# Estimates of the Neumann Problem. (MN-19)

PETER CHARLES GREINER
E. M. STEIN
Pages: 204
https://www.jstor.org/stable/j.ctt130hk1v

1. Front Matter
(pp. [i]-[ii])
2. Preface
(pp. [iii]-[iv])
(pp. [v]-[vi])
4. Introduction
(pp. 1-9)

Let M be an open relatively compact subset of a complex manifold M′ of dimension n+1, and assume that the boundary of M, bM, is smooth and strongly pseudo-convex. The$\overline{\partial }$-Neumann problem for M arises when one tries to solve the Cauchy-Riemann equations on M

(1)$\overline{\partial }\text {U}=\text{f}$,

where f is a given (0, 1) form with

(2)$\overline{\partial }\text {f}=\text{0}$,

and in particular when one wishes to have control on the behavior of U near the boundary in terms of similar control over f.

Part of the difficulty of this problem is connected with the fact that (1) is...

5. Part I Analysis on the Heisenberg group
• Guide to Part I
(pp. 10-12)

Notations. Let${\cal {D}}$be the domain$\left\{ \left( {{\text{z}}_{1}},\ldots,{{\text{z}}_{\text{n}+1}} \right),\ \operatorname{Im}{{\text{z}}_{\text{n+1}}}>{{\left| {{\text{z}}_{1}} \right|}^{2}}+{{\left| {{\text{z}}_{2}} \right|}^{2}}\cdot \cdot \ +{{\left| {{\text{z}}_{\text{n}}} \right|}^{2}} \right\}$.* The subgroup of holomorphic self-mappings of${\cal {D}}$which consist of "translations" is called the Heisenberg group Hn. Since it acts simply-transitively on the boundary of${\cal {D}}$, it can be naturally identified with this boundary.

If$(\zeta ,\text{t})\in {{\mathbb{C}}^{\text{n}}}\times \mathbb{R}$denotes a point of Hn, then it is identified with the boundary point given by$\caption {(0.1)} {{\text{z}}_{\text{n}+1}}=\text{t+i}{{\left| \zeta \right|}^{2}},\ ({{\text{z}}_{1}},{{\text{z}}_{2}},\ldots,{{\text{z}}_{\text{n}}})=\zeta$

The multiplication law on Hnis$\caption {(0.2)} (\zeta ,\text{t})\cdot ({\zeta }',\text{{t}}')\text{=}(\zeta \text{+}{\zeta }',\text{t+{t}}'+2\ \text{Im}\zeta \cdot {\bar{\zeta }}'\text{)}\text{.}$

Left-invariant vector fields on Hnare given by the Zj(see (1.1)) which are restrictions to$\text{b}{\cal {D}}$of holomorphic vector fields, tangential at$\text{b}{\cal {D}}$. (We use the notation$\text{t=}{{\text{x}}_{\text{0}}},\,{{\text{z}}_{\text{j}}}={{\text{x}}_{\text{2j-1}}}+\text{i}{{\text{x}}_{\text{2j}}}\,\text{j=1,}\ldots\text{,n}$). The...

• Chapter 1. Symbols on the Heisenberg groups
(pp. 13-26)

Let$\caption {(1.1)} {{\text{Z}}_{\text{j}}}=\frac{\partial }{\partial {{\text{z}}_{\text{j}}}}+{{\overline{\text{z}}}_{\text{j}}}\frac{\partial }{\partial \text{t}},\ \text{j=1,}\ldots\text{,n}$be the usual left invariant vector fields on the Heisenberg group Hn. Following the notation and terminology of Folland-Stein [ 9 ] we define$\caption {(1.2)} {\cal{L}}_{\alpha }=-\frac{1}{2}\sum\limits_{\text{j}=1}^{\text{n}}{({{\text{Z}}_{\text{j}}}{{\overline{\text{Z}}}_{\text{j}}}+{{\overline{\text{Z}}}_{\text{j}}}{{\text{Z}}_{\text{j}}})+\text{i}\alpha \frac{\partial }{{{\partial }{\text{t}}}}}$$=\sum\limits_{\text{j}=1}^{\text{n}}{\left( -\frac{{{\partial }^{2}}}{\partial {{\text{z}}_{\text{j}}}\partial {{\overline{\text{z}}}_{\text{j}}}}-|{{\text{z}}_{\text{j}}}{{|}^{2}}\frac{{{\partial }^{2}}}{\partial {{\text{t}}^{2}}}-\frac{\partial }{\text{i}\partial \text{t}}\left( {{\text{z}}_{\text{j}}}\frac{\partial }{\partial {{\text{z}}_{\text{j}}}}-{{\overline{\text{z}}}_{\text{j}}}\frac{\partial }{\partial {{\overline{\text{z}}}_{\text{j}}}} \right) \right)}+\text{i}\alpha \frac{\partial }{\partial \text{t}}.$

