able \( H^p \) the­ory reached its full-flower­ing a few years later. One owes this long-term achieve­ment to the work of G. Weiss, C. Fef­fer­man, Burk­hold­er, Gundy, and Coi­f­man, among oth­ers.

It be­came clear after a time that un­der­stand­ing the com­mut­at­or \( C_1 \) (and its “high­er” ana­logues) was in fact con­nec­ted with an old prob­lem that had been an ul­ti­mate, but un­reached, goal of the clas­sic­al the­ory of sin­gu­lar in­teg­rals: the bounded­ness be­ha­vi­or of the Cauchy-in­teg­ral taken over curves with min­im­al reg­u­lar­ity. The ques­tion in­volved can be for­mu­lated as fol­lows: in the com­plex plane, for a con­tour \( \gamma \) and a func­tion \( f \) defined on it, form the Cauchy in­teg­ral \[ F(z) = \frac{1}{2\pi i} \int_{\gamma} \frac{f(\zeta)}{\zeta-z}\,d\zeta, \] with \( F \) holo­morph­ic out­side \( \gamma \). Define the map­ping \( f\rightarrow \mathcal{C} ( f ) \) by \( \mathcal{C} ( f ) = F_+ + F_{-} \), where \( F_+ \) are the lim­its of \( F \) on \( \gamma \) ap­proached from either side. When \( \gamma \) is the unit circle, or real ax­is, then \( f\rightarrow C ( f ) \) is es­sen­tially the Hil­bert trans­form. Also when \( \gamma \) has some reg­u­lar­ity (e.g., \( \gamma \) is in \( C^{1+\epsilon} \)), the ex­pec­ted prop­er­ties of \( \mathcal{C} \) (i.e., \( L^2 \), \( L^p \) bounded­ness, etc.) are eas­ily ob­tained from the Hil­bert trans­form. The prob­lem was what happened when, say, \( \gamma \) was less reg­u­lar, and here the main is­sue that presen­ted it­self was the be­ha­vi­or of the Cauchy in­teg­ral when \( \gamma \) was a Lipschitz curve.

If \( \gamma \) is a Lipschitz graph in the plane, \[ \gamma = \{ x + iA(x), x\in \mathbb{R}\} ,\] with \( A^{\prime}\in L^{\infty} \), then up to a mul­ti­plic­at­ive con­stant, \begin{equation} \label{eqontwfi} \mathcal{C}(f)(x)= \textit{p.v.}\,\int^{\infty}_{-\infty} \frac{1}{x-y+i(A(x)-A(y))}\, f(y)b(y) \,dy, \end{equation} where \( b = 1 + iA^{\prime} \). The form­al ex­pan­sion \begin{equation} \label{eqontwsi} \frac{1}{x-y+iA(A(y))}=\frac{1}{x-y}\cdot \sum^{\infty}_{k=0}(-i)^k \biggl( \frac{A(x) - A(y)}{x-y}\biggr)^k\end{equation} then makes clear that the fate of Cauchy in­teg­ral \( \mathcal{C} \) is in­ex­tric­ably bound up with that of the com­mut­at­or \( C_1 \) and its high­er ana­logues \( C_k \) giv­en by \[ C_k f(x)=\frac{\textit{p.v.}}{\pi}\int^{\infty}_{-\infty} \biggl(\frac{A(x)-A(y)}{x-y}\biggr)^k \frac{f(y)}{x-y}\,dy. \] The fur­ther study of this prob­lem was be­gun by Coi­f­man and Mey­er in the con­text of the com­mut­at­ors \( C_k \), but the first break­through for the Cauchy in­teg­ral was ob­tained by Calderón [10] (us­ing dif­fer­ent meth­ods), in the case the norm \( \|A^{\prime}\|_{L^{\infty}} \) was small. His proof made de­cis­ive use of the com­plex-ana­lyt­ic set­ting of the prob­lem. It pro­ceeded by an in­geni­ous de­form­a­tion ar­gu­ment, lead­ing to a non­lin­ear dif­fer­en­tial in­equal­ity; this non­lin­ear­ity ac­coun­ted for the lim­it­a­tion of small norm for \( A^{\prime} \) in the con­clu­sion. But even with this lim­it­a­tion, the con­clu­sion ob­tained was stun­ning.

