# Analytic Pseudodifferential Operators for the Heisenberg Group and Local Solvability. (MN-37):

Daryl Geller
Pages: 504
https://www.jstor.org/stable/j.ctt7zvp08

1. Front Matter
(pp. [i]-[vi])
(pp. [vii]-2)
3. Introduction
(pp. 3-68)

The main purpose of this book is to develop a calculus of pseudodifferential operators for the Heisenberg group ℍn, in the (real) analytic setting, and to apply this calculus to the study of certain operators arising in several complex variables. Our main new application is the following theorem (Theorem 10.2 and Corollary 10.3):

1. Suppose M is a smooth, compact CR manifold of dimension 2n + 1. Suppose$\mathcal{U}\subset \text{M}$is open and is a real analytic strictly pseudoconvex CR manifold. Further suppose:

(i) There is a smooth, bounded pseudoconvex domain$\text{D} \subset {{\mathbb{C}}^{\text{n}+1}}$with boundary M. (D may be weakly pseudoconvex.) Let S...

4. 1. Homogeneous Distributions
(pp. 69-104)

We assume given an n-tuple of positive rationals$\underset{\tilde{\ }}{\mathop{\text{a}}}\,=({{\text{a}}_{1}},\ldots ,{{\text{a}}_{\text{n}}})$. We put$\text{Q}=\sum{{{\text{a}}_{\ell }}}$. For$\text{x }\!\!\varepsilon\!\!\text{ }{{\mathbb{R}}^{\text{n}}}$we put${{\text{D}}_{\text{r}}}\text{x=(}{{\text{r}}^{{{\text{a}}_{\text{1}}}}}{{\text{x}}_{\text{1}}}\text{,,}{{\text{r}}^{{{\text{a}}_{\text{n}}}}}{{\text{x}}_{\text{n}}}\text{)}$. For f a function on ℝn, r > 0, we define the functions Drf,Drf by Drf(x) = f(Drx),$({{\text{D}}^{\text{r}}}\text{F}|\text{g})={{\text{r}}^{-{{\text{Q}}_{\text{D}}}1/{{\text{r}}_{\text{f}}}}}$. For$\text{F }\!\!\varepsilon\!\!\text{ }{{\mathcal{S}}^{'}}({{\mathbb{R}}^{\text{n}}})$we define${{\text{D}}^{\text{r}}}\text{F,}{{\text{D}}_{\text{r}}}\text{F }\!\!\varepsilon\!\!\text{ }{{\text{S}}^{\text{ }\!\!'\!\!\text{ }}}$by (DrF|g) = (F|Drg), (DrF|g) = (F|Drg) for$\text{g }\!\!\varepsilon\!\!\text{ }\mathcal{S}$. (Here and elsewhere (F|g) denotes the sequilinear pairing, linear in g). For k ε ℂ we say that F is hcmogeneous of degree k if DrF = rkF for all r > 0. We let$\text{Rho}{{\text{m}}_{\text{k}}}=\text{Rho}{{\text{m}}_{\text{k}}}(\underset{\tilde{\ }}{\mathop{\text{a}}}\,)=\left\{ {{\text{K}}_\varepsilon}\text{S }\!\!'\!\!\text{ }\left| \text{K} \right. \right.$is homogeneous of degree k and is Caway from 0}....

5. 2. The Space $\text{Z}_{\text{q,j}}^{\text{q}}$
(pp. 105-145)

In this section, unless otherwise stated, q is an arbitrary real number which is greater than 1, and (1/p) + (1/q) = 1. Also, unless otherwise stated, j will be an arbitrary complex number.

The subspace$\left\{ \text{f }\!\!\varepsilon\!\!\text{ RZ}_{\text{q,j}}^{\text{q}}|\text{f}\sim 0 \right\}$is especially important, and we begin by studying it.

