数理解析研究所講究録
1001 巻 1997 年 1-21

1Some open problems on the analysis of the
Cauchy-Fueter system in several variables
Irene Sabadini
Dipartimento di Matematica
di Milano
Via C. Saldini 50, 20133 Milano, Italy.
$\mathrm{U}\mathrm{n}\mathrm{i}_{\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{S}}\mathrm{i}\mathrm{t}\mathrm{a}^{j}$

Daniele C. Struppa
Department of Mathematical Sciences
George Mason University
University Drive, Fairfax, VA 22030, USA.

2

1

Introduction

In the last few years there has been a resurgence of interest for the analysis ofregular functions of a quaternionic variable and, more generally, for the analysis of monogenic functions of Clifford variables [28], of which quaternions
are one of the simplest examples. There are many reasons for such a rebirthof a theory whose origin goes back to Fueter’s works in the first half of thiscentury (actually, as Professor Shapiro pointed out [16], the first works inthis direction are due to Moisil and Theodoresco [19], [20] . On one hand
Clifford analysis has acquired particular relevance in theoretical physics (seee.g. [5] , and, in particular, quaternionic quantum field theory is now being
actively developed (see [5]); on the other hand, Clifford analysis has become
increasily relevant to questions in classical harmonic analysis, as it has been(see also [28]) and moreclcarly demollstratcd by thc work ofandsource of interest is the very recent novel approach due toShapiro, [16].
Up to now, however, very little attention has been paid (an exception being [26] to the case of several variables (for example several quaternionicvariables); this fact is most likely due to the intrinsic difficulty of studying
objects (regular functions) which seem to lack some of the most elementary properties (as an example, the composition of two regular functions is
not even regular, nor is their product). A couple of years ago, however, a. Berenstein, P. Loustaunau and the authors)group of mathematicians (realized that an abstract algebraic treatment of the Cauchy-Fueter system
could provide the necessary tool for the study of regular functions in several
quaternionic variables.
In this paper, after a brief introduction to the subject, we will describe theresults which we have obtained so far, and which seem to suggest the existence of a quaternionic analog of the theory of hyperfunctions. The core of
the paper, however, is the last section, in which we will state several open
problems, together with some ideas which may play a role in their solution.
Acknowledgements. The first author would like to thank the Departmentof Mathematical Sciences of the George Mason University for the kind hospitality during the period in which the paper was written. The second author
$)$

$)$

$\mathrm{M}\mathrm{c}\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{o}\mathrm{s}\mathrm{h}[18]$$\mathrm{I}\backslash \perp\cdot \mathrm{a}\mathrm{v}\prime \mathrm{c}\mathrm{I}\mathrm{l}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{k}_{0}$$)$$\mathrm{C}.\mathrm{A}$3whishes to thank the R.I.M.S. of Kyoto University for the hospitality during the writing of this paper, as well as Professor Kawai, for his invitation
Analysis and Fourier”Exactto lecture on this topic at theAnalysis in the Complex Domain.
$WI_{1’}\prime B$$\mathrm{w}.\mathrm{o}\mathrm{r}_{J},\mathrm{k}\mathrm{S}\mathrm{h}_{0}\mathrm{P}$2

Regular functions of a quaternionic variable

In this section we will collect the fundamental definitions and properties on
which the theory of regular functions is based , these notations will be ofgreat importance in the next sections when we will treat the case of several
variables. Our notation will follow [32] and [12], to which the interested
details. We wi.ll denote by $H$ the four-dimensional
reader is referred forreal associative algebra of the quaternions, with the standard basis $\{1, i,j, k\}$satifying the usual relations:$\cdot$mor.e$i^{2}=j^{2}=k^{2}=-1$ ,$\dot{\iota}j=-ji=k$,$jk=-kj=i$ ,$ki=-ik=j$ .

A quaternion will be an element$q=x0+x_{1}\dot{\iota}+X2j+x3k$

so that$H= \{q+\sum_{i=0}x3ie_{i}|e_{0}=1, e_{1}=\dot{\iota}, e_{2}=j, e_{3}=k\}$.

The (left) Cauchy-Fueter operator is defined by(1)$\frac{\partial}{\partial\overline{q}}=\frac{\partial}{\partial x_{0}}+i\frac{\partial}{\partial x_{1}}+j\frac{\partial}{\partial x_{2}}+k\frac{\partial}{\partial x_{3}}=0$

and we can use it to give the following
be an open and let
Definition 2.1 Let
toin the real sense (as a function from
regular if$U\subseteq H$$U\subseteq R^{4}$$\frac{\partial f}{\partial\overline{q}}=0$

on

$U$

.

$f$

: $Uarrow H$ be differentiable). We say that is $(left)-$$R^{4}$$f$

4

We will denote by
the right fff-vector space of regular functions. A
corresponding notion of right regularity can be defined by means of the right
Cauchy-Fueter operator given by$\mathcal{R}(U)$$\frac{\partial_{r}}{\partial\overline{q}}=\frac{\partial}{\partial x_{0}}+\frac{\partial}{\partial x_{1}}i+\frac{\partial}{\partial x_{2}}j+\frac{\partial}{\partial x_{3}}k=0$.