The crown­ing res­ult came in 1982, when Coi­f­man and Mey­er hav­ing en­lis­ted the help of McIn­tosh, and re­ly­ing on some of their earli­er ideas, to­geth­er proved the de­sired res­ult without lim­it­a­tion on the size of \( \|A^{\prime}\|_{L^{\infty}} \) The meth­od (in Coi­f­man, McIn­tosh, and Mey­er [e42]) was op­er­at­or-the­or­et­ic, em­phas­iz­ing the mul­ti­lin­ear as­pects of the \( C_k \), and in dis­tinc­tion to Calderón’s ap­proach was not based on com­plex-ana­lyt­ic tech­niques.

The ma­jor achieve­ment rep­res­en­ted by the the­ory of the Cauchy in­teg­ral led to a host of oth­er res­ults, either by a rather dir­ect ex­ploit­a­tion of the con­clu­sions in­volved, or by ex­ten­sions of the tech­niques that were used. I will briefly dis­cuss two of these de­vel­op­ments.

The first was a com­plete ana­lys­is of the \( L^2 \) the­ory of “Calderón–Zyg­mund op­er­at­ors.” By this ter­min­o­logy is meant op­er­at­ors of the form \begin{equation} \label{eqontwse} T(f)(x) = \int_{\mathbb{R}^n} K (x, y) f (y)\,dy \end{equation} ini­tially defined for test-func­tion \( f \in \mathcal{S} \), with the ker­nel \( K \) a dis­tri­bu­tion. It is as­sumed that away from the di­ag­on­al \( K \) agrees with a func­tion that sat­is­fies fa­mil­i­ar es­tim­ates such as \begin{equation} \label{eqontwei} |K(x,y)| \leq A|x-y|^{-n},\quad |\nabla_{x,y}K(x,y)| \leq A|x-y|^{-n-1}. \end{equation} The main ques­tion that arises (and is sug­ges­ted by the com­mut­at­ors \( C_k \)) is what are the ad­di­tion­al con­di­tions that guar­an­tee that \( T \) is a bounded op­er­at­or on \( L^2(\mathbb{R}^n) \) to it­self. The an­swer, found by Dav­id and Journé [e44] is highly sat­is­fy­ing: A cer­tain “weak bounded­ness” prop­erty, namely \( |(Tf,g)|\leq Ar^n \) wherever \( f \) and \( g \) are suit­ably nor­mal­ized bump func­tions, sup­por­ted in a ball of ra­di­us \( r \); also that both \( T(1) \) and \( T^{\star}(1) \) be­long to BMO. These con­di­tions are eas­ily seen to be also ne­ces­sary.

The ar­gu­ment giv­ing the suf­fi­ciency pro­ceeded in de­com­pos­ing the op­er­at­or in­to a sum, \( T = T_1 + T_2 \), where for \( T_1 \) the ad­di­tion­al can­cel­la­tion con­di­tion \( T_1(1) = \smash{T_1^{\star}(1)} = 0 \) held. As a con­sequence the meth­od of al­most-or­tho­gon­al de­com­pos­i­tion, \eqref{eqontwfo}, could be suc­cess­fully ap­plied to \( T_1 \). The op­er­at­or \( T_2 \) (for which \( L^2 \) bounded­ness was proved dif­fer­ently) was of para-product type, chosen so as to guar­an­tee the needed can­cel­la­tion prop­erty.

The con­di­tions of the Dav­id–Journé the­or­em, while ap­ply­ing in prin­ciple to the Cauchy in­teg­ral, are not eas­ily veri­fied in that case. However, a re­fine­ment (the “\( T(b) \) the­or­em”), with \( b= 1 + iA^{\prime} \), was found by Dav­id, Journé, and Semmes, and this does the job needed.