We define$\text{Z}_{\text{q}}^{\text{q}}=\{\text{entire functions f on }{{\mathbb{C}}^{\text{S}}}|$for some B1,B2,C > 0 we have for$\left| \text{f( }\!\!\zeta\!\!\text{ )} \right|\textless\text{C}{{\text{e}}^{{{\text{B}}_{\text{1}}}{{\left| \text{ }\!\!\zeta\!\!\text{ } \right|}^{\text{q}}}}}$for ζ ε ℂS, while$\left| \text{f( }\!\!\xi\!\!\text{ )} \right|\text{C}{{\text{e}}^{\text{-}{{\text{B}}_{\text{2}}}{{\left| \text{ }\!\!\xi\!\!\text{ } \right|}^{\text{q}}}}}$for$\text{ }\!\!\xi\!\!\text{ }\!\!\varepsilon\!\!\text{ }{{\mathbb{R}}^{\text{S}}}\}$, If we need to make the constants B1,B2,C explicit, we shall very occasionally write$\text{f }\!\!\varepsilon\!\!\text{ Z}_{\text{q}}^{\text{q}}\text{(}{{\text{B}}_{\text{1}}}\text{,}{{\text{B}}_{\text{2}}}\text{,C)}$. This space was first investigated by Gelfand and Šilov ([21], [22]). (In the latter, and later, reference the...

6. 3. Homogeneous Partial Differential Equations
(pp. 146-167)

We proceed to examine the implications of the theory we have developed to the study of analytic hypoellipticity of homogeneous partial differential operators. As our first (well known) proposition shows, the constant-coefficient case is not the place to begin.

Proposition 3.1. Let P(∂) be a constant-coefficient differential operator on${{\mathbb{R}}^{\text{n}}}$which is homogeneous with respect to the dilations${{\text{D}}_{\text{r}}}\text{x=(}{{\text{r}}^{{{\text{a}}_{\text{1}}}}}{{\text{x}}_{\text{1}}}\text{,,}{{\text{r}}^{{{\text{a}}_{\text{n}}}}}{{\text{x}}_{\text{n}}}\text{)(}{{\text{a}}_{\text{1}}}\text{,,}{{\text{a}}_{\text{n}}}\text{ }\!\!\varepsilon\!\!\text{ }\mathbb{N})$. Then P(∂) is analytic hypoelliptic if and only if a1=…=anand P(∂) is elliptic.

Remark. In fact, by [45], Vol. II, Corollary 11.4.13, one can drop the "homogeneous" hypothesis; if P(∂) has constant coefficients, P(∂) is analytic hypoelliptic if and...

7. 4. Homogeneous Partial Differential Operators on the Heisenberg Group
(pp. 168-201)

The Heisenberg group ℍnis the Lie group with underlying manifold$\mathbb{R}\times {{\mathbb{C}}^{\text{n}}}$and multiplication$(\text{t,z) }\!\!\cdot\!\!\text{ (}{{\text{t}}^{\text{ }\!\!'\!\!\text{ }}}\text{,}{{\text{z}}^{\text{ }\!\!'\!\!\text{ }}}\text{)=(t+}{{\text{t}}^{\text{ }\!\!'\!\!\text{ }}}\text{+2 Im z}\text{.}{{\text{\bar{z}}}^{\text{ }\!\!'\!\!\text{ }}}\text{,z+}{{\text{z}}^{\text{ }\!\!'\!\!\text{ }}}\text{)}$, where$\text{z }\!\!\cdot\!\!\text{ }{{\text{\bar{z}}}^{\text{ }\!\!'\!\!\text{ }}}\text{=}\sum\limits_{\text{j=1}}^{\text{n}}{{{\text{z}}_{\text{j}}}{{{\text{\bar{z}}}}_{{{\text{j}}^{\text{ }\!\!'\!\!\text{ }}}}}}$. The dilations Drare given by Dr(t,z) = (r2t, rz); thus we are in the case p = 2 of the previous chapters. Write zj= xj+ iyj; then xj= ∂/∂xj+ 2yj∂/∂t, Yj= ∂/∂yj- 2xj∂/∂t and T = ∂/∂t give the basis of left-invariant vector fields agreeing with ∂/∂xj, ∂/∂yj, ∂/∂t at 0. The right-invariant analogues are$\text{X}_{\text{j}}^{\text{R}}=\partial /\partial {{\text{x}}_{\text{j}}}-2{{\text{y}}_{\text{j}}}\partial /\partial \text{t},\text{Y}_{\text{j}}^{\text{R}}=\partial /\partial {{\text{y}}_{\text{j}}}+2{{\text{x}}_{\text{j}}}\partial /\partial \text{t}$, and T. Put${{\text{Z}}_{\text{j}}}=(1/2)(\text{X}{}_{\text{j}}\text{-i}{{\text{Y}}_{\text{j}}})=\partial /\partial {{\text{z}}_{\text{j}}}+\text{i}{{\text{\bar{z}}}_{\text{j}}}\partial /\partial \text{t}$,${{\text{\bar{Z}}}_{\text{j}}}=\partial /\partial {{\text{\bar{z}}}_{\text{j}}}-\text{i}{{\text{z}}_{\text{j}}}\partial /\partial \text{t}$; similarly one has$\text{Z}_{\text{j}}^{\text{R}}=\partial /\partial {{\text{z}}_{\text{j}}}-\text{i}{{\text{\bar{z}}}_{\text{j}}}\partial /\partial \text{t}$and$\text{\bar{Z}}_{\text{j}}^{\text{R}}$. We have commutation...

