The fundamental solution to $\mdlgwhtsquare _b$ on quadric manifolds – Part 1. General formulas

Abstract

This paper is the first of a three part series in which we explore geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of $\mathbb{C}^n\times \mathbb{C}^m$. In this paper, we present a streamlined calculation for a general integral formula for the complex Green operator $N$ and the projection onto the nullspace of $\mdlgwhtsquare _b$. The main application of our formulas is the critical case of codimension two quadrics in $\mathbb{C}^4$ where we discuss the known solvability and hypoellipticity criteria of Peloso and Ricci [J. Funct. Anal. 203 (2003), pp. 321–355] We also provide examples to show that our formulas yield explicit calculations in some well-known cases: the Heisenberg group and a Cartesian product of Heisenberg groups.

1. Introduction

The goal of this paper is to present an explicit integral formula for the complex Green operator and the projection onto the null space of the Kohn Laplacian on quadric submanifolds of $\mathbb{C}^n\times \mathbb{C}^m$. Our result generalizes the formula of BR13 from the specific case of codimension $2$ quadrics in $\mathbb{C}^4$ to the general case of arbitrary $n$ and $m$, and we also prove a formula for the complex Green operator when it is only a relative inverse of the Kohn Laplacian and not a full inverse. Additionally, the new proof is significantly simpler and uses our calculation of the $\mdlgwhtsquare _b$-heat kernel on general quadrics BR11. We then provide several applications of our formula to the case of codimension $2$ quadrics in $\mathbb{C}^4$ and provide context for how this case fits into the solvability/hypoellipticity framework of Peloso and Ricci PR03. We also provide a calculation of the complex Green operator in several instances when it is only a relative fundamental solution: $\mdlgwhtsquare _b$ on the Heisenberg group on functions and $(0,n)$-forms, and on the Cartesian product of Heisenberg groups at the top degree. We conclude with a computation of the Szegö projection on the Cartesian product of Heisenberg groups.

This paper is the first of a series where we explore the geometry and analysis of the Kohn Laplacian $\mdlgwhtsquare _b$ and its (relative) inverse, the complex Green operator, on quadric submanifolds in $\mathbb{C}^n\times \mathbb{C}^m$. The $\mdlgwhtsquare _b$-equation, $\mdlgwhtsquare _b u =f$, governs the behavior of boundary values of holomorphic functions, and the $\mdlgwhtsquare _b$-operator is a naturally occurring, nonconstant coefficient, nonelliptic operator. Solving the equation has been tantalizing mathematicians for the better part of fifty years, and while much is known for solvability/regularity in $L^p$ and other spaces (especially $L^2$) for hypersurface type CR manifolds, there has been much less work done to determine the structure of the complex Green operator, denoted by $N$, especially on non-hypersurface type CR manifolds. The problem is that the techniques used to solve the equation are functional analytic in nature and therefore nonconstructive. Consequently, to have any hope of finding an explicit solution, we need additional structure on the CR manifold. From our perspective, the gold standard for results in this area is the calculation by Folland and Stein FS74 in which they find a beautiful, closed form expression for $N$ on the Heisenberg group.

In the decades following FS74, mathematicians developed machinery to solve the $\mdlgwhtsquare _b$-heat equation (or the heat equation associated to the sub-Laplacian) on certain Lie groups. From the formulas for the heat equations, in principle, it is only a matter of integrating the time variable out to recover the formula for $N$. The first results for the heat equation were for the sub-Laplacian on the Heisenberg group by Hulanicki Hul76 and Gaveau Gav77. More results followed in a similar vein, and the vast majority rely on the group Fourier transform and Hermite functions PR03BR09BR11YZ08CCT06BGG96BGG00Eld09. The problem with these techniques is that the formulas that they generate for the heat kernel are only given up to a partial Fourier transform that is uncomputable in practice. Consequently, any information giving precise size estimates or asymptotics, let alone a formula in the spirit of Folland and Stein, is absent.

A quadric submanifold $M\subset \mathbb{C}^n\times \mathbb{C}^m$ is a CR manifold of the form

$$\begin{equation} M= \{(z,w) \in \mathbb{C}^n \times \mathbb{C}^m: \operatorname {Im}w = \phi (z,z) \}, {\tag{1}} \end{equation}$$

where $\phi : \mathbb{C}^n \times \mathbb{C}^n \mapsto \mathbb{C}^m$ is a sesquilinear form (i.e., $\phi (z, z') = \overline{\phi (z',z)}$). The fundamental solution to $\mdlgwhtsquare _b$ or the sub-Laplacian on quadrics has been studied by many authors, including BG88BGG96BR09BR13CCT06FS74PR03. See also Part II of the series BR20, in which we find useable sufficient conditions for a map $T$ between quadrics to be a $\mdlgwhtsquare _b$-preserving Lie group isomorphism as well as establish a framework for which appropriate derivatives of the complex Green operator are continuous in $L^p$ and $L^p$-Sobolev spaces (and hence are hypoelliptic). We apply the general results to codimension $2$ quadrics in $\mathbb{C}^4$.

There are two higher codimension papers that need mentioning. First, in NRS01, Nagel et al. analyze $\mdlgwhtsquare _b$ and its geometry in the special class of decoupled quadrics where $\operatorname {Im}w_j = \sum _{k=1}^n a_k^j|z_k|^2$. However, many of the interesting cases do not fall into this category. Second, Raich and Tinker compute the Szegö kernel for the polynomial model

$$\begin{equation*} M = \{(z,w)\in \mathbb{C}\times \mathbb{C}^n : \operatorname {Im}w = (a_1,\dots ,a_{n-1},1) p(\operatorname {Re}z) \big \}, \end{equation*}$$

where $p:\mathbb{R}\to \mathbb{R}$ is a smooth function satisfying $\lim _{|x|\to \infty }\frac{p(x)}{|x|} = \infty$, and the constants $a_1$, …, $a_{n-1}$ are nonzero RT15. The authors write an explicit formula for the Szegö kernel based on an integral formula of Nagel Nag86 and show that there are significant blowups off of the diagonal. Raich and Tinker evaluate all of the integrals in the case $p(x) = x^2$. That is, the CR manifold $M$ is a quadric, but this case is very special because the tangent space (at each point) has only one complex direction, so every degree is either top or bottom. These are often the exceptional cases and can give misleading intuition.

Associated to each quadric is the Levi form $\phi (z,z')$ and for each $\lambda \in \mathbb{R}^m$, the directional Levi form in the direction $\lambda$ is $\phi ^\lambda (z,z') = \phi (z,z')\cdot \lambda$ where $\cdot$ is the usual dot product without conjugation. For each $\lambda$, there is an $n\times n$ matrix $A^\lambda$ so that

$$\begin{equation*} \phi ^\lambda (z,z') = z^* A^\lambda z' \end{equation*}$$

and we identify the eigenvalues of $\phi ^\lambda$ with the eigenvalues of the matrix $A^\lambda$. Here, the $*$ designates the Hermitian transpose.

The outline of the remainder of the paper is as follows: In Section 2, we record the main result for general quadrics and provide additional context. In Section 3, we discuss the CR geometry and Lie group structure of a general quadric. In Section 4, we prove the main result, and we devote Section 5 to explicit examples.

2. Results and discussion

Under the projection $\pi :\mathbb{C}^n\times \mathbb{C}^m \to \mathbb{C}^n\times \mathbb{R}^m$ given by $\pi (z,t+is)=(z,t)$, we may identify a quadric $M$ with $\mathbb{C}^n\times \mathbb{R}^m$. The projection induces both a CR structure and Lie group structure on $\mathbb{C}^n\times \mathbb{R}^m$, and we denote this Lie group by $G$ (or $G_M$). Thus the projection is a CR isomorphism and we refer to the pushforwards and pullbacks of objects from $M$ to $G$ without changing the notation.

2.1. The main result – general formulas for the solution of $\mdlgwhtsquare _b$ on quadrics

To state the main result, we need to introduce some notation. For each $\lambda \in \mathbb{R}^m\setminus \{0\}$, let $\mu ^\lambda _1$, …, $\mu ^\lambda _n$ be the eigenvalues of $\phi ^\lambda$ (or equivalently, the Hermitian symmetric matrix, $A^\lambda$) and $v^\lambda _1$, …, $v^\lambda _n$ be an orthonormal set of eigenvectors. This means

$$\begin{equation} \phi ^\lambda (v^\lambda _j, v^\lambda _k) = \delta _{jk} \mu _j^\lambda . {\tag{2}} \end{equation}$$

Let $\alpha = \lambda /|\lambda |$, then $\mu _j^\lambda = |\lambda |\mu ^\alpha _j$ and $v_j^\lambda = v_j^\alpha$. If $z\in \mathbb{C}^n$ is expressed in terms of the unit eigenvectors of $\phi ^\lambda$, then $z_j^\lambda = z_j^\alpha$ is given by

$$\begin{equation*} z^\alpha = Z(z, \alpha ) = ({U^\alpha })^*\cdot z, \end{equation*}$$

where $U^\alpha$ is the matrix whose columns are the eigenvectors, $v^\alpha _j$, $1 \leq j \leq n$, and $\cdot$ represents matrix multiplication with $z$ written as a column vector. Note that the corresponding orthonormal basis of $(0,1)$-covectors for this basis is

$$\begin{equation*} d \bar{Z}_j(z, \alpha ), 1 \leq j \leq n, \text{ where } d \bar{Z}(z, \alpha ) = (U^\alpha )^T \cdot d \bar{z}, \end{equation*}$$

where $d \bar{z}$ is written as a column vector of $(0,1)$-forms and the superscript $T$ stands for transpose. It is a fact that the eigenvalues, eigenvectors and hence $z^\alpha = Z(z, \alpha )$ depend smoothly on $z \in \mathbb{C}^n$. However, while the dependence of the eigenvalues is continuous (in fact Lipschitz) in $\alpha$, the eigenvectors may only be functions of bounded variation (SBV) in $\alpha$ Rai11, Theorem 9.6.