A second area that was sub­stan­tially in­flu­enced by the work of the Cauchy in­teg­ral was that of second-or­der el­lipt­ic equa­tions in the con­text of min­im­al reg­u­lar­ity. Side by side with the con­sid­er­a­tion of the di­ver­gence-form op­er­at­or \( L \) in \eqref{eqononse} (where the em­phas­is is on the min­im­al smooth­ness of the coef­fi­cients), one was led to study also the po­ten­tial the­ory of the Lapla­cian (where now the em­phas­is was on the min­im­al smooth­ness of the bound­ary). In the lat­ter set­ting, a nat­ur­al as­sump­tion to make was that the bound­ary is Lipschit­zi­an. In fact, by an ap­pro­pri­ate Lipschitz map­pings of do­mains, the situ­ation of the Lapla­cian in a Lipschitz do­main could be real­ized as a spe­cial case of the di­ver­gence-form op­er­at­or \eqref{eqononse}, where the do­main was smooth (say, a half-space).

The de­cis­ive ap­plic­a­tion of the Cauchy in­teg­ral to the po­ten­tial the­ory of the Lapla­cian in a Lipschitz do­main was in the study of the bounded­ness of the double lay­er po­ten­tial (and the nor­mal de­riv­at­ive of the single lay­er po­ten­tial). These are \( n - 1 \) di­men­sion­al op­er­at­ors, and they can be real­ized by ap­ply­ing the “meth­od of ro­ta­tions” to the one-di­men­sion­al op­er­at­or \eqref{eqontwfi}. One should men­tion that an­oth­er sig­ni­fic­ant as­pect of Lapla­cians on Lipschitz do­mains was the un­der­stand­ing brought to light by Dahl­berg of the nature of har­mon­ic meas­ure and its re­la­tion to \( A_{\mkern-2mu p} \) weights. These two strands, ini­tially in­de­pend­ent, have been linked to­geth­er, and with the aid of fur­ther ideas a rich the­ory has de­veloped, ow­ing to the ad­ded con­tri­bu­tions of Jer­is­on, Kenig, and oth­ers.

Fi­nally, we re­turn to the point where much of this began — the di­ver­gence-form equa­tion \eqref{eqononse}. Here the ana­lys­is grow­ing out of the Cauchy in­teg­ral also had its ef­fect. Here I will men­tion only the use­ful­ness of mul­ti­lin­ear ana­lys­is in the study of the case of “ra­di­ally in­de­pend­ent” coef­fi­cients; also in the work on the Kato prob­lem — the de­term­in­a­tion of the do­main of \( \sqrt{L} \) in the case the coef­fi­cients can be com­plex-val­ued.

The mod­ern the­ory of sin­gu­lar in­teg­rals, de­veloped and nur­tured by Calderón and Zyg­mund, has proved to be a very fruit­ful part of ana­lys­is. Bey­ond the achieve­ments de­scribed above, a num­ber of oth­er dir­ec­tions have been cul­tiv­ated with great suc­cess, with work be­ing vig­or­ously pur­sued up to this time; in ad­di­tion, here sev­er­al in­ter­est­ing open ques­tions present them­selves. I want to al­lude briefly to three of these dir­ec­tions and men­tion some of the prob­lems that arise.

As is well known, this meth­od con­sists of de­com­pos­ing an in­teg­rable func­tion in­to its “good” and “bad” parts; the lat­ter be­ing sup­por­ted on a dis­joint uni­on of cubes, and hav­ing mean-value zero on each cube. To­geth­er with an \( L^2 \) bound and es­tim­ates of the type \eqref{eqontwei}, this leads ul­ti­mately to the weak-type \( (1{,}1) \) res­ults, etc.