The purpose of this chapter is to compute the symbol of the fundamental solution of ℒα. Let Gαdenote a (presumptive) fundamental solution and set$\caption {(1.3)} {{\text{G}}_{\alpha }}(\text{t})=\frac{1}{2\pi}\int_{-\infty }^{\infty }{\text{G}_{\alpha }^{\Lambda }({{\xi}_{\text{0}}}){{\text{e}}^{\text{it}{{\xi}_{\text{0}}}}}\text{d}{{\xi}_{\text{0}}}.}$

Taking ℒαunder the Fourier transform we obtain the operator$\caption {(1.4)} {\cal{L}}_{\alpha }^{\Lambda }=\sum\limits_{\text{j}=1}^{\text{n}}{\left( -\frac{{{\partial }^{2}}}{\partial {{\text{z}}_{\text{j}}}\partial {{\overline{\text{z}}}_{\text{j}}}}+|{{\text{z}}_{\text{j}}}{{|}^{2}}\xi_{0}^{2}-{{\xi}_{0}}\left( {{\text{z}}_{\text{j}}}\frac{\partial }{\partial {{\text{z}}_{\text{j}}}}-{{\overline{\text{z}}}_{\text{j}}}\frac{\partial }{\partial {{\overline{\text{z}}}_{\text{j}}}} \right) \right)}-\alpha {\xi}_{0}.$

Assume${{\xi}_{\text{0}}}>0$. (The case${{\xi}_{\text{0}}}<0$will follow by replacing α by −α.) Since we want Gαto act on the Heisenberg group by convolution we shall try a kernel of the following form$\caption {(1.5)}\text{G}_{\alpha }^{\Lambda }={{\text{e}}^{{{\xi }_{0}}(\text{z}{\bar{\text w}}-\bar{\text{z}}\text {w}})}\text{W}({{\xi }_{0}}{{\left| \text{z}-\text{w} \right|}^{2}}),$...

• Chapter 2. A comparison
(pp. 27-35)

Set$\caption {(2.1)} {{\Phi }_{\alpha }}=\frac{1}{{{\text{c}}_{\alpha }}}{{\varphi }_{\alpha }}=\frac{\Gamma \left( \frac{\text{n}+\alpha }{2} \right)\Gamma \left( \frac{\text{n}-\alpha }{2} \right)}{{{2}^{2-2\text{n}}}{{\pi}^{\text{n}+1}}}{{\varphi }_{\alpha }},$where$\caption {(2.2)} {{\varphi }_{\alpha }}=\frac{1}{{{\left( |\text{z}{{|}^{2}}-\text{it} \right)}^{\frac{\text{n}+\alpha }{2}}}{{\left( |\text{z}{{|}^{2}}+\text{it} \right)}^{\frac{\text{n}-\alpha }{2}}}}$

In (2. 1) we assume$\frac{\text{n}\pm \alpha }{2}\ne 0, -1, -2, \ldots$. According to Proposition 7.1 of [ 9 ] the operator Kα, defined by${{\text{K}}_{\alpha }}\text{f=f }*\text{ }{{\Phi }_{\alpha }}$$=\text{c}_{\alpha }^{-1}\int_{{{\text{H}}_{\text{n}}}}{\text{f(y)}{{\varphi }_{\alpha }}({{\text{y}}^{-1}}\text{x})}\text{d}{{\text{H}}_{\text{n}}}(\text{y}),$$\text{f}\in \text{C}_{0}^{\infty }({{\text{H}}_{\text{n}}})$is inverse to${\cal{L}}_{\alpha }$, as long as α is admissible, i.e.,$\frac{\text{n}\pm \alpha }{2}\ne 0, -1, -2, \ldots$.