8. 5. Homogeneous Singular Integral Operators on the Heisenberg Group
(pp. 202-255)

Corollary 4.10 gives a necessary and sufficient condition for a homogeneous left-invariant differential operator L*on ℍnto be analytic hypoelliptic. What is the same thing, if L has degree k, it gives a necessary and sufficient condition for there to exist${{\text{K}}_{\text{2}}}\text{ }\!\!\varepsilon\!\!\text{ }\mathcal{A}{{\mathcal{K}}^{\text{k-2n-2}}}$such that LK2=δ. Let K1= Lδ; then${{\text{K}}_{\text{1}}}\text{ }\!\!\varepsilon\!\!\text{ }\mathcal{A}{{\mathcal{K}}^{\text{-k-2n-2}}}$, and we have the precise conditions under which there exists${{\text{K}}_{\text{2}}}\text{ }\!\!\varepsilon\!\!\text{ }\mathcal{A}{{\mathcal{K}}^{\text{k-2n-2}}}$with K2*K1=δ(since LK2= L(K2*δ) = K2*Lδ. In this chapter we answer the more general question: if$\text{k }\!\!\varepsilon\!\!\text{ }\mathbb{C}$,${{\text{K}}_{\text{1}}}\text{ }\!\!\varepsilon\!\!\text{ }\mathcal{A}{{\mathcal{K}}^{\text{-k-2n-2}}}$, when does there exist${{\text{K}}_{\text{2}}}\text{ }\!\!\varepsilon\!\!\text{ }\mathcal{A}{{\mathcal{K}}^{\text{k-2n-2}}}$with K2*K1=δ? (The...

9. 6. An Analytic Weyl Calculus
(pp. 256-284)

In this chapter, we shall interpret the results of Chapters 4 and 5 in the Schro̎dinger representation, in order to discuss an interesting calculus of pseudodifferential operators on${{\mathbb{R}}^{\text{n}}}$. The Cversion of this calculus was first investigated by Grossman, Loupias and Stein [37]. Its connection with the matters discussed in Chapters 4 and 5 was noted previously by Howe [49], Melin [60] and Taylor [79]. Further results about this calculus follow as special cases of the results of Beals [7]. We shall introduce a new analytic analogue. The results of this chapter will not be used later on....

10. 7. Analytic Pseudodifferential Operators on ℍn: Basic Properties
(pp. 285-375)

In this section, we shall examine the following situation. Say$\text{k}\varepsilon \mathbb{C},\operatorname{Re}\text{k}\geqq 0$, and let {Km} be a sequence of distributions on ℍn, with${{\text{K}}^{\text{m}}}\varepsilon \mathcal{A}{{\mathcal{K}}^{\text{k+m}}}$for all m. (m is a superscript, not a power.) Suppose further that there are constants C1, R1such that$\left| {{\partial }^{\text{ }\!\!\gamma\!\!\text{ }}}{{\text{K}}^{\text{m}}}(\text{u}) \right|\textless {{\text{C}}_{1}}\text{R}_{1}^{\text{m+}{{\left\| \text{ }\!\!\gamma\!\!\text{ } \right\|}_{{{\text{ }\!\!\gamma\!\!\text{ }}^{\text{!}}}}}}\text{ whenever }1\leqq \left| \text{u} \right|\leqq 2,\text{m }\!\!\varepsilon\!\!\text{ Z}{{\text{Z}}^{\text{+}}}\text{, }\!\!\gamma\!\!\text{ }\!\!\varepsilon\!\!\text{ (Z}{{\text{Z}}^{\text{+}}}{{\text{)}}^{\text{2n+1}}}.$ (7.1)

Here$\partial =(\partial /\partial \text{t},\partial /\partial {{\text{x}}_{1}},\ldots ,\partial /\partial {{\text{x}}_{n}},\partial /\partial {{\text{y}}_{1}},\ldots ,\partial /\partial {{\text{y}}_{n}})$. Under these hypotheses, we want to understand how the constants in Theorem 2.11 depend on m, and to also obtain similar information in a converse direction.