Let $\mathcal{I}_q = \{ L = (\ell _1,\dots ,\ell _q): 1 \leq \ell _1 < \ell _2 <\cdots < \ell _q \leq n\}$. For each $K \in \mathcal{I}_q$, we will need to express $d \bar{z}^K$, in terms of $d \bar{Z}(z, \alpha )^L$ for $L \in \mathcal{I}_q$. We have

$$\begin{equation} d \bar{z}^K = \sum _{L\in \mathcal{I}_q} C_{K,L}(\alpha ) d \bar{Z}(z, \alpha )^L, {\tag{3}} \end{equation}$$

where $C_{K,L} (\alpha )$ are the appropriate $q \times q$ minor determinants of $\bar{U}^\alpha$. Note that if $q =n$, then the above sum only has one term and $C_{K,K} (\alpha )= 1$. Additionally, when $q=0$, $\mathcal{I}_0 = \emptyset$ and the sum 3 does not appear.

Denote by $\nu (\lambda )=\nu (\alpha )$ the number of nonzero eigenvalues of $\phi ^\lambda$. For each $q$-tuple $L \in \mathcal{I}_q$, set

$$\begin{equation*} \Gamma _L = \{\alpha \in S^{m-1}: \mu ^\alpha _\ell >0 \text{ for all }\ell \in L \text{ and } \mu ^\alpha _\ell < 0 \text{ for all }\ell \not \in L\} \end{equation*}$$

and

$$\begin{equation*} \varepsilon _{j,L}^\alpha = \begin{cases} \operatorname {sgn}(\mu _j^\lambda ) & \text{if }j \in L, \\ - \operatorname {sgn}(\mu _j^\lambda ) &\text{if }j\not \in L. \end{cases} \end{equation*}$$

Remark 2.1.

If $\nu = \max _{\lambda \in \mathbb{R}^m} \nu (\lambda )$, then $\{\lambda \in \mathbb{R}^m\setminus \{0\} : \nu (\lambda ) = \nu \}$ is a Zariski open set and hence carries full Lebesgue measure. In particular, if one of the sets $\Gamma _L$ is nonempty, then $\nu =n$. When $\nu < n$, we arrange our eigenvalues so that $\mu ^\lambda _{\nu +1} = \cdots \mu ^\lambda _n=0$ and write $z'=(z_1,\dots ,z_\nu )$ and $z'' = (z_{\nu +1},\dots ,z_n)$.

Definition 2.2.

Given an index in $K \in \mathcal{I}_q$, we say that a current $N_K = \sum _{K'\in \mathcal{I}_q} \tilde{N}_{K'}(z,t) d\bar{z}^{K'}$ is a fundamental solution to $\mdlgwhtsquare _b$ on forms spanned by $d \bar{z}^K$ if $\mdlgwhtsquare _b N_K = \delta _0(z,t) d\bar{z}^K$.

$N_K$ acts on smooth forms with compact support by componentwise convolution with respect to the group structure on $G$, that is, if $f = f_0 d\bar{z}^K$, then $N_K*f = \sum _{K'\in \mathcal{I}_q} \tilde{N}_{K'} * f_0 d\bar{z}^{K'}$. Thus if $f =f_0 d \bar{z}^K$ is a smooth form with compact support, then $\mdlgwhtsquare _b \{N_K * f \} = f$. In cases where $\mdlgwhtsquare _b$ has a nontrivial kernel, we let $S_K$ be the projection (Szegö) operator onto this kernel and we say that $N_K$ is a relative fundamental solution if $\mdlgwhtsquare _b \{N_K * f \} = f-S_K(f)$ holds for all compactly supported forms spanned by $d \bar{z}^K$. On quadrics, $\mdlgwhtsquare _b$ never has closed range in $L^2$ so the complex Green operator cannot be continuous in $L^2$. As a consequence, we can only discuss a relative inverse and not the relative inverse. However, a relative inverse is called canonical if its output is orthogonal to $\ker \mdlgwhtsquare _b$ whenever it belongs to $L^2$.

We can now state our main result.

Theorem 2.3.

Suppose $M$ is a quadric CR submanifold of $\mathbb{C}^{n+m}$ given by 1 with associated projection $G$. Assume the associated heat kernels, $\tilde{H}_L(s, z^\alpha , \hat{\lambda })$ in (10) for all $L \in I_q$ are jointly integrable in the variables $s>0$ and $\lambda \in \mathbb{R}^m$. Fix $K\in \mathcal{I}_q$.

(1)

If $|\Gamma _L|=0$ for all $L \in \mathcal{I}_q$, then the fundamental solution to $\mdlgwhtsquare _b$ on forms spanned by $d\bar{z}^K$ is given by$$\begin{equation} \begin{split} &N_K (z,t)=\\ &\frac{4^n}{2(2 \pi )^{m+n}} \sum _{L \in \mathcal{I}_q} \int _{\alpha \in S^{m-1}} C_{K,L} (\alpha ) d \bar{Z}(z, \alpha )^L \int _{r=0}^1 \frac{1}{|\log r|^{n-\nu (\alpha )}} \prod _{j=1}^{\nu (\alpha )} \frac{r^{\frac{1}{2}(1-\varepsilon _{j,L}^\alpha ) |\mu _j^\alpha |}|\mu _j^\alpha |}{1- r^{|\mu _j^\alpha |}} \\ & \frac{ (n+m-2)! }{ (A_\alpha (r,z)-i \alpha \cdot t)^{n+m-1}} \frac{dr d \alpha }{r}, \end{split} {\tag{4}} \end{equation}$$

where$$\begin{equation*} A_\alpha (r,z) = \frac{2}{|\log r|}|{z''}^\alpha |^2 + \sum _{j=1}^{\nu (\alpha )} |\mu _j^\alpha | \left( \frac{1+ r^{|\mu _j^\alpha |}}{1- r^{|\mu _j^\alpha |}} \right) |z_j^\alpha |^2 . \end{equation*}$$

(2)

If $|\Gamma _L|>0$ for at least one $L \in \mathcal{I}_q$, then orthogonal projection onto the $\ker \mdlgwhtsquare _b$ applied to forms spanned by $d\bar{z}^K$ is given by convolution with the $(0,q)$-form:$$\begin{equation} \begin{split} & S_K (z,t)\\ &= \frac{4^n(n+m-1)!}{(2\pi )^{m+n}} \sum _{L \in \mathcal{I}_q} \int _{\alpha \in \Gamma _L} C_{K,L} (\alpha ) d \bar{Z}(z, \alpha )^L \frac{\prod _{j=1}^n|\mu _j^\alpha |}{(\sum _{j=1}^n |\mu _j^\alpha | |z_j^\alpha |^2 - i \alpha \cdot t)^{n+m}} d\alpha . \end{split} {\tag{5}} \end{equation}$$

In the case $K = \emptyset$, $S_\emptyset (z,t)$ is the Szegö kernel.

(3)

If $|\Gamma _L|>0$ for at least one $L \in \mathcal{I}_q$, then the canonical relative fundamental solution to $\mdlgwhtsquare _b$ given by $\int _0^\infty e^{-s\mdlgwhtsquare _b}(I-S_q) ds$ applied to forms spanned by $d\bar{z}^K$ is given by$$\begin{equation} \begin{split} N_K (z,t) &= \frac{4^n(n+m-2)!}{2(2 \pi )^{m+n}} \sum _{L \in \mathcal{I}_q} \left( \int _{\alpha \not \in \Gamma _L} C_{K,L} (\alpha ) d \bar{Z}(z, \alpha )^L \right.\\ &\quad \left. \times \int _{r=0}^1 \prod _{j=1}^n \frac{r^{\frac{1}{2}(1-\varepsilon _{j,L}^\alpha ) |\mu _j^\alpha |}|\mu _j^\alpha |}{1- r^{|\mu _j^\alpha |}} \frac{ 1}{ (A(r,z)-i \alpha \cdot t)^{n+m-1}} \frac{dr d \alpha }{r} \right. \\ &\quad + \int _{\alpha \in \Gamma _L} C_{K,L} (\alpha ) d \bar{Z}(z, \alpha )^L \\ &\quad \times \left. \int _{r=0}^1 \Big [\Big (\prod _{j=1}^n \frac{|\mu _j^\alpha |}{1-r^{|\mu _j^\alpha |}}\Big ) \frac{1}{(A_\alpha (r,z)-i\alpha \cdot t)^{n+m-1}}- \frac{\prod _{j=1}^n|\mu _j^\alpha |}{(A_\alpha (0,z)-i\alpha \cdot t)^{n+m-1}}\Big ] \frac{dr d \alpha }{r} \right), \end{split} {\tag{6}} \end{equation}$$

where$$\begin{equation*} A_\alpha (r,z) = \sum _{j=1}^{n} |\mu _j^\alpha | \left( \frac{1+ r^{|\mu _j^\alpha |}}{1- r^{|\mu _j^\alpha |}} \right) |z_j^\alpha |^2 . \end{equation*}$$

In all cases, the integrals converge absolutely.