The theory of right-regular functions is completely parallel to the theory ofleft regular ones.
Remark 2.2 The Cauchy-Fueter operator was introduced by Fueter [13]and [14] (but see also [19] and [20]) who was interested in constructing, overff;, a space of ”holomorphic” functions. As he quickly found out, a naive
definition of $lH$ -holomorphicity (or regularity as he called it) based on theexistence of the $lH$ -derivative, does not work as the only functions : $lHarrow$$IH$ which admits left (or right) $H$-derivative (i.e. $\lim_{harrow 0,h\in H}h^{-1}(f(q+h)-$$f(q)))$ are linear functions of the$+b$ (or $f(q)=aq+b$ ). His.
next idea was to construct a quaternlonlc verslon of the Cauchy-Riemannsystem: such a version is the system (1).
It can be shown that much of what is known for holomorphic functions canbe replicated for regular ones. In this section we will explore some of theseproperties. To begin with, the classical Cauchy kernel $1/z$ is replaced by theso-called Cauchy-Fueter kernel$f$$\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}.f(..q)=qa$$G(q):= \frac{q^{-1}}{|q|^{2}}=\frac{\overline{q}}{|q|^{4}}$which is easily seen to be both lcft and right,’ular.

$1^{\cdot}\mathrm{c}\mathrm{g}$Let us note that, if$Dq:=dX_{1}$ A $dx_{2}$ A $dx_{3}-id_{X}0$ A $dx_{2}$ A $dx_{3}+jdx_{0}$ A $dx_{1}$ A $d_{X_{3^{-}}}kdx_{0}$ A $dx_{1}$ A $dx_{2}$and$\nu:=dx_{0}$A$dx_{1}$A$dx_{2}$A$dx_{3}$the formula,$d(_{\mathit{9}^{Dq}}f)=dg \wedge Dqf-\mathit{9}Dq\wedge df=\{(\frac{\partial_{r}g}{\partial\overline{q}}\mathrm{I}^{f}+_{\mathit{9}}(\frac{\partial_{l}f}{\partial\overline{q}})\}\nu$allows us to prove the following two theorems.

5: $U\subseteq Harrow H$ be a rcgular
Theorem 2.3 (Cauchy-Fueter ). Let
is any open set with smooth boundary, such thatfunction on U. If, then$I$$f$$V\subseteq U$$\overline{V}\subseteq U$$\int_{\partial V}Dqf(q)=0$: $U\subseteq Harrow H$ be a regular
Theorem 2.4 (Cauchy-Fueter II). Letfunction on U. Let $V\subset U$ be an open set with smooth boundary, relatively
compact in U. If $q_{0}\in V$ , then$f$$f(q \mathrm{o})=\frac{1}{2\pi^{2}}\int_{\partial V}G(q-q\mathrm{o})Dqf(q)$Remark 2.5 Let$\triangle$

be the laplacian in$R^{4}$.

. Then$\triangle==\underline{\partial}\underline{\partial}\underline{\partial}\underline{\partial}$’$\partial q\partial\overline{q}$$\partial\overline{q}\partial q$

so the Cauchy-Fueter system is elliptic.
From this last property follows that all the main results on holomorphic functions like the Liouville Theorem or the Poincar\’e Lemma or the Maximum
Modulus Principle can be also proved in this more general setting. An important difference with the theory of holomorphic functions is that it is notpossible to express a regular function in terms of powers of . In fact, theis not regular not even for $n=1$ . A Taylor series can be written
functionin terms of particular homogeneous polynomials as follows. Let$q$$q^{n}$,$P_{\nu}(q)=\underline{1}$$n!_{1\leq\lambda_{1,\ldots n}} \sum_{\leq\lambda 3},(X0e_{1}-X_{\lambda})1\ldots(X_{0}e_{\lambda_{n}}-X_{\lambda_{n}})$elements equal
possible permutations of
where the sum is on all
to $i=1,2,3$ . We have the following theorem:. Let $q_{0}\in U$ andTheorem 2.6 Let : $U\subseteq Harrow H,$. Then, $if|q-q_{01}<\delta$ , we havedist
$n_{i}$$\frac{n!}{n_{1}!n_{2}!n_{3}!}$$i,$$\delta<$$f\in \mathcal{R}(U)$$f$$(q0, \partial U)$$f(q)= \sum_{n=0\nu\in}^{+\infty}\sum P_{\nu}(\sigma nq-q0)a_{\nu}$

,

where
$a_{\nu}=(-1)n \partial_{\nu}f(q\mathrm{o})=\frac{1}{2\pi^{2}}\int_{1q-}q_{0}|=\mathit{5}0G_{\nu}(q-q)Dqf(q)$with$\partial_{\nu}=\partial^{\nu}/\partial x_{1}^{n_{1}}\partial x_{2}\partial n2X_{3}^{n}3$.