Our motivation is that we shall soon be considering series of the type$\sum\limits_{\text{m}=0}^{\infty }{{{\text{K}}^{\text{m}}}},{{\text{K}}^{\text{m}}}$as above. (7.1) is an extremely natural condition in this context. As...

11. 8. Analytic Parametrices
(pp. 376-422)

In any good analytic calculus there must be a simple criterion for inversion of operators, modulo operators with analytic kernels. What we are going to show now is that if${{\text{K}}_{\text{1}}}\sim \sum{\text{K}_{\text{1}}^{\text{m}}\varepsilon {{\mathcal{C}^{\text{k}}}(\text{U})}$if${{\text{U}}^{'}}$is open,${{\text{\bar{U}}}^{'}}$compact and${{\text{\bar{U}}}^{'}}\subset \text{U}$; and if there exists$\text{K}_{2\text{u}}^{\text{O}}\varepsilon {\mathcal{A}}{{\mathcal{K}}^{{{\text{k}}_{2}}}}$satisfying (7.98) for$\text{u}\varepsilon {{\text{U}}^{'}}$, then there exists${{\text{K}}_{\text{2}}}\text{:}\sum{\text{K}_{\text{2}}^{\text{m}}}\text{ }\!\!\varepsilon\!\!\text{ }{{\mathcal{C}}^{{{\text{k}}_{\text{2}}}}}\text{(}{{\text{U}}^{\text{ }\!\!'\!\!\text{ }}}\text{)(}{{\text{k}}_{\text{1}}}\text{+}{{\text{k}}_{\text{2}}}\text{+2n+2=-2n-2)}$so that (7.99) is also satisfied on${{\text{U}}^{'}}$. Thus,${{\mathcal{K}}_{\text{1}}}=\mathcal{O}({{\text{K}}_{\text{1}}})$is analytic hypoelliptic in the sense of Theorem 7.11(d). Thus for operators whose "principal cores"$\text{K}_{\text{1}}^{\text{O}}$satisfy (7.98) for some$\text{K}_{\text{2}}^{\text{O}}$, we have analytic hypoellipticity. Further, the necessary and sufficient conditions for (7.98) to...

12. 9. Applying the Calculus
(pp. 423-452)

In this chapter we derive a number of other results about the calculus, which are useful in applications. The chapter ends with a proof that the Kohn Laplacian, and a parametrix for it, lie in the systems analogue of the calculus after a contact transformation, under natural hypotheses.

1. When using the calculus, it is frequently simplest to work with formal sums. To make this easier to do, we add some elementary facts to (8.4) and Proposition 8.3, for later reference.

(i) Suppose$\text{K }\!\!\varepsilon\!\!\text{ }{{C}^{\text{k}}}(\text{U}),K=O(\text{K}),{{\text{U}}_{\text{O}}}\subset \text{U},{{\text{U}}_{\text{O}}}$open, andKf = 0 on UOfor all$\text{f }\!\!\varepsilon\!\!\text{ C}_{\text{C}}^{\infty }({{\text{U}}_{\text{O}}})$. Then Ku≡ 0 for all...

13. 10. Analytic Pseudolocality of the Szego̎ Projection and Local Solvability
(pp. 453-488)

Let M be a smooth compact CR manifold of dimension 2n+l. Suppose$\mathcal{U}\subset \text{M}$is open, real analytic, and strictly pseudoconvex. Under a further "global" assumption on M, which is automatically satisfied if M is the boundary of a bounded smooth pseudoconvex domain in (${{\mathbb{C}}^{\text{n}+1}}$, or more generally, if the range of${{\bar{\partial }}_{\text{b}}}:{{\text{C}}^{\infty }}(\text{M})\to {{\Lambda }^{0,1}}(\text{M})$is closed in the Ctopology, we shall show that the Szego̎ projection S on M is "analytic pseudolocal" on$\mathcal{U}$. That is:

(i) If$\text{V}\subset U$is open, and f is real analytic on V, then Sf is also real analytic on V.

Among other consequences,...

14. References
(pp. 489-495)