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%% Webster, Sidney M.
%% 
%% {\it Hypo-analytic structures, local theory}, by Francois Treves, 
%%   Princeton University Press, Princeton, NJ, 1992, xvii + 497 pp., 
%%   \$65.00. ISBN 0-691-08744-X
%% 
%% publ:  Bull. Amer. Math. Soc. (N.S.) 30(1994) no. 2
%% pp:    290-292
%% type:  Book Review                  markup: amstex    file size: 13K
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%% copyright: American Math. Society copyright; see end of article
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\topmatter
\cvol{30}
\cvolyear{1994}
\cmonth{April}
\cyear{1994}
\cvolno{2}
\cpgs{290-292}

\revtop {\it Hypo-analytic structures, local theory}, by 
Francois Treves,
Princeton University Press, Princeton, NJ, 1992, xvii + 
497 pp., \$65.00. ISBN
0-691-08744-X \endrevtop
\reviewer Sidney M. Webster  \endreviewer
\affil University of Chicago\endaffil
\endtopmatter

\document               

\par Since the 1950s there has been a significant 
interplay between
multivariable complex analysis and certain parts of 
partial differential
equations theory. As is typical, problems of complex 
analysis have been treated
by PDE methods, and in turn complex analysis has contributed
concepts and driven certain lines of development in PDE 
theory. The literature
is now enormous. Among the textbooks dealing specifically 
with this interplay we
mention those by Morrey, Folland-Kohn, Ehrenpreis, several 
by Hormander, and
some of the previous ones by Treves himself. His present 
book, which I shall
refer to as ``the text'', focuses on that part of the 
theory which may be
characterized as arising from the study of real 
submanifolds of a complex
manifold. Its emphasis is more on the basic local complex 
analytic and
geometric foundations and concepts, which are rather 
diverse. It deals less
with more advanced real analysis or functional analysis 
techniques. Thus, the
``hypoanalytic'' of the title is essentially a geometric 
concept, not to be
confused with the more familiar term ``hypoelliptic'', 
which is analytic in
nature.
\par If we except the complex Monge-Ampere equation with 
its considerable
success in the compact Kahler case, the predominant  role 
of PDE theory in
complex manifold theory is in the linear realm, via the 
Cauchy-Riemann
equations. In the compact case the theory is wholly 
elliptic and well
understood. In the noncompact case the theory is much more 
difficult and less
complete. One must usually either deal with a boundary or 
place growth
conditions at infinity.  When the complex dimension 
exceeds one, the boundary
of a bounded domain in complex space becomes of paramount 
importance. In
favorable cases it is either a smooth real hypersurface or 
contains smooth real
submanifolds. The trace of an analytic function on such a 
submanifold $M$ is
annihilated by all complex vectors of type $(0,1)$ tangent 
to $M$, i.e., is a
CR function. The study of such functions is very important 
to the function
theory of the domain.
\par Real submanifolds $M$ may be divided into two 
classes: (i) those for which
the tangent $(0,1)$ vectors form a vector subbundle (of 
constant rank) of the
complexified tangent bundle of $M$, the CR submanifolds; 
and (ii) the rest,
those with CR singularities. The text very wisely leaves 
aside case (ii), for
which very little analytic theory has been developed, and 
begins with formally
integrable, or involutive, structures,   
i.e., subbundles $V$ of the complex
tangent bundle of a real manifold which satisfy the 
Frobenius integrability
condition (smooth sections of $V$ are closed under Lie 
bracket). Associated to
$V$ is the natural deRham-Dolbeault complex with operator 
$d''$.
\par There are two main problems in this generality: (a) 
the integrability
problem: find sufficiently many solutions with good 
properties to the
homogeneous equations $d''f=0$; and (b) the inhomogeneous 
problem: solve
$d''f=g$ for the form $f$, given that the form $g$ 
satisfies the compatibility
condition $d''g=0$, again with some control of $f$. 
Generally, the first
problem is the more immediate and geometric one. The 
second is more analytic in
nature and is used in solving the first. The ideal way of 
handling it is
perhaps by means of a ``homotopy formula'', $g=d''Pg+
Qd''g$, where $P$ and $Q$
are manageable operators. For (domains on) complex 
manifolds the emphasis is
usually global, e.g., as in solving the Levi problem. For 
more general
involutive structures the primary emphasis in (a) is on 
the Complex Frobenius
problem of existence of a complete, or maximally 
independent, set of local
integrals. However, a general solution may not be a power 
series in these local
integrals. This necessitates the finer concept of 
hypoanalytic structure: a
covering of the manifold by open sets with integrals which 
are analytically
related on the intersections.
\par The local integrability problem has ancient roots. 
The most important case
concerns the existence of complex coordinates for an 
almost complex manifold.
The one- (complex)dimensional case is the problem of 
isothermal coordinates or
the local conformal mapping of a curved surface to the 
plane, a fundamental
problem of cartography. It was solved in the real analytic 
case by Gauss in
about 1820. The smooth case had to wait another one 
hundred years for the
necessary analytic machinery to be developed. It was 
solved by Korn and
Lichtenstein. By the late 1940s and early 1950s the 