6
The existence of such an expansion is essentially due to the fact that it
is possible to multiply two given regular functions using the so-called C-K
product to get a regular function, at least when they are entire. An analogous
C-K product is not known in the case of several variables; this fact prevents. We
us from defining a Taylor series for regular functions defined on
would like to point out that it is not known if it is possible to prove ananalogous theorem to the case of regular functions of several quaternionicvariables. The theory for regular functions in one variable is so similar to thecorresponding theory in the complex case that it is also possible to prove afor every open set
Mittag-Leffler theorem, that is the vanishing of. This allows the introduction of the notion of hyperfunctions:$lH^{n}$$H^{1}(U, \mathcal{R})$$U$

Definition 2.7 Let be an open set in $\tilde{H}=\{q=\Sigma_{i=0^{X_{i}e}}^{3}i|x_{0}=0\}$ andis relativefy closed in V. The rightletbe an open set in $JH$ such that
vector space on $lH$ defined by
$U$

$U$

$V$

$\mathcal{F}^{\cdot}(U):=\frac{\mathcal{R}_{l}(V\backslash U)}{\mathcal{R}_{l}(V)}$is said to be the space of Rf-hyperfunctions.

the vector space
The correspondence that assigns to every open set in$F(U)$ defines a flabby sheaf on. The classical Fantappi\’e-K\"othe-Martineau
duality holds also in this case, namely$U$$\tilde{H}$$J\tilde{H}$$\mathcal{F}_{\mathrm{A}’}\cong\frac{\mathcal{R}\iota(U\backslash Ic)}{\mathcal{R}_{l}(U)}\cong(\mathcal{G}(K))’$is the space of $lH$ -hyperfunctions with support in $K$ andwheredenotes the space of left linear continuos functionals on
is reflexive it
and the duality is topological as was proved in [29]. Since
holds also$(\mathcal{G}(K))’$$F_{h’}$$\mathcal{G}(K)=\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{l}\mathrm{i}\mathrm{m}U\supset K\mathcal{R}r(U)$$\mathcal{F}_{Ii^{r}}$$\mathcal{G}(K)\cong(\mathcal{F}_{\mathrm{A}}’)’$

3

The Cauchy-Fueter system in several variables

As we have seen in section 2, the construction of a hyperfunction theory
based on boundary values of regular functions of one quaternionic variable

7
is an extremely natural affair, but the methods used do not lend themselves
to easy multi-dimensional generalizations.
The first step consists in suitably defining regular functions in several
quaternionic variables. Unfortunately, the non-conrmutativity of quaternions
plays there a rather strong role. Let us consider two quaternionic variables
and . If $f=f(q_{1})$ and $g=g(q_{2})$ are, respectively, left-regular in . and, then the product $fg$ is not left-regular in both of them, since$q_{1}$$q_{1}$$q_{2}$$q_{2}$$\frac{\partial(fg)}{\partial\overline{q}_{1}}=\frac{\partial f}{\partial\overline{q}_{1}}g+\sum_{0i=}^{3}e_{i}f\frac{\partial g}{\partial x_{i}}$$\frac{\partial(fg)}{\partial\overline{q}_{2}}=\frac{\partial f}{\partial\overline{q}_{2}}g+\sum_{0i=}^{3}e_{i}f\frac{\partial g}{\partial y_{i}}$fails. For this reason, Nono defined inand therefore left-regularity inwhich requires to be[22] a notion of biregularity for functions
left-regular in and right-regular in . A reasonably simple theory can beestablished for these functions; however no generalization along these lines ispossible to three or more variables. We have taken, on the other hand, thedirection of defining regular functions of several variables as those functions: $H^{n}arrow H$ who are left-regular in each variable or, which is equivalent,
which satisfy$q_{1},$$q_{2}$$f(q_{1}, q_{2})$$f$$q_{2}$$q_{1}$$f$$\frac{\partial f}{\partial\overline{q}_{1}}=\ldots=\frac{\partial f}{\partial\overline{q}_{n}}=0$.

Constructing examples for such functions is not as simple as for biregularfunctions, since even simple products of regular polynomials are not regular.
For example if we set $q_{1}=x_{0}+iX_{1}.+jX2+kx_{3}$ and $q_{2}=y0+\dot{i}y_{1}+jy_{2}+ky_{3}$ ,we have that even$(x_{1}-iX\mathrm{o})(y_{2^{-}\dot{J}}y\mathrm{o})+(y2-jy0)(X_{1}-ix_{0})$is not regular regular in$q_{1}$

and$q_{2}$. On the other hand,$(x_{1}-i_{X}\mathrm{o})(y1-\dot{i}y_{0})+(y1-iy\mathrm{o})(x_{1}-i_{X}0)$

is regular and, more generally, examples can be constructed by using synlmetric products of regular polynomials in which the factors involve the same
imaginary unit. This is equivalent to the request that the two polynomials