Remark 2.4.

In many of the most important cases, the functions $C_{K,L}(\alpha ) = \delta _{KL}$ and formulas from the theorem simplify. There are several cases when this simplifcation occurs. The first is when $q=0$ or $q=n$. The second is when the orthonormal basis $\{ v^\lambda _j \}$ is independent of $\lambda$. This independence happens both when $m=1$ (the hypersurface type case) and in the sum of squares case considered by Nagel, Ricci, and Stein NRS01, discussed in Section 1.

Remark 2.5.

It is a straightforward exercise to recover the classical complex Green operator on the Heisenberg group from 4 BR13. Additionally, BR13, Theorem 2 is now a simple and immediate application of 4.

2.2. Solvability, hypoellipticity, and $\phi ^\lambda$

In PR03, Peloso and Ricci say that

(1)

$\mdlgwhtsquare _b$ is solvable if given any smooth $(0,q)$-form $\psi$ on $G$ with compact support, there exists a $(0,q)$-current $u$ on $G$ so that $\mdlgwhtsquare _b u = \psi$;

(2)

$\mdlgwhtsquare _b$ is hypoelliptic if given any $(0,q)$-current $\psi$ on $G$, $\psi$ is smooth on any open set on which $\mdlgwhtsquare _b\psi$ is smooth.

Peloso and Ricci are able to characterize solvability and hypoellipticity of $\mdlgwhtsquare _b$.

Theorem 2.6 (PR03).

Let $n^+(\lambda )$, resp., $n^-(\lambda )$, be the number of positive, resp., negative eigenvalues of $\phi ^\lambda$. Then

(1)

$\mdlgwhtsquare _b$ is solvable on $(0,q)$-forms if and only if there does not exist $\lambda \in \mathbb{R}^m\setminus \{0\}$ for which $n^+(\lambda ) = q$ and $n^-(\lambda ) = n-q$.

(2)

$\mdlgwhtsquare _b$ is hypoelliptic on $(0,q)$-forms if and only if there does not exist $\lambda \in \mathbb{R}^m\setminus \{0\}$ for which $n^+(\lambda ) \leq q$ and $n^-(\lambda ) \leq n-q$.

Remark 2.7.

The condition $|\Gamma _L|>0$ is equivalent to the nontriviality of $S_K(z,t)$ and is easy to check. By combining the solvability criteria of PR03 (Theorem 2.6) and the formula for the $\mdlgwhtsquare _b$-heat kernel from BR11 (Theorem 3.1), it must be the case that $\nu = n$ (see 10) as $S_L(z,t) = \lim _{s\to \infty } H_L(s,z,t)$ and solvability is equivalent $S_L(z,t)=0$. The latter statement follows from the fact the condition in part (1) of Theorem 2.6 is an open condition, that is, when solvability fails, $\Gamma _L$ is a (union of) cones, at least one of which will be open and hence has nonzero measure.

3. The Kohn Laplacian on quadrics

For a discussion of the group theoretic properties of $G$, please see BR11 or PR03. By definition, the operator $\mdlgwhtsquare _b$ is defined on $(0,q)$ forms as $\mdlgwhtsquare _b = \bar{\partial }_b \bar{\partial }_b^* + \bar{\partial }_b^* \bar{\partial }_b$ without reference to any particular coordinate system. However in order to do computations, we need formulas for $\mdlgwhtsquare _b$ with respect to carefully chosen coordinates.

For $v \in \mathbb{R}^{2n} \approx \mathbb{C}^n$, let $\partial _v$ be the real vector field given by the directional derivative in the direction of $v$. Then the right invariant vector field at an arbitrary $g=(z,w) \in M$ corresponding to $v$ is given by

$$\begin{equation*} X_v (g)= \partial _v + 2 \operatorname {Im}\phi (v,z) \cdot D_t =\partial _v - 2 \operatorname {Im}\phi (z,v) \cdot D_t. \end{equation*}$$

Let $Jv$ be the vector in $\mathbb{R}^{2n}$ which corresponds to $iv$ in $\mathbb{C}^n$ (where $i=\sqrt {-1}$). The CR structure on $G$ is then spanned by vectors of the form:

$$\begin{equation} Z_v(g)=(1/2)(X_v-iX_{Jv}) =(1/2)(\partial _v - i \partial _{Jv}) -i \overline{\phi (z,v)} \cdot D_t {\tag{7}} \end{equation}$$

and

$$\begin{equation} \bar{Z}_v(g)=(1/2)(X_v+iX_{Jv}) =(1/2)(\partial _v + i \partial _{Jv}) +i \phi (z,v) \cdot D_t. {\tag{8}} \end{equation}$$

Let $v_1$, …, $v_n$ be any orthonormal basis for $\mathbb{C}^n$. Let $X_j= X_{v_j}$, $Y_j=X_{Jv_j}$, and let $Z_j=(1/2) (X_j-iY_j)$, $\bar{Z}_j = (1/2) (X_j+iY_j)$ be the right invariant CR vector fields defined above (which are also the left invariant vector fields for the group structure with $\phi$ replaced by $-\phi$). A $(0,q)$-form can be expressed as $\sum _{K\in \mathcal{I}_q} \phi _K d\bar{z}^K$. An explicit formula for $\mdlgwhtsquare _b$ on quadrics is written down by Peloso and Ricci PR03 (see also BR11) which takes the following form: if $\phi = \sum _K \phi _K d \bar{z}^K$ is a $(0,q)$-form, then

$$\begin{equation} \mdlgwhtsquare _b ( \sum _{K\in \mathcal{I}_q} \phi _K d\bar{z}^K ) = \sum _{K,L\in \mathcal{I}_q} \mdlgwhtsquare _{LK}^v \phi _K d \bar{z}^L, {\tag{9}} \end{equation}$$

where

$$\begin{equation*} \mdlgwhtsquare _{LL}^v = (1/2) \sum _{\ell =1}^n\big ( Z_\ell \bar{Z}_\ell + \bar{Z}_\ell Z_\ell \big ) +(1/2) \left( \sum _{\ell \in L} [ Z_\ell , \bar{Z}_\ell ]- \sum _{k \not \in L} [Z_k, \bar{Z}_k] \right). \end{equation*}$$

If $L \not =K$, then $\mdlgwhtsquare _{LK}^v$ is zero unless $|L \cap K| = q-1$, in which case

$$\begin{equation*} \mdlgwhtsquare _{LK}^v=(-1)^{d_{kl}} [Z_k, \bar{Z}_\ell ], \end{equation*}$$

where $k \in K$ is the unique element not in $L$ and $\ell \in L$ is the unique element not in $K$ and $d_{kl}$ is the number of indices between $k$ and $\ell$. The notation $\mdlgwhtsquare _{LK}^v$ indicates the dependency of this differential operator on the particular orthonormal basis $v = (v_1, \dots , v_n)$ chosen and the resulting basis (i.e., $Z_1$, …, $Z_n$) and the associated dual basis of $(0,1)$-forms (i.e., $d \bar{z}_1$, …, $d \bar{z}_n$).

Note that if $|L \cap K| = q-1$, then $\mdlgwhtsquare _{LK}^v$ is quite simple since it is a linear combination over $\mathbb{C}$ of $t_j$ derivatives. In the next section, we will use the coordinates $z^\alpha = Z(z, \alpha )$ derived from the basis $v^\alpha _1$, …, $v^\alpha _n$ used in Section 2, and we will see that we can ignore the $\mdlgwhtsquare _{LK}^{v^\alpha }$ when $L \not = K$.

3.1. Fourier transform of $\mdlgwhtsquare _b$

Since the quadric defining equations are independent of $t \in \mathbb{R}^m$, we can use the Fourier transform in the $t$-variables:

$$\begin{equation*} \hat{f}( \lambda ) = \frac{1}{(2\pi )^{\frac{m}{2}}} \int _{\mathbb{R}^m} f(t) e^{-i \lambda \cdot t} dt. \end{equation*}$$

In the case that $f$ is a function of $(z,t)$, we use the notation $f(z,\hat{\lambda })$ to denote the partial Fourier transform of $f$ in the $t$-variables. We transform $\mdlgwhtsquare _b$ via the Fourier transform and consider the fundamental solution to the heat operator in the transformed variables. We then use the $z^\alpha$ coordinates relative to the basis $v_j^\alpha$ chosen in Section 2 for the $z$-variable in $f(z^\alpha , \hat{\lambda })$ with $\alpha = \frac{\lambda }{|\lambda |}$. Thus, $\lambda$ plays two roles - first as the Fourier transform variable and second, as the label for the coordinates relative to the basis $v_j^\alpha$ which diagonalizes $\phi ^\lambda$. Also note that the operation of Fourier transform in $t$ and the operation of expressing $z$ in terms of the $z^\alpha$ coordinates are interchangeable (i.e., these operations commute).