2.4 Theorem. Let α be admissible. Then$\sigma ({{\text{K}}_{\alpha }})(\text{x}, \xi)=\sigma ({{\text{G}}_{\alpha }})(\text{x}, \xi),$ξ ≠ 0, where$\sigma({{\text{G}}_{\alpha }})(\text{x},\xi)$is given by (1.22). In other words,$\caption {(2.5)} ({{\text{K}}_{\alpha }}\text{f})(\text{x})={{(2\pi )}^{-2\text{n}-1}}\int_{{{\mathbb{R}}^{2\text{n}+1}}}{{{\text{e}}^{\text{i}<\text{x},\xi >}}\sigma ({{\text{G}}_{\alpha }})(\text{x},\xi )\hat{\text{f}}(\xi )\text{d}\xi }$whenever$\text{f}\in \cal{S}$,

Proof. Starting with$\caption {(2.6)} ({{\text{K}}_{\alpha }}\text{f})(\text{x})=\text{c}_{\alpha }^{-1}{{(2\pi )}^{-2\text{n}-1}}\int_{{{\text{H}}_{\text{n}}}}{{{\varphi }_{\alpha }}({{\text{y}}^{-1}}\text{x})\text{d}{{\text{H}}_{\text{n}}}(\text{y})\int_{{{\mathbb{R}}^{2\text{n}+1}}}{{{\text{e}}^{\text{i}<\text{y},\xi >}}}\hat{\text{f}}(\xi )\text{d}\xi }$we shall compute$\caption {(2.7)} \sigma ({{\text{K}}_{\alpha }})(\text{x},\xi )=\text{c}_{\alpha }^{-1}{{\text{e}}^{-\text{i}<\text{x},\xi >}}\int_{{{\mathbb{R}}^{2\text{n}+1}}}{{{\varphi }_{\alpha }}({{\text{y}}^{-1}}\text{x}){{\text{e}}^{\text{i}<\text{y},\xi >}}\text{dy}{{2}^{-\text{n}}}}.$

We shall do the computation only if${{\xi}_{\text{0}}}>0$. It is similar if${{\xi}_{\text{0}}}<0$. We use the notation${{\text{z}}_{\text{j}}}={{\text{x}}_{2\text{j}-1}}+\text{i}{{\text{x}}_{2\text{j}}},\ {{\text{w}}_{\text{j}}}={{\text{y}}_{2\text{j}-1}}+\text{i}{{\text{y}}_{2\text{j}}},\ \text{j=1,}\ldots\text{,n,}$and recall that$\text{d}{{\text{H}}_{\text{n}}}(\text{x})={{2}^{-\text{n}}}\text{dx}.$

First we compute...

• Chapter 3. □b on functions and the solvability of the Lewy equation
(pp. 36-43)