8
are holomorphic with respect to two complex variables
belonging to the same complex plane.
The point of view that we have taken, however, is to use the ellipticityof the Cauchy-Fueter system and to consider regular functions as vector
solutions of the multi-dimensional Cauchy-Fueter system. From this pointof view, a regular function : $H^{n}arrow H$ is a differentiable function :$R^{4n}arrow R^{4}$ which satisfies a $4n\cross 4$ system$(x_{i}+e_{i^{X_{0})}}, (y_{i}+e_{i}y_{0})$$\vec{f}$$f$$[_{\partial/\partial X}^{\partial/\partial X}\partial/\partial.X_{3}^{\cdot}n1\partial/\partial X\partial/\partial x\partial\partial/\partial x_{2n}\partial/.\partial x/\partial X0_{n}1n30_{1}2111$$\partial/\partial x\partial/\partial\partial/\partial x_{\partial}-\partial\partial/\partial x01-\partial/-\partial/x_{2n}-\partial/\partial/\partial x_{2}x_{0}\partial 3n3xX_{1n}n1111$. $]=0arrow$$-\partial^{/}-\partial-\partial/-\partial/\partial/\partial\partial/.\partial.X\partial//\partial.x/\partial\partial\partial x_{0n}\partial X\partial\partial x_{01}X_{1n}x_{3n}x_{11}2n3121$$-\partial/\partial X-\partial/\partial x-\partial/\partial x_{3}-\partial\partial/\partial_{X}.0_{n}\partial\partial\partial/\partial X/\partial x_{2n}/\partial x0/\partial X1^{\cdot}n31121n11$(2)

where we have written $q_{t}=x_{0t}+iX_{1t}+_{\dot{J}2t}x+kx3t$ and $f=f\mathrm{o}+if1+_{J}f_{2}+kf_{3}$ .
Note that in view of the ellipticity of the Cauchy-Fueter system, it does
not matter whether one consider in $A$ (real analytic functions),
functions), $D’$ (distributions) or even in (hyperfunctions). We will denoteby
the sheaf of regular functions. The advantage of this point of view isthat it allows us to disengage from the use of quaternions, and enables us tobring in the powerful algebraic treatment championed in [24]. For the sakelet us review the fundamentals of such an approach. Letof$R=C[z_{1}, \ldots , z_{n}]$ be the ring of polynomials with complex coefficients inmatrix in so that $P(D)=[P_{ij}(D)]$variables, and let $P=[P_{ij}]$ be an
is a matrix of linear constant coefficients differential operators, where $D=$
. If
is a sheaf of generalized functions (in the case weare interested in, will either be the sheaf of hyperfunctions, or the sheaf
the kernel ofof infinitely differentiable functions), we will denote by
the map$f$$\mathcal{E}(C^{\infty}$$B$$\mathcal{R}$$\mathrm{C}\mathrm{o}\mathrm{m}_{\mathrm{P}^{\mathrm{l}\mathrm{e}})}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{S}$

$n$

$R$

$r_{1}\cross r_{0}$$(-i \frac{\partial}{\partial x_{1}}, \ldots , -i\frac{\partial}{\partial x_{n}})$

$S$

$S$

$B$

$S^{P}$$\mathcal{E}$$P(D):S^{r_{0}}arrow S^{r_{1}})$

i.e. the sheaf of -solutions to the system$S$$P(D)\vec{f}=0arrow$

.

(3)

9
Now the multiplication by the matrix
its cokernel$P^{t}$

maps$R^{r_{1}}$to$R^{r_{0}}$and we know that

$M= \frac{R^{r_{0}}}{P^{t}R^{r_{1}}}$.,

contains all the relevant information on (3), in view of the well known isomorphism$\mathcal{H}_{om_{R}}(M,S)\cong SP$.

In particular we know that Hilbert’s syzygy theorem (proved in its mostgeneral form by Palamodov [24] shows that there is a finite free resolution$)$$P^{t}$$Marrow.R^{\tau_{0}}arrow.$$0.arrow$.

R$T_{1}arrow R^{r_{2}}P_{2}^{t}.arrow$

.

.. .

$arrow R^{r_{m-1}}arrow R^{r_{m}}P_{m}^{t}arrow 0$(4)

for which $m\leq n$ .
of (4) is the complex
The$\mathrm{d}\mathrm{u}\mathrm{a}\mathrm{l}\dot{\mathrm{i}}\mathrm{z}\mathrm{e}\mathrm{d}$$0arrow R^{r_{0}}arrow R^{r_{1}}Parrow RP_{2\gamma_{2}}arrow\ldotsarrow R^{r_{m-1}}arrow R^{r_{m}}P_{m}arrow 0$(5)whose cohomology groups are denoted by Ext $(M, R).$ IntereStingl..v enough.
even though (4) and (5) are not uniquely defined, the $Ext$ groups (they are
actually -modules) are uniquely determined by $M$ .
Computing the resolution (4) amounts to computing the generators for the
kernel of polynomial homomorphisms or, in algebraic jargon, their syzygies.
This can be done explicitely (at least in principle) by using Gr\"obner bases
techniques, [3], and if the module $M$ is graded (i.e. if the matrix has homogeneous columns) one can actually compute a minimal lenght free resolution.
There are several reasons why (5) is an important object, and among them
the following result of Ehrenpreis et al. (see [15] for the complete references).
$R$

Theorem 3.1 If is a convex open (or convex compact) set $mR^{J1}$ , andthen the sequenceis a sheaf on$\Omega$$S$$JR^{n}$)$0arrow S^{P}(\Omega)arrow S(\Omega)T0s(P(arrow D)\Omega)^{r_{1}}P_{2}(D).P_{m}(Darrow..arrow S)(\Omega)^{r_{m}}arrow 0$

is $exact_{y}$ which in particular means that $P_{2}(.D)$ is a compatibility $sy_{St}.em$ for$P(D)$ .