It was re­cog­nized quite early that this meth­od al­lowed sub­stan­tial ex­ten­sion. The gen­er­al­iz­a­tions that were un­der­taken were not so much pur­sued for their own sake but rather were mo­tiv­ated in each case by the in­terest of the ap­plic­a­tions. Roughly, in or­der of ap­pear­ance, here were some of the main in­stances:

1. The heat equa­tion and oth­er para­bol­ic equa­tions. This began with the work of F. Jones [e20] for the heat equa­tion, with the Calderón–Zyg­mund cubes re­placed by rect­angles whose di­men­sions re­flec­ted the ho­mo­gen­eity of the heat op­er­at­or. The the­ory was ex­ten­ded by Fabes and Rivière to en­com­pass more gen­er­al sin­gu­lar in­teg­rals re­spect­ing “non­iso­trop­ic” ho­mo­gen­eity in Eu­c­lidean spaces.

2. Sym­met­ric spaces and semisimple Lie groups. To be suc­cinct, the cru­cial point was the ex­ten­sion to the set­ting of nil­po­tent Lie groups with dila­tions (“ho­mo­gen­eous groups”), mo­tiv­ated by prob­lems con­nec­ted with Pois­son in­teg­rals on sym­met­ric spaces, and con­struc­tion of in­ter­twin­ing op­er­at­ors.

3. Sev­er­al com­plex vari­ables and subel­lipt­ic equa­tions. Here we re­turn again to the source of sin­gu­lar in­teg­rals, com­plex ana­lys­is, but now in the set­ting of sev­er­al vari­ables. An im­port­ant con­clu­sion at­tained was that for a broad class of do­mains in \( \mathbb{C}^n \), the Cauchy–Szegö pro­jec­tion is a sin­gu­lar in­teg­ral, sus­cept­ible to the above meth­ods. This was real­ized first for strongly pseudo-con­vex do­mains, next weakly pseudo-con­vex do­mains of fi­nite type in \( \mathbb{C}^2 \); and more re­cently, con­vex do­mains of fi­nite-type in \( \mathbb{C}^n \). Con­nec­ted with this is the ap­plic­a­tion of the above ideas to the \( \bar{\partial} \)-Neu­mann prob­lem, and its bound­ary ana­logue for cer­tain do­mains in \( \mathbb{C}^n \), as well as the study solv­ing op­er­at­ors for subel­lipt­ic prob­lems, such as Kohn’s Lapla­cian, Hörmander’s sum of squares, etc.; these mat­ters also in­volved us­ing ideas ori­gin­at­ing in the study of nil­po­tent groups as in (ii).

The three kinds of ex­ten­sions men­tioned above are prime ex­amples of what one may call “one-para­met­er” ana­lys­is. This ter­min­o­logy refers to the fact that the cubes (or their con­tain­ing balls), which oc­cur in the stand­ard \( \mathbb{R}^n \) set-up, have been re­placed by suit­able one-para­met­er fam­ily of gen­er­al­ized “balls,” as­so­ci­ated to each point. While the gen­er­al one-para­met­er meth­od clearly has wide ap­plic­ab­il­ity, it is not suf­fi­cient to re­solve the fol­low­ing im­port­ant ques­tion:

Prob­lem: De­scribe the nature of the sin­gu­lar in­teg­rals op­er­at­ors which are giv­en by Cauchy–Szegö pro­jec­tion, as well as those that arise in con­nec­tion with the solv­ing op­er­at­ors for the \( \bar{\partial} \) and \( \bar{\partial}_b \) com­plexes for gen­er­al smooth fi­nite-type pseudo-con­vex do­mains in \( \mathbb{C}^n \).

Some spec­u­la­tion about what may be in­volved in resolv­ing this ques­tion can be found be­low.