For a general orthonormal basis $v =\{v_1, \dots , v_n\}$, let $\mdlgwhtsquare _{LL}^{v,\hat{\lambda }}$ be the partial Fourier transform in $t$ of the sub-Laplacian $\mdlgwhtsquare _{LL}^v$. When $v=v^\alpha$, we have (from BR11):

$$\begin{equation*} \mdlgwhtsquare _{LL}^{v^\alpha , \hat{\lambda }} = -\frac{1}{4} \Delta +2i \sum _{k=1}^n \mu _k^\lambda \operatorname {Im}\{ z_k^\alpha \partial _{z_k^\alpha } \} + \sum _{k=1}^n (\mu _k^\lambda )^2 |z_k^\alpha |^2 - \left( \sum _{k \in L} \mu _k^\lambda - \sum _{k \not \in L} \mu _k^\lambda \right), \end{equation*}$$

where $\Delta$ is the ordinary Laplacian in $z=z^\alpha$ coordinates.

Also note that

$$\begin{align*} \mdlgwhtsquare ^{v^\alpha }_{LK} &= (-1)^{d_{k \ell }} [Z_{v_k^\alpha } , \bar{Z}_{v_\ell ^\alpha }] \\ &=2i (-1)^{d_{k \ell }} \operatorname {Re} \phi (v_k^\alpha , v_\ell ^\alpha ) \cdot D_t \text{ using \xhref[disp-formula]{#Z-formula}{7} and \xhref[disp-formula]{#Zbar-formula}{8}} . \end{align*}$$

Using 2, we conclude that the Fourier transform of $\mdlgwhtsquare ^{v^\alpha }_{LK}$ is

$$\begin{equation*} \mdlgwhtsquare ^{v^\alpha , \hat{\lambda }}_{LK} = -2 (-1)^{d_{k \ell }} \operatorname {Re} \phi (v_k^\alpha , v_\ell ^\alpha ) \cdot \lambda =0 \end{equation*}$$

when $L \not =K$ (i.e., when $\ell \not =k$). The significance of this calculation is that the partial Fourier transform of $\mdlgwhtsquare _{LK}$ (expressed in global coordinates) is incorporated into the operators $\mdlgwhtsquare ^{v^\alpha ,\hat{\lambda }}_{LL}$.

Next, we recall the heat kernel and Szegö kernel for the $\mdlgwhtsquare _{LL}^{v^\alpha , \hat{\lambda }}$ heat equation. Let $\tilde{H}_L (s, z^\alpha ,\hat{\lambda })$ be the “heat kernel”, i.e., the solution to the following boundary value problem:

$$\begin{align*} \left[ \frac{\partial }{\partial s} + \mdlgwhtsquare _{LL}^{v^\alpha , \hat{\lambda }} \right] \{\tilde{H}_L(s, z^\alpha ,\hat{\lambda }) \} &= 0 \text{ for } s>0, \\ \tilde{H}_L(s=0, z^\alpha , \hat{\lambda }) &=(2 \pi )^{-m/2} \delta _0 (z^\alpha ), \\ &= (2 \pi )^{-m/2} \delta _0 (z), \end{align*}$$

where $\delta _0$ is the Dirac-delta function centered at the origin in the $z$ variables. Let $\tilde{S}_L (z^\alpha , \hat{\lambda })$ be the Szegö kernel which represents orthogonal projection of $L^2 (\mathbb{C}^n)$ onto the kernel of $\mdlgwhtsquare _{LL}^{v^\alpha , \hat{\lambda }}$. Note that the tilde over the $\tilde{H}_L$ and $\tilde{S}_L$ indicates that these terms are functions rather than differential forms. By contrast, $N_K$ and $S_K$ in Theorem 2.3 do not have tildes and they are differential $(0,q)$-forms.

Theorem 3.1 (BR11).

Let $L \in \mathcal{I}_q$ be a given multiindex of length $q$ and fix a nonzero $\lambda \in \mathbb{R}^m$ and $\alpha = \frac{\lambda }{|\lambda |}$. Then

(1)

The heat kernel which solves the above boundary value problem is$$\begin{equation} \tilde{H}_L (s, z^\alpha , \hat{\lambda }) =\frac{2^{n-\nu (\alpha )}}{(2 \pi )^{m/2+ n}s^{n-\nu (\alpha )}} e^{-\frac{|z''\{\alpha \}|^2}{s}} \prod _{j=1}^{\nu (\alpha )} \frac{2e^{s \varepsilon _{j,L}^\alpha |\mu _j^\lambda | }| \mu _j^\lambda | }{\sinh (s |\mu _j^\lambda | )} e^{- |\mu _j^\lambda | \coth (s |\mu _j^\lambda | ) |z_j^\alpha |^2}. {\tag{10}} \end{equation}$$

(2)

If $\alpha \in \Gamma _L$, then the projection onto $\ker \mdlgwhtsquare _{LL}^{v^\alpha , \hat{\lambda }}$ is given by$$\begin{equation} \tilde{S}_L (z^\alpha , \hat{\lambda })=\lim _{s \to \infty } \tilde{H}_L(s, z^\alpha ,\hat{\lambda }) = \frac{4^n}{(2\pi )^{n+m/2}} \prod _{j=1}^n |\mu _j^\lambda | e^{- |\mu _j^\lambda | |z_j^\alpha |^2}, {\tag{11}} \end{equation}$$

otherwise $\tilde{S}_L (z^\alpha , \hat{\lambda })=0$.

(3)

The connection between the fundamental solution to the heat equation and the canonical relative fundamental solution to $\mdlgwhtsquare _{LL}^{v^\alpha , \hat{\lambda }}$, denoted by $\tilde{N}_L(z^\alpha , \hat{\lambda })$, is given as follows:$$\begin{equation} \tilde{N}_L(z^\alpha ,\hat{\lambda }) = \int _0^\infty \left[ \tilde{H}_L(s, z^\alpha ,\hat{\lambda })- \tilde{S}_L (z^\alpha , \hat{\lambda }) \right] ds. {\tag{12}} \end{equation}$$

In particular,$$\begin{equation} \mdlgwhtsquare _{LL}^{v^\alpha , \hat{\lambda }} \{ \tilde{N}_L(z^\alpha ,\hat{\lambda }) \} = (2 \pi )^{-m/2} (\delta _0(z) - \tilde{S}_L(z^\alpha , \hat{\lambda }) ), {\tag{13}} \end{equation}$$

Both of the kernels $\tilde{H}_L(s,\cdot ,\hat{\lambda })$ and $\tilde{S}_L(\cdot ,\hat{\lambda })$ act on $L^2 (\mathbb{C}^n)$ via a twisted convolution, $*_\lambda$, where $(f*_\lambda g)(z) = \int _{w \in \mathbb{C}^n} f(w)g(z-w) e^{-2i \lambda \cdot \operatorname {Im}\phi (z,w)} dw$, as defined in BR11, Section 5.4, but this plays no role here.

Let $\mathcal{F}_\lambda ^{-1}$ denote the inverse Fourier transform in $\lambda$ - that is, if $\tilde{f}(z, \lambda )$ is an integrable function of $\lambda \in \mathbb{R}^m$, then

$$\begin{equation*} \mathcal{F}_\lambda ^{-1} (\tilde{f}(z, \lambda ))(t) \coloneq \frac{1}{2^{m/2}} \int _{\lambda \in \mathbb{R}^m} \tilde{f}(z, \lambda ) e^{i \lambda \cdot t} d \lambda . \end{equation*}$$

Now we can formulate our relative solution to $\mdlgwhtsquare _b$ and Szegö kernel in terms of the inverse Fourier transform.

Proposition 3.2.

For a given index $K \in \mathcal{I}_q$, the relative fundamental solution to $\mdlgwhtsquare _b$ applied to a form spanned by $d\bar{z}^K$ given by $\int _0^\infty e^{-s\mdlgwhtsquare _b}(I-S_K) ds$ is

$$\begin{equation} N_K (z,t) = \mathcal{F}_\lambda ^{-1} \left\{ \sum _{L \in \mathcal{I}_q} C_{K,L} (\alpha ) \tilde{N}_L (z^\alpha , \hat{\lambda }) d \bar{Z}(z, \alpha )^L\right\}(t). {\tag{14}} \end{equation}$$

Moreover, the orthogonal projection onto the $\ker \mdlgwhtsquare _b$ applied to forms spanned by $d\bar{z}^K$ is given by convolution with the $(0,q)$-form

$$\begin{equation} S_K(z,t) = \mathcal{F}_\lambda ^{-1} \left\{ (2 \pi )^{-m/2}\sum _{L \in \mathcal{I}_q} C_{K,L} (\alpha ) \tilde{S}_L (z^\alpha ,\hat{\lambda }) d \bar{Z}(z, \alpha )^L\right\}(t). {\tag{15}} \end{equation}$$

Proof.