10
On the other hand, the $Ext$ modules also carry much relevant information:, i.e. the columns of are linis$Ext0(M, R)=0$ means thatearly independent over (and the system corresponding to is determined[15] . Also, it is known that Ext $(M, R)=0$ is equivalent to the fact that(i.e. for any compact set
the Hartogs’ phenomenon holds foris connected, every element insuch thatelement in $S^{P}(R^{n}))$ ; thanks to the pioneering work of Palamodov, we also
$P$

$P$

$\mathrm{o}\mathrm{n}\mathrm{e}-\mathrm{t}_{0^{-}}\mathrm{o}\mathrm{n}\mathrm{e}$

$P$

$R$

$)$

$K\subseteq R^{n}$$S^{P}$$S^{P}(R^{n}\backslash K)\mathrm{e}\mathrm{x}1()[\iota(]_{\mathrm{L}}\mathrm{b}(_{()}t1\mathrm{l}1$$R^{n}\backslash K$

know that there are several important analytic consequences of the vanishingof the higher $Ext$ -modules (still related to various results on the removability of special singularities); the reader may refer to [11] and [24] for moredetails. Palamodov’s work also makes evident the strong geometric flavor ofthis entire construction; in fact, if is the characteristic variety associatedto the operator $P(D)$ , i.e. if is the set of conlmon zeroes of the maximal, then $dim(V)=n-p-1$ if and only if Ext $(M, R)=0$
size minors offor all $j\leq p$ and $Ext^{p+1}(M, R)\neq 0$ .
theBut let us now go back to the specific case in which
Cauchy-Fueter system given by (2). In this case the polynomial matrix is
given by$V$

$V$$[P_{ij}]$$P(D)\vec{f}=0\mathrm{i}\mathrm{s}arrow$.

$[_{\xi_{3}}^{\xi}\xi.2^{\cdot}n1\xi_{1}\xi_{\mathrm{o}n}\xi 3\xi\xi 01211nn1$$.\xi_{3n}\xi_{0}.n\xi_{3}\xi 0_{1}1--\xi--\xi.2n\xi 21\xi 111n$$-\cdot\xi-\xi_{2n}--\xi_{21}\xi_{1n}\xi 0^{\cdot}.n\xi 11\xi_{01}\xi_{31}3^{\cdot}n$.

$-\xi_{1}..n-\xi 3-\xi 11-\xi_{3}1\xi_{0}n\xi\xi 0\xi_{2}2\gamma n_{1}11$$]$

and in a series of papers [1], [4], [6], we have extensively studied the resolution).
for the -dimensional Cauchy-Fueter system (or, better, for its module
Let us summarize here our most significant results in this direction. We
refer the reader to our papers for their proofs.
$\mathcal{M}_{n}$$\mathrm{n}$Theorem 3.2 The projective dimension of the Cauchy-Fueter module
has a free resolution of length $2n-1$ and it is not possible
is $2n-1,$ $i.e$ .
.
to find a shorter resolution for$\mathcal{M}_{n}$$\mathcal{M}_{n}$$\mathcal{M}_{n}$Theorem 3.3 The characteristic variety$Ch(\mathrm{A}t_{n})$has dimension $2n+1$ and11

therefore

Ext $(\mathcal{M}_{n}, R)=0$$j=0,1,$$\ldots,$$2n-2$

while$Ext^{2n-1}(\mathcal{M}_{n}, R)\neq 0$.

Since, as it is well known, the Cauchy-Fueter system is elliptic, one candeduce the following important consequences:
Theorem 3.4 The sheaf
mension equal to $2n-1$ .
Theorem 3.5 If

is any open set in$U$of regular functions has flabby di-$\mathcal{R}=B^{P}=\mathcal{E}^{P}$$H^{i}(U, \mathcal{R})=0$,$H^{n}$, then$j\geq 2n-1$ .

The reader can see that these last two statements are the quaternionic analogues of well known results of Malgrange [30].
Another general consequence of the ellipticity of the system is the following result (see also [31]):
Theorem 3.6 Let $K$ be a compact convex set in$n$$[H^{0}(K, \mathcal{R})]’\cong H_{I\backslash }2’-1$

For the casesystem associated to$n=1_{f}B^{Q}\cong \mathcal{R}$( ffln,$IH^{n}$$\beta Q$

Then

).

isnothing but the, and for the case $n=2,$. The situation is more complex for $n\geq 3$ .
$Q$$\partial/\partial q_{2}-\partial/\partial q_{1}$As we pointed out earlier, the vanishing of the $Ext$-modules has relevant
applications to the removability of singularities of regular functions. We.
have the following two results:, and let $K$ be a
Theorem 3.7 Let be a convex connected open set inand setbe closed half-spaces incompact subset of . Let
extends. Then every regular function inU.. .
, which coincides with into a regular function infor $K’a$
compact subset of .
$H^{n}$$\Omega$$\Omega$$\Sigma=\Sigma_{1}$$\Sigma_{1)}\ldots,$$R^{4n}$$\Sigma_{2n-2}$$f$$\cup\Sigma_{2n-2}$$\Omega\backslash \Sigma$$f$$\Omega\backslash (K\cup\Sigma)$$\Omega\backslash (K’\cup\Sigma)$$\Omega$