The meth­od of ro­ta­tions is both simple in its con­cep­tion, and far-reach­ing in its con­sequences. The ini­tial idea was to take the one-di­men­sion­al Hil­bert trans­form, in­duce it on a fixed (sub­group) \( \mathbb{R}^1 \) of \( \mathbb{R}^n \), ro­tate this \( \mathbb{R}^1 \) and in­teg­rate in all dir­ec­tions, ob­tain­ing in this way the sin­gu­lar in­teg­ral \eqref{eqonsi} with odd ker­nel, which can be writ­ten as \begin{equation} \label{eqontwni} T_{\Omega} (f)(x)=\textit{p.v.}\,\int_{\mathbb{R}^n}\frac{\Omega (y)}{|y|^n}f(x-y)\,dy, \end{equation} where \( \Omega \) is ho­mo­gen­eous of de­gree 0, in­teg­rable in the unit sphere, and odd.

In much the same way the gen­er­al max­im­al op­er­at­or \begin{equation} \label{eqonthze} M_{\Omega}(f)(x)=\sup_{r > 0}\frac{1}{r^n}\biggl|\,\int_{|y|\leq r} \Omega (y) f(x-y) \,dy\biggr| \end{equation} arises from the one-di­men­sion­al Hardy–Lit­tle­wood max­im­al func­tion.

This meth­od worked very well for \( L^p \), \( 1 < p \) es­tim­ates, but not for \( L^1 \) (since the weak-type \( L^1 \) “norm” is not sub­ad­dit­ive). The ques­tion of what hap­pens for \( L^1 \) was left un­re­solved by Calderón and Zyg­mund. It is now to a large ex­tent answered: we know that both \eqref{eqontwni} and \eqref{eqonthze} are in­deed of weak-type \( (1,1) \) if \( \Omega \) is in \( L( \log L) \). This is the achieve­ment of a num­ber of math­em­aticians, in par­tic­u­lar Christ and Ru­bio de Fran­cia. When the meth­od of ro­ta­tions is com­bined with the sin­gu­lar in­teg­rals for the heat equa­tion (as in 1(i) above), one ar­rives at the “Hil­bert trans­form on the para­bola.” Con­sid­er­a­tion of the Pois­son in­teg­ral on sym­met­ric spaces leads one also to in­quire about some ana­log­ous max­im­al func­tions as­so­ci­ated to ho­mo­gen­eous curves. The ini­tial ma­jor break­throughs in this area of re­search were ob­tained by Na­gel, Rivière, and Wainger. The sub­ject has since de­veloped in­to a rich and var­ied the­ory: be­gin­ning with its trans­la­tion in­vari­ant set­ting on \( \mathbb{R}^n \) (and its re­li­ance on the Four­i­er trans­form), and then promp­ted by sev­er­al com­plex vari­ables, to a more gen­er­al con­text con­nec­ted with os­cil­lat­ory in­teg­rals and nil­po­tent Lie groups, where it was re­christened as the the­ory of “sin­gu­lar radon trans­forms.”

A com­mon un­re­solved en­igma re­mains about these two areas which have sprung out of the meth­od of ro­ta­tions. This is a ques­tion which has in­trigued work­ers in the field, and whose solu­tion, if pos­it­ive, would be of great in­terest.

Prob­lem:

1. Is there an \( L^1 \) the­ory for \eqref{eqontwni} and \eqref{eqonthze} if \( \Omega \) is merely in­teg­rable? 2

2. Are the sin­gu­lar Radon trans­forms, and their cor­res­pond­ing max­im­al func­tions, of weak-type \( (1,1) \)?

To over­sim­pli­fy mat­ters, one can say that “product the­ory” is that part of har­mon­ic ana­lys­is in \( \mathbb{R}^n \) which is in­vari­ant with re­spect to the \( n \)-fold dila­tions: \( x=(x_1,x_2,\ldots, \) \( x_n)\rightarrow (\delta_1x_1,\delta_2 x_2,\ldots, \) \( \delta_n x_n) \), \( \delta_j > 0 \). An­oth­er way of put­ting it is that its ini­tial con­cern is with op­er­at­ors that are es­sen­tially products of op­er­at­ors act­ing on each vari­able sep­ar­ately, and then more gen­er­ally with op­er­at­ors (and as­so­ci­ated func­tion spaces) which re­tain some of these char­ac­ter­ist­ics. Re­lated to this is the mul­ti­para­met­er the­ory, stand­ing part-way between the one-para­met­er the­ory dis­cussed above and product the­ory: here the em­phas­is is on op­er­at­ors which are “in­vari­ant” (or com­pat­ible with) spe­cified sub­groups of the group of \( n \)-para­met­er dila­tions.

