#### by Elias M. Stein

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.

#### 1.1: Zygmund’s vision: 1927–1949

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.

##### 1.1.1: The conjugate function and its basic properties

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.

##### 1.1.2: The theory of the Hardy spaces __\( H^p \)__

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.

##### 1.1.3: The Littlewood–Paley theory

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.

##### 1.1.4: The boundary behavior of harmonic functions

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.

#### 1.2: Calderón and Zygmund: 1950–1957

##### 1.2.1

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.

##### 1.2.2

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.

##### 1.2.3

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.

#### 1.3: Acceptance: 1957–1965

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.

##### 1.3.1

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.*

##### 1.3.2

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].

##### 1.3.3

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.

#### 1.4: Calderón’s new theory of singular integrals: 1965–

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.

##### 1.4.1

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.

##### 1.4.2

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.

##### 1.4.3

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.

#### 1.5: Some perspectives on singular integrals: past, present, and future

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.

##### 1.5.1: Method of the Calderón–Zygmund lemma

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:

*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.*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.*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.

##### 1.5.2: The method of rotations

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:**

Is there an

__\( L^1 \)__theory for__\eqref{eqontwni}__and__\eqref{eqonthze}__if__\( \Omega \)__is merely integrable? 2Are the singular Radon transforms, and their corresponding maximal functions, of weak-type

__\( (1,1) \)__?

##### 1.5.3: Product theory and multiparameter analysis

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.

#### 1.6: Bibliographical notes

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.

##### 1.1

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].

##### 1.2.2

##### 1.2.3

##### 1.3.1

##### 1.4.1

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.

##### 1.4.2

##### 1.4.3

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.

##### 1.5.1

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}__.

##### 1.5.2

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.