With the definitions of $N_K$ and $S_K$ given by 14 and 15, respectively, we shall show $\mdlgwhtsquare _b N_K=I-S_K$. On the transform side, we have

$$\begin{align*} \mdlgwhtsquare _b^{v^\alpha , \hat{\lambda }} \left\{ N_K(z^\alpha , \hat{\lambda }) \right\} &= \sum _{L \in \mathcal{I}_q} C_{K,L}(\alpha ) \mdlgwhtsquare _b^{v^\alpha , \hat{\lambda }} \left\{ \tilde{N}_L (z^\alpha , \hat{\lambda }) d \bar{Z}(z, \alpha )^L \right\} \\ &=\sum _{L \in \mathcal{I}_q} C_{K,L}(\alpha ) \mdlgwhtsquare _{LL}^{\hat{\lambda }} \big \{ \tilde{N}_L (z^\alpha , \hat{\lambda })\big \} d \bar{Z}(z, \alpha )^L \\ &= (2 \pi )^{-m/2} \sum _{L \in \mathcal{I}_q} C_{K, L} (\alpha ) \big [\delta _0(z^\alpha )- \tilde{S}_L(z^\alpha ,\hat{\lambda })\big ] d \bar{Z}(z, \alpha )^L\quad \text{ from \xhref[disp-formula]{#NLsolve}{13}} \\ &= (2 \pi )^{-m/2}\delta _0(z) \otimes 1_\lambda d \bar{z}^K \\ & \quad -(2 \pi )^{-m/2} \sum _{L \in \mathcal{I}_q} C_{K, L} (\alpha )\tilde{S}_L(z^\alpha ,\hat{\lambda }) d \bar{Z}(z, \alpha )^L \quad \text{ from \xhref[disp-formula]{#texmlid1}{3}}, \end{align*}$$

where the function $1_\lambda$ is the constant function which is 1 in the $\lambda$ coordinates. Now take the inverse Fourier transform (in $\lambda$) of both sides. The left side becomes $\mdlgwhtsquare _b N_K$ and then use the fact that $F_\lambda ^{-1} \{ (2 \pi )^{-m/2} 1_\lambda \}(t) = \delta _0(t)$ and we obtain

$$\begin{align*} \mdlgwhtsquare _b \{N_K(z,t) \} &\!=\! \delta _0(z) \delta _0 (t) d \bar{z}^K - \mathcal{F}_{\lambda }^{-1}\bigg \{(2 \pi )^{-m/2}\sum _{L \in \mathcal{I}_q} C_{K, L} (\alpha )\tilde{S}_L(z^\alpha ,\hat{\lambda }) d \bar{Z}(z, \alpha )^L\bigg \} \\ &= \delta _0(z) \delta _0 (t) d \bar{z}^K - S_K(z,t) \text{ using } \xhref[disp-formula]{#szego}{15} \end{align*}$$

as desired.

■

4. A new derivation of the integral formula – Proof of Theorem 2.3

Proof of Theorem 2.3.

We first assume the Szegö kernel is zero, that is, $|\Gamma _L|=\emptyset$ for all $L\in \mathcal{I}_q$. Consequently, it follows from 12 that $\tilde{N}_L(z^\alpha ,\hat{\lambda }) = \int _0^\infty \tilde{H}_L(s, z^\alpha ,\hat{\lambda }) ds$. To prepare for the calculation of $\tilde{N}_L(z, t)$, we use polar coordinates and write $\lambda = \alpha \tau$ where $\alpha$ belongs to the unit sphere $S^{m-1}$ and $\tau >0$. We observe

$$\begin{equation*} \frac{2e^{s \varepsilon _{j,L}^\alpha | \mu _j^\lambda |} | \mu _j^\lambda |}{\sinh (s |\mu _j^\lambda | )} = \frac{4e^{s \tau \varepsilon _{j, L}^\alpha |\mu _j^\alpha | } |\mu _j^\alpha | \tau }{e^{s| \mu _j^\alpha | \tau } - e^{-s |\mu _j^\alpha | \tau }} = \frac{4e^{s \tau ( \varepsilon _{j,L}^\alpha -1) |\mu _j^\alpha | }|\mu _j^\alpha | \tau }{1 - e^{- 2s |\mu _j^\alpha | \tau }} \end{equation*}$$

and

$$\begin{equation*} \coth (s |\mu _j^\lambda |) = \frac{e^{s \tau |\mu _j^\alpha |} + e^{-s \tau |\mu _j^\alpha |}}{e^{s \tau |\mu _j^\alpha | } - e^{-s \tau |\mu _j^\alpha |}} = \frac{1 + e^{-2s \tau |\mu _j^\alpha |}}{1 - e^{-2s \tau |\mu _j^\alpha |}}. \end{equation*}$$

We now recover $N_K (z,t)$ from 14 by computing the inverse Fourier transform using polar coordinates ($\lambda = \alpha \tau , \alpha \in S^{m-1}, \tau >0$).

$$\begin{equation*} \begin{split} & N_K ( z,t)\\ & = (2 \pi )^{-m/2} \sum _{L \in \mathcal{I}_q} \int _{s=0}^\infty \int _{\lambda \in \mathbb{R}^m} C_{K,L}(\alpha ) d \bar{Z}(z, \alpha )^L \tilde{H}_L (s, z, \hat{\lambda }) e^{it \cdot \lambda } d \lambda ds \\ & = (2 \pi )^{-m/2} \sum _{L \in \mathcal{I}_q}\int _{s=0}^\infty \int _{\tau =0}^\infty \int _{\alpha \in S^{m-1}} C_{K,L}(\alpha ) d \bar{Z}(z, \alpha )^L \tilde{H}_L (s, z, \widehat{\alpha \tau }) e^{i t \cdot \alpha \tau } \tau ^{m-1} d \alpha d \tau ds, \end{split} \end{equation*}$$

where $d\alpha$ is the surface volume form on the unit sphere in $\mathbb{R}^m$. Since $\tilde{H}_L(s, z^\alpha , \hat{\lambda })$ is assumed to be jointly integrable in the variables $s>0$ and $\lambda$, we can integrate the $s, \tau , \alpha$ variables in any order.

Let $r=e^{-2s \tau }$ in the $s$-integral and so $ds=-dr/(2 \tau r)$ and the oriented $r$-limits of integration become $1$ to $0$. We obtain

$$\begin{align*} N_K (z,t) &= \frac{4^n}{2(2 \pi )^{m+n}} \sum _{L \in \mathcal{I}_q} \int _{r=0}^1 \int _{\alpha \in S^{m-1}} C_{K,L}(\alpha ) d \bar{Z}(z, \alpha )^L \\ & \int _{\tau =0}^\infty \frac{1}{|\log r|^{n-\nu (\alpha )}} \prod _{j=1}^{\nu (\alpha )} \frac{r^{\frac{1}{2}(1-\varepsilon _{j,L}^\alpha ) |\mu _j^\alpha |}|\mu _j^\alpha |}{1- r^{|\mu _j^\alpha |}} e^{-\tau (A_\alpha ( r,z)-it\cdot \alpha )} \tau ^{n+m-2} d \tau d \alpha \frac{dr}{r}, \end{align*}$$

where

$$\begin{equation*} A_\alpha (r,z) = \frac{2}{|\log r|}|{z''}^\alpha |^2 + \sum _{j=1}^{\nu (\alpha )} |\mu _j^\alpha | \left( \frac{1+ r^{|\mu _j^\alpha |}}{1- r^{|\mu _j^\alpha |}} \right) |z_j^\alpha |^2 . \end{equation*}$$

We now perform the $\tau$-integral by using the following formula:

$$\begin{equation*} \int _{\tau =0}^\infty \tau ^p e^{-a \tau } d \tau = \frac{p!}{a^{p+1}} \text{ for } \operatorname {Re}a >0 \end{equation*}$$

which concludes the proof for 4.

Repeating this argument for the Szegö kernel using 15, we have

$$\begin{align*} S_K(z,t) &= \frac{4^n}{(2\pi )^{m+n}} \sum _{L \in \mathcal{I}_q} \int _{\alpha \in \Gamma _L} C_{K,L}(\alpha ) d \bar{Z}(z, \alpha )^L \\ & \quad \int _0^\infty \Big (\prod _{j=1}^n|\mu _j^\alpha |\Big ) \tau ^{n+m-1} e^{-\tau (\sum _{j=1}^n |\mu _j^\alpha | |z_j^\alpha |^2 - i \alpha \cdot t)} d\tau d\alpha \\ & = \frac{4^n(n+m-1)!}{(2\pi )^{m+n}} \sum _{L \in \mathcal{I}_q} \int _{\alpha \in \Gamma _L} C_{K,L}(\alpha ) d \bar{Z}(z, \alpha )^L \frac{\prod _{j=1}^n|\mu _j^\alpha |}{(\sum _{j=1}^n |\mu _j^\alpha | |z_j^\alpha |^2 - i \alpha \cdot t)^{n+m}} d\alpha \end{align*}$$

which concludes the proof for 5.