The product the­ory of \( \mathbb{R}^n \) began with Zyg­mund’s study of the strong max­im­al func­tion, con­tin­ued with Mar­cinkiewicz’s proof of his mul­ti­pli­er the­or­em, and has since branched out in a vari­ety of dir­ec­tions where much in­ter­est­ing work has been done. Among the things achieved are an ap­pro­pri­ate \( H^p \) and BMO the­ory, and the many prop­er­ties of product (and mul­ti­para­met­er) sin­gu­lar in­teg­rals which have came to light. This is due to the work of S.-Y. Chang, R. Fef­fer­man, and J.-L. Journé, to men­tion only a few of the names.

Fi­nally, I want to come to an ex­ten­sion of the product the­ory (more pre­cisely, the in­duced “mul­ti­para­met­er ana­lys­is”) in a dir­ec­tion which has par­tic­u­larly in­ter­ested me re­cently. Here the point is that the un­der­ly­ing space is no longer Eu­c­lidean \( \mathbb{R}^n \), but rather a nil­po­tent group or an­oth­er ap­pro­pri­ate gen­er­al­iz­a­tion. On the basis of re­cent, but lim­ited, ex­per­i­ence I would haz­ard the guess that mul­ti­para­met­er ana­lys­is in this set­ting could well turn out to be of great in­terest in ques­tions re­lated to sev­er­al com­plex vari­ables. A first vague hint that this may be so, came with the real­iz­a­tion that cer­tain bound­ary op­er­at­ors arising from the \( \bar{\partial} \)-Neu­mann prob­lem (in the mod­el case cor­res­pond­ing to the Heis­en­berg group) are ex­cel­lent ex­amples of mul­tiple-para­met­er sin­gu­lar in­teg­rals (see Müller, Ricci, and Stein [e58]). A second in­dic­a­tion is the de­scrip­tion of Cauchy–Szegö pro­jec­tions and solv­ing op­er­at­ors for \( \bar{\partial}_b \) for a wide class of quad­rat­ic sur­faces of high­er codi­men­sion in \( \mathbb{C}^n \), in terms of ap­pro­pri­ate quo­tients of products of Heis­en­berg groups (see Na­gel, Ricci, and Stein [e63]). And even more sug­gest­ive are re­cent cal­cu­la­tions (made jointly with A. Na­gel) for such op­er­at­ors in a num­ber of pseudo-con­vex do­mains of fi­nite type. All this leads one to hope that a suit­able ver­sion of mul­ti­para­met­er ana­lys­is will provide the miss­ing the­ory of sin­gu­lar in­teg­rals needed for a vari­ety of ques­tions in sev­er­al com­plex vari­ables. This is in­deed an ex­cit­ing pro­spect.

I wish to provide here some ad­di­tion­al cita­tions of the lit­er­at­ure closely con­nec­ted to the ma­ter­i­al I have covered. However, these notes are not meant to be in any sense a sys­tem­at­ic sur­vey of rel­ev­ant work.

Zyg­mung [e36] is a greatly re­vised and ex­pan­ded second edi­tion of his 1935 book. His ini­tial work on the strong max­im­al func­tion is in Zyg­mung [e6]. His views about the cent­ral role of com­plex meth­ods in Four­i­er ana­lys­is are ex­plained in Zyg­mung [e14]. A his­tor­ic­al sur­vey of square func­tions and an ac­count of Zyg­mund’s work in this area can be found in Stein [e43].