concepts of complex and
almost-complex manifolds were well established, and the 
higher-dimensional real
analytic case was solved via the complex analytic 
Frobenius theorem. In 1957
Newlander and Nirenberg settled the smooth case. Their 
theorem, which is of
fundamental importance in several regards, has since 
received several other
proofs. One is given in the text based on the homotopy 
formula of Leray and
Koppleman. The result is then extended to ``elliptic'' 
structures, i.e., those
for which the annihilator of $V$ contains no real 
one-forms. By a further
extension the ``rigid structures'' studied by Rothschild 
and others is handled.
\par For more general structures the problem is much more 
subtle in the smooth
case, with the formal integrability condition not always 
being sufficient. For
the case of CR structures of nondegenerate real 
hypersurface type,
counterexamples, due to Nirenberg and Jacobowitz-Treves, 
are given in the text.
A major achievement here was the local CR embedding 
theorem of Kuranishi,
extended by Akahori, for the strictly pseudoconvex case in 
seven or more
dimensions. The 5-dimensional case still remains open. 
Currently, progress is
being made on the regularity of the solution in 
yet-to-be-published work of Ma
and Michel.
\par All these results rely on the solution of the 
inhomogeneous problem. In
one complex variable it was first considered around the 
turn of the century, but
it rarely occurs in the standard presentations of the 
subject. It has been
ubiquitous in several variables since the work of Oka and 
Cartan. A landmark
event was H. Lewy's discovery in 1956 of his locally 
unsolvable equation. His
operator is the tangential Cauchy-Riemann operator on the 
ordinary 3-sphere in the space of two complex variables.
Another example was found by Mizohata and Grushin. It 
involves the operator in
the $xy$-plane associated to the ``folded complex 
structure'', where the basic
analytic function is $x+iy^2$. Both operators play an 
important role in linear
PDE theory. The text devotes a chapter to such examples of 
nonsolvability and
nonintegrability, including the interesting phenomenon of 
one-sided
integrability discovered by C. D. Hill. Another chapter 
concerns necessary and
sufficient conditions for a Poincar\'e lemma.
\par Chapter 2 presents one of the most important results 
of the theory,
namely, the Baouendi-Treves approximation theorem, a very 
natural analogue of
the Weierstrass approximation theorem. When there exists a 
complete set of first
integrals, then every other solution to the homogeneous 
problem can be
approximated uniformly on suitably small domains by 
polynomials in these
integrals. It is both elementary to prove, following from 
a Fourier integral
representation, and extremely useful. There are now many 
results which depend
on it and many others which are greatly simplified by it. 
Local CR extension
theorems are a good class of examples. It also underscores 
the appropriateness
of the concept of hypoanalyticity.
\par Another strength of the text is a chapter on the FBI
(Fourier-Bros-Iagolnitzer) transform. This is a 
generalization of the Fourier
transform which is particularly suited to the study of 
hypoanalytic structures.
The behavior of this transform bears on regularity 
properties, holomorphic
extendability, and hypoanalyticity of distributions. It 
also is used to
establish propagation of hypoanalyticity in certain cases. 
We are also assured
that it plays a significant role in the microlocal theory. 
A final chapter
deals with nonlinear equations. The main emphasis is on 
uniqueness in the
Cauchy problem and on the approximation of not-so-smooth 
solutions by
smoother ones.
\par My initial reaction, especially after learning from 
the preface that a
second volume will be necessary for the microlocal theory, 
was that the text at
497 pages was perhaps too long. (I am always glad to find 
a book like this
which contains information which I need to know, but I 
always hope to have to read
it as little as possible!) After some experience with it, 
however, my opinion
has changed. As the author warns us, the considerable 
amount of foundational
material covered does not always make for the best sort of 
reading. However,
many examples are spread throughout the book, and it is 
not clear that a
shorter book would have been a better one. There are a few 
misprints
(unavoidable in a work of this length), but none at all 
serious. The exposition
is exceptionally clear and careful.
\par How far will the theory go and in what direction? It 
is risky to
speculate, but it may be useful to consider some points. 
The theorem on
isothermal coordinates and the Newlander-Nirenberg theorem 
have genuinely useful
applications. But, while the real Frobenius theorem has a 
host of applications
in elementary differential geometry, its complex analogue 
has scarcely any. It
is more a theorem for its own sake at this stage. The 
technical difficulties of
even the simplest proofs of the centrally important 
Kuranishi embedding theorem
may indicate limits in this direction for the theory. 
Also, they show that even
the simplest useful domains on CR manifolds tend to have 
characteristic points.
Thus, the troublesome submanifolds with CR singularities 
must eventually be
dealt with. In any event, it seems clear that this text of 
Treves will
definitely play a positive role in future developments of 
the subject.
\enddocument
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