A q-form f on Hnis a sum$\caption {(3.1)} \text{f}=\sum\limits_{\left| \text{J} \right|=\text{q}}{{{\text{f}}_{\text{J}}}\text{d}{{{\bar{\text{z}}}}^{\text{J}}}}$where fJare complex-valued functions on Hnindexed by$\text{J=(}{{\text{j}}_{\text{1}}},{{\text{j}}_{\text{2}}},\ldots,{{\text{j}}_{\text{q}}}),\ {{\text{j}}_{\text{1}}}<{{\text{j}}_{\text{2}}}<\ldots<{{\text{j}}_{\text{q}}},\ \text{and}\ \text{d}{{\overline{\text{z}}}^{\text{J}}}=\text{d}{{\overline{\text{z}}}_{{{\text{j}}_{1}}}}\wedge \text{d}{{\overline{\text{z}}}_{{{\text{j}}_{2}}}}\wedge \ldots\wedge \text{d}{{\overline{\text{z}}}_{{{\text{j}}_{\text{q}}}}}.$The${{\overline{\partial }}_{\text{b}}}$operator (mapping q-forms to q+1-forms) is then defined by$\caption {(3.2)} {{\overline{\partial }}_{\text{b}}}\text{f=}\sum\limits_{\text{j,J}}{{{\overline{\text{Z}}}_{\text{j}}}({{\text{f}}_{\text{J}}})\text{d}{{\overline{\text{z}}}_{\text{j}}}\wedge \text{d}{{\overline{\text{z}}}^{\text{J}}},}$where${{\text{Z}}_{\text{j}}}=\frac{\partial }{\partial {{\text{z}}_{\text{j}}}}+\text{i}{{\overline{\text{Z}}}_{\text{j}}}\frac{\partial }{\partial \text{t}},\ \text{j=1,}\ldots\text{,n}$, see (1.1), and the formal adjoint${{\vartheta }_{\text{b}}}$is given by$\caption {(3.3)} {{\vartheta }_{\text{b}}}\text{f}=-\sum\limits_{\text{j},\text{J}}{{{\text{Z}}_{\text{j}}}({{\text{f}}_{\text{J}}})}\text {d}{{\overline{\text{z}}}_{j}}\left. {\underline {\, {} \,}}\! \right| \text{d}{{\overline{\text{z}}}^{\text{J}}}$The interior product ┘ is defined after formula (6. 10). One then defines the Laplacian corresponding to this complex; it is$\caption {(3.4)} {{\square}_{\text{b}}}={{\overline{\partial }}_{\text{b}}}{{\vartheta }_{\text{b}}}\ +\ {{\vartheta }_{\text{b}}}{{\overline{\partial }}_{\text{b}}}.$

We denote the restriction of □bto q-forms by${{\square }_{\text{b}}}^{(\text{q})}$. It turns out that on the Heisenberg group${{\square }_{\text{b}}}^{(\text{q})}$takes a particularly elegant form.${{\square }_{\text{b}}}^{(\text{q})}$is diagonal, more precisely$\caption {(3.5)} {{\square }_{\text{b}}}^{(\text{q})}( \sum\limits_{\text{J}}{{{\text{f}}_{\text{J}}}\text{d}{{\overline{\text{z}}}^{\text{J}}}} )=\sum\limits_{\text{J}}{{{\cal{L}}_{\alpha }}({{\text{f}}_{\text{J}}})\text{d}{{\overline{\text{z}}}^{\text{J}}}}$with α...

6. Part II Parametrix for the $\overline{\partial }$-Neumann problem
• Guide to Part II
(pp. 44-47)

The purpose of this part is to reduce the$\overline{\partial }$-Neumann problem to the inversion of a pseudo-differential operator □+defined on the boundary; thereby one obtains a parametrix, an approximate "Neumann" operator, which gives a solution for our original problem, modulo controllable error terms.

Since the gist of the method is to reduce the question to the boundary, bM, and there to approximate by the Heisenberg group, certain preliminary problems must be dealt with, which we now describe.

Under the assumption that bM is strongly pseudo-convex we can introduce, for each fixed ξ ∈ bM, a basic mapping η...

• Chapter 4. Admissible coordinates on strongly pseudo-convex CR manifolds
(pp. 48-63)

Letmbe a CR manifold, i.e., a real oriented Cmanifold of dimension 2n+1, n=1, 2, 3, …, together with a subbundle T1,0of the complexified tangent bundle ℂTmsatisfying

(a) dimT1,0= n,

(b)${{\text{T}}^{1,0}}\ \cap \ {{\overline{\text{T}}}^{1,0}}=\{0\},$

(c) T1,0is integrable in the sense of Frobenius, i.e., if Z1, Z2are sections of T1,0, then so is their Lie bracket [Z1, Z2].

Now we restrict our attention to a local coordinate patch U. Let V0, V1, …, V2ndenote a basis for the tangent bundle Tmon U, such that Z1, …, Znyield a basis for T1,0, where$\caption {(4.1)} {{\text{Z}}_{\text{j}}}=\frac{1}{2}\left( {{\text{V}}_{\text{j}}}-\text{i}{{\text{V}}_{\text{j}+\text{n}}} \right),\ \text{j}=1,\ldots,\text{n}.$...

