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 H_n. 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 H_n, 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 H_nis\[\caption {(0.2)} (\zeta ,\text{t})\cdot ({\zeta }',\text{{t}}')\text{=}(\zeta \text{+}{\zeta }',\text{t+{t}}'+2\ \text{Im}\zeta \cdot {\bar{\zeta }}'\text{)}\text{.}\]

Left-invariant vector fields on H_nare given by the Z_j(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...

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 H_n. 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 ℒ_α. Let G_αdenote 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 ℒ_αunder 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 G_αto act on the Heisenberg group by convolution we shall try a kernel of the following form\[\caption {(1.5)}\text{G}_{\alpha }^{\Lambda }={{\text{e}}^{{{\xi }_{0}}(\text{z}{\bar{\text w}}-\bar{\text{z}}\text {w}})}\text{W}({{\xi }_{0}}{{\left| \text{z}-\text{w} \right|}^{2}}),\]...

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 K_α, 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...

A q-form f on H_nis 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 f_Jare complex-valued functions on H_nindexed 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...

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...

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

(a) dim_ℂT^1,0= n,

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

(c) T^1,0is integrable in the sense of Frobenius, i.e., if Z₁, Z₂are sections of T^1,0, then so is their Lie bracket [Z₁, Z₂].

Now we restrict our attention to a local coordinate patch U. Let V₀, V₁, …, V_2ndenote a basis for the tangent bundle Tmon U, such that Z₁, …, Z_nyield a basis for T_1,0, where\[\caption {(4.1)} {{\text{Z}}_{\text{j}}}=\frac{1}{2}\left( {{\text{V}}_{\text{j}}}-\text{i}{{\text{V}}_{\text{j}+\text{n}}} \right),\ \text{j}=1,\ldots,\text{n}.\]...

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 T^1,0(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 T^1,0(bM) coincides with the Levi form (i.e., (5.7) is satisfied). We call such a metric...

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, …, Z_n+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 C^∞functions f. (6.3) implies that in local coordinates${{\overline{\text{Z}}}_{\text{j}}}$is the complex...

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) [u_t]_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...

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...

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...

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...

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...

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 C^∞up 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\[\caption {(11.2)} \nu (\text{u}){{|}_{\text{bM}}}=0,\quad \text{and}\quad \nu (\overline{\partial }\text{u}){{|}_{\text{bM}}}=0\]...

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 B^p(ℝ^m) consists of all f ∈ L^pfor 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 ∈ B^p(ℝ^m); then there exists a family {f_ε}_{0 < ε ≤ 1}of smooth functions so that f_ε→ f in L^pnorm 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,...

We let Λ_α(ℝ^m) denote the standard Lipschitz spaces, α > 0, as described in Chapter V, §4 of [33]. Thus a bounded function f belongs to Λ_α, 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 ∈ Λ_α(ℝ^m) ⇔ f ∈ L^∞and$\frac{\partial \text{f}}{\partial {{\text{x}}_{\text{j}}}}\in {{\Lambda }_{\alpha -1}}({{\mathbb{R}}^{\text{m}}})$, 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 ℝ^m+1so as...

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 B^p(bM), Λ_α(bM); also$\text{L}_{\text{k}}^{\text{p}}(\overline{\text{M}})$and${{\Lambda }_{\alpha }}(\overline{\text{M}})$. In fact suppose, for example f ∈ B^p(ℝ^m). Then if φ is a local diffeomorphism of ℝ^mand$\psi \in \text{C}_{0}^{\infty }$whose support is contained where φ is regular, then ψ(x)f(φ(x)) ∈ B^p(ℝ^m). This follows immediately for the characterization given in Lemma 12.1. This allows one to define B^p(bM) in terms of a finite patching of coordinate neighborhoods of bM. Similarly one can...

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 <...

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...

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...