Finally, if $S_L(z,t)\neq 0$, then using 14 and 12

$$\begin{align*} & N_K(z,t)\\ &= \frac{1}{(2\pi )^{\frac{m}{2}}} \int _{s=0}^\infty \sum _{L \in \mathcal{I}_q}\\ & \left( \int _{\alpha \in \Gamma _L} C_{K,L}(\alpha ) d \bar{Z}(z, \alpha )^L \int _{\tau =0}^\infty \Big (\tilde{H}_L(s,z,\widehat{\tau \alpha }) - \tilde{S}_L(z,\widehat{\tau \alpha })\Big )e^{i\tau (\alpha \cdot t)} \tau ^{m-1} d\tau d\alpha \right) ds \\ & + \frac{1}{(2\pi )^{\frac{m}{2}}} \int _{s=0}^\infty \sum _{L \in \mathcal{I}_q} \!\!\left( \int _{\alpha \in \Gamma _L} \!\!\int _{\alpha \not \in \Gamma _L} \!\!\! C_{K,L}(\alpha ) d \bar{Z}(z, \alpha )^L \int _{\tau =0}^\infty \!\! \tilde{H}_L(s,z,\widehat{\tau \alpha })e^{i\tau (\alpha \cdot t)} \tau ^{m-1} d\tau d\alpha \right) ds \\ &= I_K +II_K. \end{align*}$$

The second set of integrals is virtually identical to what we computed earlier and we get

$$\begin{multline*} II_K = \frac{4^n(n+m-2)!}{2(2 \pi )^{m+n}} \int _{r=0}^1 \sum _{L \in \mathcal{I}_q} \left( \int _{\alpha \not \in \Gamma _L} C_{K,L}(\alpha ) d \bar{Z}(z, \alpha )^L\right.\\ \left.\prod _{j=1}^n \frac{r^{\frac{1}{2}(1-\varepsilon _{j,L}^\alpha ) |\mu _j^\alpha |}|\mu _j^\alpha |}{1- r^{|\mu _j^\alpha |}} \frac{d \alpha }{ (A(r,z)-i \alpha \cdot t)^{n+m-1}} \right) \frac{dr}{r}, \end{multline*}$$

where

$$\begin{equation*} A_\alpha (r,z) = \sum _{j=1}^n |\mu _j^\alpha | \left(\frac{1+ r^{|\mu _j^\alpha |}}{1- r^{|\mu _j^\alpha |}} \right) |z_j^\alpha |^2 . \end{equation*}$$

For the first set of integrals, we observe that

$$\begin{align*} I_K &= \frac{4^n}{2(2\pi )^{m+n}} \sum _{L \in \mathcal{I}_q} \left( \int _{r=0}^1 \int _{\alpha \in \Gamma _L} C_{K,L}(\alpha ) d \bar{Z}(z, \alpha )^L \int _{\tau =0}^\infty \right. \\ & \left. \Big [\Big (\prod _{j=1}^n \frac{|\mu _j^\alpha |}{1-r^{|\mu _j^\alpha |}}\Big ) e^{-\tau (A_\alpha (r,z)-it\cdot \alpha )} - \prod _{j=1}^n|\mu _j^\alpha |e^{\tau (A_\alpha (0,z)-it\cdot \alpha )}\Big ] \tau ^{n+m-2} d\tau d\alpha \frac{dr}{r} \right) \\ &= \frac{4^n(n+m-2)!}{2(2\pi )^{m+n}} \sum _{L \in \mathcal{I}_q} \left( \int _{r=0}^1 \int _{\alpha \in \Gamma _L} C_{K,L}(\alpha ) d \bar{Z}(z, \alpha )^L \right. \\ & \left. \Big [\Big (\prod _{j=1}^n \frac{|\mu _j^\alpha |}{1-r^{|\mu _j^\alpha |}}\Big ) \frac{1}{(A_\alpha (r,z)-it\cdot \alpha )^{n+m-1}} - \frac{\prod _{j=1}^n|\mu _j^\alpha |}{(A_\alpha (0,z)-it\cdot \alpha )^{n+m-1}}\Big ] d\alpha \frac{dr}{r} \right). \end{align*}$$

This completes the proof of 6.

That the convergence of the resulting integrals is absolute follows from a straightforward Taylor expansion argument around $r=0$ and $r=1$, the only possible points where the integrand appears to blow up.

■

5. Examples

We analyze three examples in this section, all of which fall into the cases discussed in Remark 2.4 so the formulas from Theorem 2.3 are slightly simpler. We discuss codimension $2$ quadrics in $\mathbb{C}^4$ when $q=0,2$, the Heisenberg group (so $m=1$), and the product the Heisenberg groups (so we fall into the sum of squares case).

5.1. Codimension $2$ quadrics in $\mathbb{C}^4$

When $n=m=2$, we wrote down the formulas for $N$ in the case of three canonical examples BR13:

•

$M_1$ where $\phi (z,z)=(|z_1|^2, |z_2|^2)^T$

•

$M_2$ where $\phi (z,z) = (2 \operatorname {Re}(z_1 \bar{z}_2), |z_1|^2-|z_2|^2)^T$

•

$M_3$ where $\phi (z,z) = (2 |z_1|^2, 2 \operatorname {Re}(z_1 \bar{z}_2))^T$

These examples are canonical in the sense that any quadric in $\mathbb{C}^2 \times \mathbb{C}^2$ whose Levi form has image which is not contained in a one-dimensional cone is biholomorphic to one of these three examples (see Bog91). Additionally, these three examples perfectly demonstrate the three possibilities for solvability/hypoellipticity of $\mdlgwhtsquare _b$ on quadrics.

The quadric $M_1$ is simply a Cartesian product of Heisenberg groups and both solvability and hypoellipticity are impossible for any degree. In this case,

$$\begin{equation*} A^\lambda = \begin{pmatrix} \lambda _1 & 0 \\ 0 & \lambda _2 \end{pmatrix}, \end{equation*}$$

so the eigenvalues of $A^\lambda$ are $\lambda _1$ and $\lambda _2$, so

$$\begin{equation*} \{(n^+(\lambda ),n^-(\lambda ))\} = \{(2,0),(1,0),(1,1),(0,1),(0,2)\}. \end{equation*}$$

For $M_2$, it follows from Peloso and Ricci PR03 that solvability and hypoellipticity occur for $(0,q)$ forms if and only if $q=0$ or $q=2$. In this case,

$$\begin{equation*} A^\lambda = \begin{pmatrix} \lambda _2 & \lambda _1 \\ \lambda _1 & -\lambda _2 \end{pmatrix} \end{equation*}$$

which gives us eigenvalues $\pm |\lambda |$, so that for all $\lambda \in \mathbb{R}^2\setminus \{0,0\}$, $(n^+(\lambda ),n^-(\lambda )) = (1,1)$. Additionally, we showed that the complex Green operator is given by (group) convolution with respect to the kernel

$$\begin{equation*} N_2 (z,t) = C (|z|^4+|t|^2)^{-3/2}, \end{equation*}$$

where $C$ is a constant BR13, Theorem 3.

For $M_3$, $\mdlgwhtsquare _b$ is solvable if and only if $q=0$ or $2$ and is never hypoelliptic. In this case,

$$\begin{equation*} A^\lambda = \begin{pmatrix} 2\lambda _1 & \lambda _2 \\ \lambda _2 & 0 \end{pmatrix} \end{equation*}$$

so that the eigenvalues are $\lambda _1 \pm |\lambda |$. Thus $\{(n^+(\lambda ),n^-(\lambda ))\} = \{(1,0),(1,1),(0,1)\}$ with the degenerate values occurring when $\lambda _2=0$. In Corollary 5.1, we give a more useful formula for $N$ on $M_3$ BR13. The analysis of the operator is extremely complicated and delicate and is the subject of a later work in the series BR. We must mention the paper of Nagel, Ricci, and Stein which analyzes $L^p$ estimates on a class of higher codimension quadrics in $\mathbb{C}^n \times \mathbb{C}^m$ which depend only on $|z_j|^2$, $1 \leq j \leq m$ NRS01. However, their result applies to neither $M_2$ nor $M_3$ for these quadrics cannot be described in this manner.