• Chapter 5. Levi metrics
(pp. 64-69)

Let M be a sub-domain with smooth boundary bM of a complex manifold M′. Then to each P point of bM one can assign a Levi-form, a Hermitian form on T1,0(bM)|P. (This Levi-form is not unique, but is determined up to a positive multiple.) The assumption that M is strongly pseudo-convex means that this form is strictly positive definite at each point P ∈ bM. The purpose of this chapter is to give an explicit construction of a Hermitian metric on M, which restricted to T1,0(bM) coincides with the Levi form (i.e., (5.7) is satisfied). We call such a metric...

• Chapter 6. □ on (0, 1)-forms
(pp. 70-78)

From now on we assume a fixed Hermitian metric$\left( {{\text{g}}_{\text{i}\overline{\text{j}}}} \right)$on M which satisfies (5.7) on bM, (a "Levi" metric). The following analysis is done in a fixed boundary coordinate neighborhood U. Let ρ denote geodesic distance from bM, at least in some sufficiently small neighborhood of bM, ρ > 0 in M and ρ < 0 outside of$\overline{\text{M}}$. Let$\caption {(6.1)} {{\omega}_{1}},\ldots,{{\omega}_{\text{n}}},{{\omega}_{\text{n}+1}}=\sqrt{2}\,\partial \rho$denote an orthonormal basis for T(1,0)(M)* in U and let Z1, …, Zn+1denote the dual basis for T(1,0)(U). We have$\caption {(6.2)} \text{df}=\sum\limits_{\text{j}=1}^{\text{n}+1}{({{\text{Z}}_{\text{j}}}\text{f}){{\omega}^{\text{j}}}}+\sum\limits_{\text{j}=1}^{\text{n}+1}{({{\overline{\text{Z}}}_{\text{j}}}\text{f}){{\overline{\omega}}^{\text{j}}}},$or, equivalently$\caption {(6.3)} {{\text{Z}}_{\text{j}}}\text{f}=\left\langle \text{df},{{\omega}^{\text{j}}} \right\rangle ,\ \text{j}=1,\ldots,\text{n}+1$for Cfunctions f. (6.3) implies that in local coordinates${{\overline{\text{Z}}}_{\text{j}}}$is the complex...

• Chapter 7. Local solution of the Dirichlet problem for □
(pp. 79-100)

Let${{\omega}_{\text{n}+1}}=\sqrt{2}\,\partial \rho$be the invariantly defined "complex normal" holomorphic covector near bM. Here ρ stands for geodesic distance from bM. Let$\text{u}\ \in \text{C}_{(0,1)}^{\infty }(\text{{M}}')$. Set$\caption {(7.1)} \text{u}={{\text{u}}_{\text{t}}}+{{\overline{\omega}}_{\text{n}+1}}\nu (\text{u})$in a neighborhood of bM, where$\caption {(7.2)} \nu (\text{u})={{\overline{\omega}}_{\text{n}+1}}\left. {\underline {\, {} \,}}\! \right| \ \text{u}.$

Clearly${{[{{\text{u}}_{\text{t}}}]}_{\text{bM}}}\in \text{C}_{(0,1)}^{\infty }(\text{bM})$, where [ ]bMstands for the restrictions to bM. Let$\text{u}\ \in \text{C}_{(0,1)}^{\infty }(\overline{\text{M}})$solve the following "Dirichlet" problem

(7.3) □(u) = f in M

(7.4) [ut]bM= h,

(7.5) [ν(u)]bM= g.

The purpose of this chapter is to construct, locally. Green's operator G: (f;h, g) → u. The construction is quite technical, therefore we begin with a quick sketch of the main idea behind it (and...

• Chapter 8. Reduction to the boundary
(pp. 101-109)

Here we compute the$\overline{\partial }$-Neumann boundary value of the Poisson operator P. Recall that P is defined by (7.52) on$\text{C}_{(0,1),0}^{\infty }({{\mathbb{R}}^{2\text{n}+1}})\oplus \text{C}_{0}^{\infty }({{\mathbb{R}}^{2\text{n}+1}})$. We change the meaning of P slightly keeping the same notation, and hopefully not introducing ambiguity.