For the real vari­able the­ory of the Hil­bert trans­form see Be­sicov­itch [e1], Titch­marsh [e5], and Mar­cinkiewicz [e9].

Two pa­pers of Calderón and Zyg­mund deal­ing with the sym­bol­ic cal­cu­lus of op­er­at­ors \eqref{eqonse} ([5] and [6]).

Fur­ther work of Calderón, ap­ply­ing sin­gu­lar in­teg­rals to par­tial dif­fer­en­tial equa­tions, is con­tained in [8] and [9].

The the­ory of para-products was de­veloped later in Bony [e41]. The gen­er­al form of Cot­lar’s lemma, \eqref{eqontwfo}, as well as the ap­plic­a­tion to in­ter­twin­ing op­er­at­ors, may be found in Knapp and Stein [e32]; the ap­plic­a­tion to pseudo-dif­fer­en­tial op­er­at­ors is in Calderón and Vail­lan­court [11]. The re­la­tion between the bounded­ness of the usu­al sin­gu­lar in­teg­rals on Hardy spaces and equi­val­ences like \eqref{eqontwth} is in Stein [e28], chapter 7. An ac­count of the real-vari­able \( H^p \) the­ory can be found in Stein [e56], chapters 3 and 4.

For sys­tem­at­ic present­a­tions of top­ics such as com­mut­at­ors, the Cauchy in­teg­ral, mul­ti­lin­ear ana­lys­is, and the \( T(b) \) the­or­em, the read­er should con­sult Coi­f­man and Mey­er [e39], Mey­er [e52], and Mey­er and Coi­f­man [e52].

In Kenig [e57] the read­er will find an ex­pos­i­tion of the area deal­ing with the op­er­at­or \eqref{eqononse} as well as the Lapla­cian on do­mains with Lipschitz bound­ary.

In con­nec­tion with (ii), the read­er may con­sult Stein [e31]. For the oc­cur­rence of Calderón–Zyg­mund-type sin­gu­lar in­teg­rals on strictly-pseudo-con­vex do­mains, see Kor­a­nyi-Vági [e29], C. Fef­fer­man [e34], and Fol­land and Stein [e33]. Some cor­res­pond­ing res­ults for do­mains in \( \mathbb{C}^2 \) of fi­nite type may be found in Christ [e49], Mache­don [e48], Na­gel, Rosay, Stein, and Wainger [e51]. For the Cauchy–Szegö pro­jec­tion on con­vex do­mains in \( \mathbb{C}^n \), see McNeal and Stein [e60].

Re­gard­ing the Calderón–Zyg­mund lemma, two fur­ther sources should be cited. In Coi­f­man and Weiss [e30] the use of this meth­od on spaces of a gen­er­al char­ac­ter is sys­tem­at­ized. The work of C. Fef­fer­man [e27] con­tains an im­port­ant de­par­ture re­gard­ing the Calderón–Zyg­mund meth­od, in­volving cer­tain ad­di­tion­al \( L^2 \) ar­gu­ments, and al­low­ing him to prove a num­ber of subtle weak-type res­ults. This meth­od has proved to be rel­ev­ant in vari­ous oth­er in­stances, in par­tic­u­lar to the study of op­er­at­ors of the type \eqref{eqontwni} and \eqref{eqonthze}.

The meth­od of ro­ta­tions ori­gin­ates in Calderón and Zyg­mund [4]. For \eqref{eqontwni} and \eqref{eqonthze} see also See­ger [e59] and Tao [e62]. For sin­gu­lar Radon trans­forms, see Stein and Wainger [e37], Phong and Stein [e47], and Christ, Na­gel, Stein, and Wainger [e61], where oth­er ref­er­ences can be found.

Among the pa­pers that may be con­sul­ted for the product and mul­ti­para­met­er the­ory in the Eu­c­lidean set-up are: S.-Y. Chang and R. Fef­fer­man [e40], Journé [e46], Ricci and Stein [e55].