5.2. Example $M_3$.

Let $q=0$. As defined in Section 1,

$$\begin{equation*} M_3= \{ (z,w) \in \mathbb{C}^2 \times \mathbb{C}^2: \operatorname {Im}w_1 = 2|z_1|^2, \operatorname {Im}w_2 = 2 \operatorname {Re}(z_1 \bar{z}_2) \}. \end{equation*}$$

Here, $m=n=2$, and for $\alpha =(\cos \theta , \sin \theta )$, we easily compute $\mu _1^\alpha = 1+\cos \theta$, $\mu _2^\alpha = \cos \theta -1$. The function $\phi$ satisfies both $A_0$ and $A_2$, though we will concentrate on the case $q=0$ (the case for $q=2$ is similar). Since $L=\{0\}$, $\varepsilon _j^\alpha = - \operatorname {sgn}(\mu _j^\alpha )$ and so $\varepsilon _1^\alpha = -1$ and $\varepsilon _2^\alpha = +1$ (except when $\theta =0$ or $1$ which is a set of measure zero). We obtain

$$\begin{equation} \begin{split} N(z,t) =(2 \pi )^{-4} \int _{\theta =0}^{2 \pi } &\int _{r=0}^1 4^2\frac{ r^{\cos \theta } \sigma _1 (\theta ) \sigma _2 (\theta )}{(1-r^{\sigma _1 (\theta )})(1-r^{\sigma _2 (\theta )})} \\ &\quad \times \frac{dr d \theta }{ (-i \alpha (\theta ) \cdot t + \sigma _1 (\theta ) E_1 (r, \theta ) |z_1^\theta |^2 + \sigma _2 (\theta ) E_2 (r, \theta ) |z_2^\theta |^2)^3}, \end{split} {\tag{16}} \end{equation}$$

where

$$\begin{equation*} \alpha (\theta ) = ( \cos \theta , \sin \theta ), \sigma _1 (\theta ) = 1+ \cos \theta , \sigma _2 (\theta ) = 1- \cos \theta , E_j(r, \theta ) = \frac{1+r^{\sigma _j (\theta )}}{ 1- r^{\sigma _j (\theta )}}. \end{equation*}$$

We first wrote this formula in BR13. We wish to express it in a more useful and computable form which we will use in BR.

We let $t=(t_1, t_2)$ which gives $\alpha (\theta ) \cdot t =t_1 \cos \theta +t_2 \sin \theta$. We also let

$$\begin{equation*} x=r^{\sigma _1}, \text{ so } dx = \sigma _1 r^{\sigma _1-1} dr \text{ and } \sigma = \frac{\sigma _2}{\sigma _1} = \frac{1- \cos \theta }{1+\cos \theta }, d \theta = \frac{d \sigma }{(\sigma +1) \sqrt {\sigma }} \end{equation*}$$

and obtain

$$\begin{equation*} \cos \theta = \frac{1- \sigma }{1+ \sigma } \text{ and } \sin \theta = \frac{\pm 2 \sqrt {\sigma }}{1+ \sigma }, \end{equation*}$$

where $\pm$ is $+$ for $\theta \in [0, \pi ]$ and $-$ for $\theta \in (\pi , 2 \pi ]$. Also the interval $0 \leq \theta \leq \pi$ corresponds to the oriented $\sigma$ interval $[0 , \infty )$ and the interval $\pi \leq \theta \leq 2\pi$ corresponds to $(\infty , 0]$. In Theorem 2.3, the point $z$ is expressed in terms of the eigenvectors of $\phi ^\lambda$. To this end, we set

$$\begin{align*} z_1\{\sqrt \sigma \} &= \frac{1}{\sqrt {1+\sigma }} \big (z_1 + \sqrt {\sigma } z_2 \big ), \\ z_2\{\sqrt \sigma \} &= - \frac{1}{\sqrt {1+\sigma }} \big (\sqrt \sigma z_1 -z_2 \big ), \end{align*}$$

and

$$\begin{align*} \tilde{z}_1\{\sqrt \sigma \} &= -z_1\{\sqrt \sigma \} = \frac{1}{\sqrt {1+\sigma }} \big (-z_1 + \sqrt {\sigma } z_2 \big ), \\ \tilde{z}_2\{\sqrt \sigma \} &= -z_2\{\sqrt \sigma \} = - \frac{1}{\sqrt {1+\sigma }} \big (\sqrt \sigma z_1 +z_2 \big ). \end{align*}$$

We then obtain Corollary 5.1 to Theorem 2.3:

Corollary 5.1.

The fundamental solution to $\mdlgwhtsquare _b$ for $M_3$ on functions is given by convolution with the kernel

$$\begin{equation*} \begin{split} N (z,t) &= 2{(2 \pi )^{-4} } \int _{\sigma =0}^{ \infty } \int _{x=0}^1 \frac{ \sqrt {\sigma } (\sigma +1) }{ (1-x)(1-x^\sigma )}\\ & \quad \frac{ dx d \sigma }{\left[ - i \left( t_1 \frac{1-\sigma }{2} + t_2 \sqrt {\sigma } \right) + \left(\frac{1+x}{1-x} \right) |z_1\{\sqrt \sigma \}|^2 + \sigma \left( \frac{1+x^\sigma }{1-x^\sigma } \right) |z_2\{\sqrt \sigma \}|^2 \right]^3} \\ & + 2(2 \pi )^{-4} \int _{\sigma =0}^{ \infty } \int _{x=0}^1 \frac{ \sqrt {\sigma } (\sigma +1) }{ (1-x)(1-x^\sigma )}\\ & \quad \frac{ dx d \sigma }{\left[ - i \left( t_1 \frac{1-\sigma }{2} - t_2 \sqrt {\sigma } \right) + \left(\frac{1+x}{1-x} \right) |\tilde{z}_1\{\sqrt \sigma \}|^2 + \sigma \left( \frac{1+x^\sigma }{1-x^\sigma } \right) |\tilde{z}_2\{\sqrt \sigma \}|^2 \right]^3}. \end{split} \end{equation*}$$

This formula is the launching point for BR.

5.3. The Heisenberg group

Denote the Heisenberg group $\mathcal{H}^n \cong \mathbb{R}^{2n}\times \mathbb{R}$. The Kohn Laplacian $\mdlgwhtsquare _b$ has a nontrivial kernel in the case that $L = \emptyset$ or $L = \{1,\dots ,n\}$. The calculation for these two cases is identical and we prove the details in the case $L = \{1,\dots ,n\}$. A derivation of a related formula from the classical methods appears in Ste93, pp.615-617. We set

$$\begin{equation} \log \Big (\frac{|z|^2-it}{|z|^2+it}\Big ) = \log (|z|^2-it) - \log (|z|^2+it) {\tag{17}} \end{equation}$$

for all $z\in \mathbb{C}^n$ and $t\in \mathbb{R}$ and assume that the logarithm is defined via the principal branch.

Theorem 5.2.

On the Heisenberg group $\mathcal{H}^n$,

(1)

The relative fundamental solution $e^{-s\mdlgwhtsquare _b}(I-S_0)$ to $\mdlgwhtsquare _b = \bar{\partial }^*_b\bar{\partial }_b$ on functions is given by the integration kernel$$\begin{equation*} N_{\emptyset }(z,t) = \frac{2^{n-2}(n-1)!}{\pi ^{n+1}} \frac{1}{(|z|^2+it)^n}\Big [ \log \Big (\frac{|z|^2+it}{|z|^2-it}\Big )- \sum _{j=1}^{n-1}\frac{1}{j}\Big ]. \end{equation*}$$

(2)

The relative fundamental solution $e^{-s\mdlgwhtsquare _b}(I-S_n)$ to $\mdlgwhtsquare _b = \bar{\partial }_b\bar{\partial }^*_b$ on $(0,n)$-forms is given by the integration kernel$$\begin{equation*} N_{\{1,\dots ,n\}}(z,t) = \frac{2^{n-2}(n-1)!}{\pi ^{n+1}} \frac{1}{(|z|^2-it)^n}\Big [ \log \Big (\frac{|z|^2-it}{|z|^2+it}\Big )- \sum _{j=1}^{n-1}\frac{1}{j}\Big ]. \end{equation*}$$

Remark 5.3.

(1)

Up to a function in $\ker \mdlgwhtsquare _b$, our formula appears to be the complex conjugate of the formula in Ste93, Chapter XIII, Equation (51). This is a consequence of the fact that our computations are taken with respect to right invariant vector fields and not left invariant vector fields.

(2)

For a discussion regarding the consequences of the existence of a relative fundamental solution, we again refer the reader to Ste93, Chapter XIII, Section 4.2. It is easy to see that the convolution $N$ with a Schwartz function will be an object in $L^2$ and hence orthogonal to $\ker \mdlgwhtsquare _b$.

Proof.