8.1 Definition. We define the operator$\text{P}:\text{C}_{(0,1),0}^{\infty }({{\mathbb{R}}^{2\text{n}+1}})\to \text{C}_{(0,1)}^{\infty }(\mathbb{R}_{+}^{2\text{n}+2})$by$\text{P}(\text{g})=\text{P}(\text{g}\oplus 0),\ \text{g}\in \text{C}_{(0,1),0}^{\infty }(\mathbb {R}^{2\text{n}+1}).$

Now let$\text{u}\in \text{C}_{(0,1),0}^{\infty }(\overline{\mathbb{R}_{+}^{2\text{n}+2}}\cap \text{U})$be given by$\caption {(8.2)} \text{u}=\sum\limits_{\text{j}=1}^{\text{n}+1}{{{\text{u}}_{\text{j}}}{{\overline{\omega}}^{\text{j}}}}={{\text{u}}_{\text{t}}}+(\nu \text{u}){{\overline{\omega}}_{\text{n}+1}}.$

The first$\overline{\partial }$-Neumann boundary condition is

(8.3) [νu]0= 0.

According to (6.16), (and (7.1)) the second condition is the vanishing of$\caption {(8.4)} {{[\nu \overline{\partial }(\text{u})]}_{0}}={{[({{\overline{\text{Z}}}_{\text{n}+1}}{{\text{I}}_{\text{n}}}+{{\overline{\text{S}}}_{\text{n}+1}}){{\text{u}}_{\text{t}}}]}_{0}}.$

We therefore introduce the short-hand of the operator${{\text{B}}_{\overline{\partial }}}$as follows.

$\caption {(8.5)} {{\text{B}}_{\overline{\partial }}}(\text{u})={{[({{\overline{\text{Z}}}_{\text{n}+1}}{{\text{I}}_{\text{n}}}+{{\overline{\text{S}}}_{\text{n}+1}}){{\text{u}}_{\text{t}}}]}_{0}}={{[\nu \overline{\partial }(\text{u})]}_{0}}$for$\text{u}\in \text{C}_{(0,1)}^{\infty }$with [νu]0= 0.

Here, with a...

• Chapter 9. A parametrix for □ near bM; n > 1
(pp. 110-117)

In this chapter we shall construct an approximate local left inverse for the$\overline{\partial }$-Neumann problem when n > 1, i.e., we shall obtain a local representation of$\text{u}\in \text{C}_{(0,1)}^{\infty }(\overline{\text{M}})$in terms of f, if

(9.1) □(u) = f in M,$\caption {(9.2)} {{[\nu (\text{u)}]}_{\text{bM}}}={{[\nu \overline{\partial }(\text{u})]}_{\text{bM}}}=0.$

We shall next describe heuristically this local inverse (or approximate "Neumann operator"). In the brief description that follows we shall disregard error terms, (which will turn out to be smoothing operators), and pay no attention to the host of cut-off functions that must be used (which introduce additional error terms of smoothing operators). Thus according to Theorem 7.66 we have...

• Chapter 10. The parametrix for □ near bM; n=1
(pp. 118-129)

In this case$\caption {(10.1)} \begin{array}{lll} {{\square }_{\text{b}}}(\varphi ) & = & -\frac{1}{2}({{\text{Z}}_{1}}{{\overline{\text{Z}}}_{1}}+{{\overline{\text{Z}}}_{1}}{{\text{Z}}_{1}})(\varphi )-\text{iT}(\varphi ) \\ {} & {} & +\varepsilon ({{\text{Z}}_{1}},{{\overline{\text{Z}}}_{1}},\varphi ) \\ {} & = & -{{\overline{\text{Z}}}_{1}}{{\text{Z}}_{1}}(\varphi )+\varepsilon ({{\text{Z}}_{1}},{{\overline{\text{Z}}}_{1}},\varphi ) \\ \end{array}$

Here we have the added complication that □bhas no inverse; i.e., parametrix. However, we can make use of the result concerning the solvability of the Lewy equation discussed in Chapter 3. The idea, stated somewhat imprecisely, is as follows:

We let${{\overline{\text{C}}}_{\text{b}}}$denote the (presumptive) projection on the boundary values of anti-holomorphic functions. Then on the orthogonal complement, (whose "projection" is given by$\text{I}-{{\overline{\text{C}}}_{\text{b}}}$), □bhas an inverse; namely by the results of Chapter 3 we can find an integral operator$\overline{\text{K}}$so that$\overline{\text{K}}{{\square }_{\text{b}}}=\text{I}-{{\overline{\text{C}}}_{\text{b}}}$, approximately; and so □+has an inverse on...