Theorem 3.8 Let be a subspace of $H^{n}=R^{4n}$ of dimension $2n+2$ . Thenfor every compact-set $K\subseteq L$ , and every connected open set , relatively
every regular function defined in the neighborhood ofcompact in
can be extended to a regular function defined in a neighborhood of $K$ .
$L$$\Omega$$K_{f}$$K\backslash \Omega$

12Partial improvements of these general results have recently been obtainedin [21], where some classical ideas of Severi are used to remove compact
singularities in $IH\cross R$ and $lH\cross C$ ; also, in [21] we have shown how some
admit stronger singularity removability properties. It
subclasses of
may be worth pointing out that Theorems 3.7 and 3.8 can be improved for a
special subclass of regular functions defined in the recent [21]. These abstract
methods also provide a concrete way to represent all regular functions on any. Indeed, the Ehrenpreis-Palamodov Fundamental
open convex set
Principle implies the following representation:$\mathcal{R}(H^{2})$$\Omega\subseteq lH^{n}$

Then every regular

be an.. open convex set.
Theorem 3.9 Let
function in can be represented as$\Omega\subseteq H^{n}$$f$$\Omega$$f(q_{1}, \ldots, q_{n})=\int_{c}h(\mathcal{M}_{n})\mathrm{x}\mathrm{e}\mathrm{p}(\xi\cdot q)\cdot\overline{\xi}d\mu$

and $d\mu=$ .
where
densities supported on$(d\mu_{1}, \ldots , d\mu_{n})$$\xi\in \mathcal{H}^{n}f$$Ch(\mathcal{M}_{n})$andis a vector of quaternionic valued,, for every compact$K\subseteq\Omega$$sati_{S}fy\ddot{\iota}ng$$\int_{c}h(\mathcal{M}_{n})I)\exp(\max_{\mathrm{i}’}(\xi\cdot q)|d\mu|<+\infty$.

Note that this representation corresponds to a particular choice for the. Additional represennoetherian operator associated to the module
tations for regular functions in several quaternionic variables are also givenin the recent [25].
$A4_{n}$4Open problems

which appears in Theorem 3.2 is particularly important
4.1 The matrix
because it allow.s us toly.the . . atibility conditions for the
non-homogeneous system$P_{2}$

writ.

$\mathrm{c}\mathrm{o}\mathrm{m}$$\mathrm{p}$$\mathrm{e}\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{l}\mathrm{i}_{\mathrm{C}}\mathrm{i}\mathrm{t}.\mathrm{e}.$$.$$\frac{\partial f}{\partial\overline{q}_{1}}=g_{1}$

$\{$

.
.

$\cdot$$\frac{\partial f}{\partial\overline{q}_{n}}=g_{n}$,When $n=2$ , these conditions were first explicitely computed using
and are provided in [1], where they were expressed in terms of the 8 real$\mathrm{c}_{0}\mathrm{c}_{0}\mathrm{A}$13
coordinates; this gives an 8
matrix, explicitely described in [I]. It is
however possible to read these conditions in purely quaternionic terms. In
this case the 8
real matrix becomes the very simple 2
quaternionic.
matrix$\cross 8$$\cross 8$$\cross 2$

’

$‘$

.
$[ \frac{\partial^{2}}{\frac{\partial\overline{q}_{2\partial^{2}}\partial q1}{\partial\overline{q}_{2}\partial q_{2}}}$$\cdot$.

$- \frac{\frac{\partial^{2}}{\partial\overline{q}_{1\partial^{2}}\partial q_{1}}}{\partial\overline{q}_{1}\partial rq2}-]$

which can be interpreted by saying that, on every convex set
system$\Omega\subseteq H^{2}$, the$\frac{\partial f}{\partial\overline{q}_{1}}=g_{1}$$\{$$\frac{\partial f}{\partial\overline{q}2}=g_{2}$has a solution if and only if

It is possible to obtain such conditions from a purely analytic point of view
(but, as far as we can see now, only their necessity can be established insuch fashion). For $n=3$ , the matrix
is a 12matrix, whose entries
are homogeneous second degree polynomials, and which we have calculated
once again using. This means that we now expect 10 quaternioniccompatibility conditions, represented by second order differential operators,
but, up to now, we have been unable to describe such conditions on a purely
analytic basis. In particular, it is known that [17] for arbitrary , the matrix
is a $4n\cross p(n)$ matrix, with
$P_{2}$$\cross 40$$\mathrm{c}_{0}\mathrm{c}_{0}\mathrm{A}$$n$$P_{2}$$p(n)=2+4$ .