Since $L = \{1,\dots ,n\}$, the Szegö kernel $S(z,\hat{\lambda }) = S_L(z,\hat{\lambda })$ has support $\operatorname {supp}S_L(z,\hat{\lambda }) =[0,\infty )$ which means (suppressing $L$)

$$\begin{equation*} N(z,t) = \frac{1}{\sqrt {2\pi }} \int _0^\infty \int _0^\infty \big (H(s,z,\hat{\lambda })-S(z,\hat{\lambda })\big )e^{it\lambda } ds d\lambda + \frac{1}{\sqrt {2\pi }} \int _{-\infty }^0 \int _0^\infty H(s,z,\hat{\lambda }) e^{it\lambda } ds d\lambda . \end{equation*}$$

Equation 6 yields

$$\begin{equation} \begin{split} &N(z,t)\\ &= \frac{4^n(n-1)!}{2(2\pi )^{n+1}}\int _0^1 \!\frac{1}{r}\Big [\frac{1}{(1-r)^n} \frac{1}{(\frac{1+r}{1-r}|z|^2-it)^n} - \frac{1}{(|z|^2-it)^n}\Big ] \!+\! \frac{r^{n-1}}{(1-r)^n} \frac{1}{(\frac{1+r}{1-r}|z|^2+it)^n} dr. \end{split} {\tag{18}} \end{equation}$$

Set $a = \frac{|z|^2-it}{|z|^2+it}$ and for $\delta >0$, $a_{\delta } = \frac{|z|^2+\delta -it}{|z|^2+\delta +it}$ (so $|a|=|a_{\delta }|=1$). The reason that we introduce $a_{\delta }$ is that a logarithm appears in the integral, and $\log a$ is not well defined with the principal branch if $|z|^2=0$. By introducing $\delta$, it is immediate that for any $a_{\delta }$

$$\begin{equation*} \log a_{\delta } = \log (|z|^2+\delta -it) - \log (|z|^2+\delta +it) \end{equation*}$$

and by sending $\delta \to 0$, we obtain $\log a$ as in 17. Ignoring the constants, we compute

$$\begin{align*} I_\delta &= \int _0^1 \frac{1}{r} \Big [\frac{1}{((1+r)(|z|^2+\delta )it(1-r))^n} - \frac{1}{(|z|^2+\delta -it)^n}\Big ]\\ & \qquad \quad + \frac{r^{n-1}}{((1+r)(|z|^2+\delta )+it(1-r))^n} dr \\ & = \int _0^1 \frac{1}{r}\Big [\frac{1}{((|z|^2+\delta +it)r+|z|^2+\delta -it)^n} - \frac{1}{(|z|^2+\delta -it)^n}\Big ]\\ & \qquad \quad + \frac{r^{n-1}}{((|z|^2+\delta -it)r + (|z|^2+\delta +it))^n} dr \\ & = \frac{1}{(|z|^2+\delta +it)^n} \bigg [\int _0^1\Big ( \frac{1}{(r+a_{\delta })^n} -\frac{1}{a_{\delta }^n}\Big ) \frac{dr}{r} + \int _0^1 \frac{r^{n-1}}{(a_{\delta } r + 1)^n} dr \bigg ]. \end{align*}$$

For the second integral, we change variables $r = \frac{1}{s}$ and compute

$$\begin{equation*} \int _0^1 \frac{s^{n-1}}{(a_{\delta } s+1)^n} ds = \int _1^\infty \frac{1}{(r+a_{\delta } )^n}\frac{dr}{r}. \end{equation*}$$

Thus,

$$\begin{equation*} (|z|^2+\delta +it)^n I_\delta = \lim _{\epsilon \to 0} \bigg [\int _\epsilon ^\infty \frac{1}{(r+a_{\delta } )^n}\frac{dr}{r} +\frac{1}{a_{\delta } ^n}\log \epsilon \bigg ]. \end{equation*}$$

A geometric series argument shows that

$$\begin{equation*} \frac{1}{r(r+a_{\delta } )^n} = \frac{1}{a_{\delta } ^n r} - \frac{1}{a_{\delta } ^n(r+a_{\delta } )} - \sum _{j=1}^{n-1} \frac{1}{a_{\delta } ^{n-j}(r+a_{\delta } )^{j+1}}. \end{equation*}$$

Therefore

$$\begin{equation*} \begin{split} &(|z|^2+\delta +it)^n I_\delta \\ &\quad = \lim _{\epsilon \to 0} \bigg [\int _\epsilon ^\infty \frac{1}{a_{\delta } ^n r} - \frac{1}{a_{\delta } ^n(r+a_{\delta } )} dr -\sum _{j=1}^{n-1} \int _\epsilon ^\infty \frac{1}{a_{\delta } ^{n-j}(r+a_{\delta } )^{j+1}} dr +\frac{1}{a_{\delta } ^n}\log \epsilon \bigg ] \\ &\quad = \lim _{\epsilon \to 0}\bigg [\frac{\log (a_{\delta } +\epsilon )}{a_{\delta } ^n} -\sum _{j=1}^{n-1}\frac{1}{ja_{\delta } ^{n-j}(a_{\delta } +\epsilon )^j}\bigg ] = \frac{1}{a_{\delta } ^n}\Big (\log a_{\delta } - \sum _{j=1}^{n-1}\frac{1}{j}\Big ). \end{split} \end{equation*}$$

Thus, if we set $N_\delta (z,t)$ to equal the right hand side of 18 except with $|z|^2$ replaced by $|z|^2+\delta$, then $\frac{|z|^2+\delta -it}{|z|^2+\delta -it}$ stays away from the branch cut and

$$\begin{align*} &N_\delta (z,t)\\ &\quad = \frac{2^{n-2}(n-1)!}{\pi ^{n+1}} \frac{1}{(|z|^2+\delta -it)^n}\Big [ \log \Big (\frac{|z|^2+\delta -it}{|z|^2+\delta +it}\Big )- \sum _{j=1}^{n-1}\frac{1}{j}\Big ]\\ &\quad = \frac{2^{n-2}(n-1)!}{\pi ^{n+1}} \frac{1}{(|z|^2+\delta -it)^n}\Big [ \log (|z|^2+\delta -it) - \log (|z|^2+\delta +it) - \sum _{j=1}^{n-1}\frac{1}{j}\Big ]. \end{align*}$$

This function is continuous in $\delta$, thus we may send $\delta \to 0$ and obtain the theorem.

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5.4. The Cartesian product of Heisenberg groups

In contrast to the explicit computability of the Heisenberg group case, if

$$\begin{equation*} M = \{(z,w)\in \mathbb{C}^2\times \mathbb{C}^2 : \operatorname {Im}w_j = |z_j|^2\}, \end{equation*}$$

$L = \{1,2\}$, and $\alpha = (\cos \theta ,\sin \theta )$, then $\Gamma ^\alpha _{\{1,2\}}$ is the first quadrant and from Theorem 2.3, we have

$$\begin{equation*} \begin{split} N_{\{1,2\}}(z,t) &=\frac{1}{\pi ^4}\int _{r=0}^1 \int _{\frac{\pi }{2}}^\pi |\cos \theta \sin \theta | \frac{r^{|\cos \theta |}}{(1-r^{|\cos \theta |})(1-r^{|\sin \theta |})} \frac{1}{(A_{\alpha }(r)-i(t_1\cos \theta +t_2\sin \theta ))^{n+m-1}} d\theta \frac{dr}{r} \\ &\qquad +\frac{1}{\pi ^4}\int _{r=0}^1 \int _\pi ^{\frac{3\pi }{2}} |\cos \theta \sin \theta | \frac{r^{|\cos \theta +\sin \theta |}}{(1-r^{|\cos \theta |})(1-r^{|\sin \theta |})} \frac{1}{(A_{\alpha }(r)-i(t_1\cos \theta +t_2\sin \theta ))^{n+m-1}} d\theta \frac{dr}{r}\\ & \qquad +\frac{1}{\pi ^4}\int _{r=0}^1 \int _{\frac{3\pi }{2}}^{2\pi } |\cos \theta \sin \theta | \frac{r^{|\sin \theta |}}{(1-r^{|\cos \theta |})(1-r^{|\sin \theta |})}\frac{1}{(A_{\alpha }(r)-i(t_1\cos \theta +t_2\sin \theta ))^{n+m-1}} d\theta \frac{dr}{r}\\ & \qquad +\frac{1}{\pi ^4} \int _{r=0}^1 \int _0^{\frac{\pi }{2}} \cos \theta \sin \theta \left[\frac{1}{(1-r^{\cos \theta })(1-r^{\sin \theta })}\frac{1}{(A_{\alpha }(r)-i(t_1\cos \theta +t_2\sin \theta ))^{n+m-1}} -\frac{1}{(A_{\alpha }(0)-i(t_1\cos \theta +t_2\sin \theta ))^{n+m-1}}\right] d\theta \frac{dr}{r} \end{split} \end{equation*}$$

where

$$\begin{equation*} A_{\alpha }(r) = |\cos \theta | \left(\frac{1+ r^{|\cos \theta |}}{1- r^{|\cos \theta |}} \right) |z_1|^2 + |\sin \theta | \left(\frac{1+ r^{|\sin \theta |}}{1- r^{|\sin \theta |}} \right) |z_2|^2. \end{equation*}$$

On the other hand, using 5, we compute the Szegö kernel

$$\begin{equation*} \begin{split} S_{\emptyset }(z,t) &= \frac{6}{\pi ^4}\int _\pi ^{\frac{3\pi }{2}} \frac{\cos \theta \sin \theta }{((|z_1|^2+it_1)\cos \theta +(|z_2|^2+it_2)\sin \theta )^4} d\theta \\ & = \frac{1}{\pi ^4(|z_1|^2+it_1)^2(|z_2|^2+it_2)^2}. \end{split} \end{equation*}$$