7. Part III The estimates
• Guide to Part III
(pp. 130-131)

Here we deal with the regularity properties of the solutions of the$\overline{\partial }$-Neumann problem, (11.1) and (11.2) below.

The L2theory of Kohn allows one to write down an "abstract" solution to the problem, in terms of a "Neumann operator" N. The theory proceeds by using L2Sobolev estimates, and culminates in the assertion that$\text{Nf}\in {{\text{C}}^{\infty }}(\overline{\text{M}})$if$\text{f}\in {{\text{C}}^{\infty }}(\overline{\text{M}})$. These results are reviewed, with no proofs given, in Chapter 11. We then turn to estimates for N in other function spaces.

Using the construction of approximate Neumann operators carried out in Chapters 9 and 10, the problem can be...

• Chapter 11. Review of the L2 theory
(pp. 132-133)

We shall now summarize the results of Kohn [21], (see also the exposition in [8]), concerning the L2and regularity theory for the solution of the$\overline{\partial }$Neumann problem.

As before, M is an open sub-domain in a larger complex manifold M′; M has a smooth boundary, bM, which is strongly pseudo-convex.$\text{C}_{(0,1)}^{\infty }(\overline{\text{M}})$denotes the (0, 1) forms in$\overline{\text{M}}$which are Cup to the boundary;$\text{L}_{(0,1)}^{2}(\overline{\text{M}})$denotes its closure in the L2norm (using the Levi-metric of Chapter 5). We are concerned, in effect, with the problem of solving

(11.1) □u = f with the boundary conditions$\caption {(11.2)} \nu (\text{u}){{|}_{\text{bM}}}=0,\quad \text{and}\quad \nu (\overline{\partial }\text{u}){{|}_{\text{bM}}}=0$...

• Chapter 12. The Besov spaces Bp(ℝm)
(pp. 134-140)

We shall consider the Besov spaces, denoted by$\Lambda _{\text{1-1/p}}^{\text{p,p}}({\mathbb{R}}^{\text{m}})$in [33], Chapter V, §5.*Here 1 < p < ∞. We shall always write${{\text{B}}^{\text{p}}}({{\mathbb{R}}^{\text{m}}})=\Lambda _{\text{1}-1/\text{p}}^{\text{p,p}}({{\mathbb{R}}^{\text{m}}})$. For simplicity of notation we have written m = 2n+1.

The space Bp(ℝm) consists of all f ∈ Lpfor which the norm${{\left\| \text{f} \right\|}_{\text{p}}}+(\int_{{{\mathbb{R}}^{\text{m}}}}{\frac{\left\| \text{f}(\text{x}-\text{t})-\text{f}(\text{x}) \right\|_{\text{p}}^{\text{p}}}{{{\left| \text{t} \right|}^{\text{m}+\text{p}-1}}}\text{dt}{{)}^{1/\text{p}}}}$is finite. (Here$\left\| \text{f}(\text{x}-\text{t})-\text{f}(\text{x}) \right\|_{\text{p}}^{\text{p}}=\int_{{{\mathbb{R}}^{\text{m}}}}{{{\left| \text{f}(\text{x}-\text{t})-\text{f}(\text{x}) \right|}^{\text{p}}}\text{dx}}$.)

For our applications we shall need an equivalent characterization.

Suppose f ∈ Bp(ℝm); then there exists a family {fε}0 < ε ≤ 1of smooth functions so that fε→ f in Lpnorm as ε → 0, at a definite rate, while$\parallel \nabla {{\text{f}}_{\varepsilon }}{{\parallel }_{\text{p}}}=\ \parallel {{(\sum\limits_{\text{j}=1}^{\text{m}}{|\frac{\partial {{\text{f}}_{\varepsilon }}}{\partial {{\text{x}}_{\text{j}}}}{{|}^{2}}})^{1/2}}}{{\parallel }_{\text{p}}}$can be controlled appropriately,...