$+$Up to now, only
such syzygies can be obtained by purely analyticalmethods.
Problem: Give an analytic interpretation for the matrixof the firstsyzygies, for any $n\geq 2$ .
$P_{2}$

4.2 In all the examples which we have explicitely computed, we have seenthat (with the exception of ) all the matrices appearing in Theorem 3.2
have entries which are degree one polynomials. However we have been unable$P_{2}$

14
to prove this fact in general, nor do we understand its analytical meaning.
may cause
In [21] we have shown how smallof the module
this phenomenon to disappear.
Problem: Prove that all the matrices which appear in Theorem 3.2 haveentries which are degree one polynomials, with the exception of .
$\mathcal{M}_{n}$$\mathrm{m}.\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{i}_{\mathrm{C}}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}\mathrm{S}}$$P_{2}$

4.3 When $n=2$ , the syzygies are expressed by the following equations:$=0arrow$$=0arrow$(which are explicitely constructed in [1], andso that the matrices andexpressed in real terms) originate, respectively, the sheaf of regular functions
is not soand the sheaf of anti-regular functions. For $n=3$ the matrix
simple anymore, and in general we do not know the relationship between the, with the exception of the interesting fact thatandandsheaveshave the same characteristic variety; this is an immediate consequence of ourTheorem 3.3 and the general theory of Palamodov [24].
.
Problem: Establish a relationship between
and
$P$

$Q$

$Q$

$\mathcal{R}$$B^{Q}$$P$$Q$$B^{Q}$$\mathcal{R}$

4.4 The results of section 3 point to the possibility of developing a quaternionic hyperfunction theory, and in particular to the possibility to construct,bisa (flabby?) sheaf via the relative cohomology of . In order
to be developed, we need to prove a quaternionic analog of the classical theorem on the pure -codimensionality ofin , with respect to ; such atheorem is equivalent to the fact that$\mathcal{R}$$IR^{n}$$n$$H_{R^{n}}^{j}(\sigma n, \mathcal{O})=0$$\mathrm{f}\mathrm{o}1^{\cdot}1$$(_{}\}_{1\mathrm{e}}\mathrm{o}1^{\cdot}.\mathrm{y}\cdot$$C^{n}$$\mathcal{O}$$\forall j\neq n$and$H_{R^{n}}^{n}(\sigma^{n}, \mathcal{O})\neq 0$.

The role of the pairis clear, but we do not have a quaternionicanalogue, partly because we do not have a full understanding of the CauchyKowalewski phenomenon for general matrix systems. For the case $n=1$ , wehave proved [12], that, for $j\neq 1$ andis in fact asheaf of hyperfunctions.
$(C^{n}, R^{n})$$H_{\overline{H}}^{j}(H, \mathcal{R})=0$$\cdot$$H_{\tilde{H}}^{1}(H, \mathcal{R})$15
Problem: Find a $(2n+1)$ -dimensional variety$H_{S}^{j}(ffl^{n}, \mathcal{R})=0$$S$

in $H^{n}=R^{4n}$ such that$\forall j\neq 2n-1$

and$fI_{s^{n-}}^{21}( \int fIn, \mathcal{R})\neq 0$.

4.5 When $n=2$ , Theorem 3.8 shows that compact singularities containedin subspaces of dimension 6, 7, 8 can always be removed. On the other hand,
it is obviously impossible to extend this result to singularities contained insubspaces of dimension 4 (just consider the origin in). The caseof dimension 5 cannot be solved by purely algebraic methods, because
moduleassociated to the Cauchy-Fueter system has
but. In [21] we have shown that at least for some subclasses
of , even the case of dimension 5 can be answered in the affirmative.
Problem: Can Theorem 3.8 be extended, when $n=2$ , to include all subspaces.of dimension 5?
$IH\llcorner_{arrow}H^{2}$$\mathrm{t},\mathrm{t}\downarrow\in$)$Ext^{2}(\mathrm{A}\not\in_{2}, R)=0$$\mathcal{M}_{2}$$Ext^{3}(\mathcal{M}_{2}, R)\neq 0$$\mathcal{R}$

4.6 Theorem 3.1 holds when the open set
is convex. This hypothesis,that is requested to assure that the space
is LAU, is too strong. In fact,(as it was proved in [11]), when is not convex the theorem still holds under
the weaker hypothesis of -convexity. This is equivalent to a suitable relation
between the system $P$ and the topological structure of . It has not been
investigated, up to now, what is the characterization of $P$ -convexity whenis the Cauchy-Fueter system; this characterization will probably depend on
possibility to dcfinc a notion of a tangential Cauchy-Fueter operator.
Problem: Give a geometric characterization of the $P$ -convex sets.
$\Omega\subseteq IR^{n}$$S(\Omega)$$\Omega$$P$$\Omega$$P$$\mathrm{t}1_{1}\mathrm{e}$4.7 Another interesting problem is related to a possible construction of a
Weyl algebra, where and are suitable variables and operators
respectively. The answer is not trivial, in fact we have, apparently, a double
non commutativity in both the coefficients and the variables. In the literature, as far as we know, some examples of Weyl algebras on a general ringare provided. Unfortunately it seems that they do not work in our case, infact we have to keep in mind that in order to study systems of differential
equations, we need to have a Weyl algebra $A_{1}(H)$ such that the spaces
and $F(U)$ can be considered as $A_{1}(H)$ -modules.
Problem: Construct a quaternionic Weyl algebra.
$H[x, \partial]$$x$$\partial$$\mathcal{R}(U)$16
4.8 The theory of regular functions in one quaternionic variable seems to becompletely understood. In spite of this, the notion of exponential function
seems to cause some troubles when one tries to restore a Fourier transformtheory for (compactly supported) $lH$ -hyperfunctions. It is well known thatin order to have a regular function, it is necessary to define the exponential
as follows$\exp(q):--\sum_{n=0}^{+}\sum\infty\nu P_{\nu}(q)$.