• Chapter 13. The spaces Λα(ℝm) and ${{\Lambda }_{\alpha }}(\mathbb{R}_{+}^{\text{m}+1})$
(pp. 141-148)

We let Λα(ℝm) denote the standard Lipschitz spaces, α > 0, as described in Chapter V, §4 of [33]. Thus a bounded function f belongs to Λα, where 0 < α < 1, when ∥f(x−t) − f(x)∥≤ A|t|α. (Here ∥ · ∥denotes the sup norm.) For α = 1, we require ∥f(x+t) + f(x−t) − 2f(x)∥≤ A|t|, and when α > 1, we proceed inductively, i.e., f ∈ Λα(ℝm) ⇔ f ∈ Land$\frac{\partial \text{f}}{\partial {{\text{x}}_{\text{j}}}}\in {{\Lambda }_{\alpha -1}}({{\mathbb{R}}^{\text{m}}})$, j=1, …, m.

On$\mathbb{R}_{+}^{\text{m}+1}$we define the space${{\Lambda }_{\alpha }}(\overline{\mathbb{R}}_{+}^{\text{m}+1})$to consist of all functions on$\overline{\mathbb{R}}_{+}^{\text{m}+1}$which can be extended to ℝm+1so as...

• Chapter 14. The spaces Bp, $\text{L}_{1}^{\text{P}}$, Λα on M and bM
(pp. 149-152)

We come closer to our ultimate applications.

M is a domain in a complex manifold with smooth boundary bM. What we have done above will make it easy to define the spaces Bp(bM), Λα(bM); also$\text{L}_{\text{k}}^{\text{p}}(\overline{\text{M}})$and${{\Lambda }_{\alpha }}(\overline{\text{M}})$. In fact suppose, for example f ∈ Bp(ℝm). Then if φ is a local diffeomorphism of ℝmand$\psi \in \text{C}_{0}^{\infty }$whose support is contained where φ is regular, then ψ(x)f(φ(x)) ∈ Bp(ℝm). This follows immediately for the characterization given in Lemma 12.1. This allows one to define Bp(bM) in terms of a finite patching of coordinate neighborhoods of bM. Similarly one can...

• Chapter 15. Main results
(pp. 153-171)

Let N denote the (exact) Neumann operator described in Theorem 11.3. Our purpose here will be to prove the regularity of N in terms of function spaces$\text{L}_{\text{k}}^{\text{p}}(\overline{\text{M}})$,${{\Lambda }_{\alpha }}(\overline{\text{M}})$, and others that will be defined below. For simplicity of notation we are using$\text{L}_{\text{k}}^{\text{p}}(\overline{\text{M}})$to denote not only the previously defined space of scalar-valued functions, but also its analogue of (0, 1) forms on M whose components belong to$\text{L}_{\text{k}}^{\text{p}}(\overline{\text{M}})$; similarly for the other spaces studied in Chapter 14. But this abuse of notation should not lead to any confusion. In all our theorems we have 1 <...

• Chapter 16. Solution of $\overline{\partial }\text{U}=\text{f}$
(pp. 172-176)

We begin by pointing out that our solution of the$\overline{\partial }$-Neumann problem is, strictly in the interior of M, elliptic in the sense that there is a gain of two in the usual sense. This of course follows by the general "interior regularity" of solutions of elliptic equations, but in our case it is a consequence of Theorems 15.1, part (b), and Theorem 15.33, since allowable vector fields are not restricted away from the boundary.

Another fact is that the "normal" component of the solution behaves in an elliptic way even up to the boundary. This can be made...

• Chapter 17. Concluding Remarks
(pp. 177-189)

In this chapter we point out some further results in order to round out the picture we have presented above. We give only an indication of the proofs, since the reader who has followed us this far should have no difficulty in filling out the required details.

The first question we pose is that of giving a characterization of those u, in terms of regularity conditions on u and boundary conditions, so that u belongs to the self-adjoint extension □eof □ (described in Chapter 11) i.e., when is u = N(f),$\text{f}\in {{\text{L}}^{2}}(\overline{\text{M}})$; or more generally when is u...

8. Summary of Notation
(pp. 190-191)
9. References
(pp. 192-194)
10. Back Matter
(pp. 195-195)