By using this function (that can be considered as the C-K product of $\exp(z_{1}e_{1})$ ,, it is possible to define the Fourier transform. . ., $\exp(z_{3}e_{3}),$function or a distribution (it suffices to follow theof a rapidly decreasingideas in [7] and also to prove the analogue of the Paley-Wiener theorem.
The usual properties of the Fourier transform hold up to some ”reflection operators” The Fourier transform of a $IH$ -hyperfunction seems to be a quite
complicated object, and the Paley-Wiener theorem (that is a key tool in esswith support
tablish the isomorphism between the space ofof entire functions of exponential
in the compact $K$ and thetype) has not been proved.
Problem: Prove a Paley-Wiener Theorem for the Fourier transform of $H-$
hyperfunctions.
$z_{i}=x_{i}-e_{i}x\mathrm{o})$$C^{\infty}$$)$$H-\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\tilde{\mathrm{n}}$$\mathrm{s}_{\mathrm{P}\mathrm{P}}.\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{E}\mathrm{X}(K)$

4.9 In Quaternionic Quantum Mechanics, physicists use a different notion of
the exponential function that is the formal analogue of $\exp(z)$ . Obviously thisis not a regular function, but this is the function used to construct scatteringmatrices (see e.g. [5]). It is known that, at least in the cornplex case, the elements of the scattering matrix are boundary values of holomorphic functions
and can be considered as microfunctions whose support gives the directionof causality of the interaction. To extend this approach in the quaternioniccase would be very helpful, since it allows to consider various directions of
the interactions but, at the same time, we have that the non-regularity ofthe exponential function used in this case is an obstruction to restore thetheory.
Problem: It is possible to prove that the elements of the quaternionic scatsense of thetering. matrix are boundary values of regular func.t ions (in?
hyperfunctions which we have introduced$\mathrm{t}1_{1}\mathrm{c}$$\mathrm{i}\mathrm{n}[12])$4.10 Both physical and mathematical considerations seem to imply that it

17
may be interesting to carry out our kind of analysis in situations that, atleast formally, seem to be analogous to the quaternionic case. In [16] Shapiro
and Kravchenko study the so-called Moisil-Theodoresco system which isthe 3-dimensional analogue (and historical precursor) of the Cauchy-Fueter
system. Since functions in the kernel of the Moisil-Theodoresco system havean intrinsic physical interest and since they have a natural regular extension
to all of $H$ , we are interested in the following problems :Problem: Apply our algebraic analysis to the Moisil-Theodoresco system.
Problem: Is it possible to use these results to get information on the
Cauchy-Fueter system?
4.11 Recently, the authors with a group of collaborators have started to
study some hypercomplex operators that are the linearization of some ultrahyperbolic operators. The importance of these operators (wave operators) inPhysics is well known. In [8] we have introduced an operator whose kernel
contains, as a particular case the electromagnetic fields, i.e. functions withvalues in the space of biquaternions, solutions of the Maxwell’s system. In
the forthcoming [9] we provide the analysis of an operator that linearizes
the Klein-Gordon equation using octonions. In this setting there are several
open problems.
Problem: Given a hypercomplex algebra study all the possible families
operators linearizing the wave operator.
Problem: Study the fundamental solution of such operators
provideintegral formulas for the functions in their kernel.
$0\dot{\mathrm{f}}$a.n$\mathrm{d}$4.12 The original idea of Fueter was to consider holomorphic functions of two
variables as special cases of regular functions of one quaternionic variable.
An analogous approach can be taken by considering regular functions of twoquaternionic variables as special cases of regular functions of some other kindof variable. For example, regular functions in two quaternionic variables canbe seen as regular functions of one biquaternionic variable ([8]) or as regularfunctions of one .uitable Clifford variable [27]. So wethe following
general.
Problem: Given an hypercomplex algebra $A$ and the notion of $A$-regularity,
notion of -regularity
find a hypercomplex algebra and asuch that every $A$-regular function of several variables a particular casf 01-regular function of one variable.

can. .ose$\mathrm{s}$$\mathrm{p}$$\cdot$$C$$C$$\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{s}_{\mathrm{P}^{\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\tilde{\mathrm{n}}}\mathrm{g}}$$\dot{1}\mathrm{S}$

$C$

18
4.13 An octonionic analogue of the theory of regular functions has beendeveloped by Sce and Dentoni in [10] and more recently revisited by Nonoin [23]. There are at least two possible ways in which this approach canbe followed up. On one hand, Sce, Dentoni and Nono defined a CauchyFueter-Cayley operator which linearizes the laplacian in analogy of what
happen in and $lH$ . In [8] however we have shown how biquaternions can beutilized to factorize more interesting ultra-hyperbolic operators and therefore
it seems reasonable to introduce a notion of bioctonions to linearize similar
higher dimension operators. Such an approach is currently being taken in[9]. On the other hand, it has been suggested by Ehrenpreis that it may beinteresting to utilize our approach to develop a theory of regular functionsof several octonion variables.
Problem: Develop such a theory, compute the flabby dimension of the sheafof octonionic regular functions, and construct an analogue of the MoisilTheodoresco operator in such a context.
$C$

The list of problems in this framework is far from being complete. It is ourhope that this paper will stimulate further researches in the area.

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