On the weighted $L^2$ estimate for the $k$-Cauchy-Fueter operator and the weighted $k$-Bergman kernel

TL;DR

This paper develops weighted $L^2$ estimates for the $k$-Cauchy-Fueter operator, introduces the weighted $k$-Bergman space and projection, and analyzes the asymptotic behavior of the associated kernel function, extending complex analysis concepts to quaternionic settings.

Contribution

It establishes weighted $L^2$ estimates for the $k$-Cauchy-Fueter operator and introduces the $k$-Bergman projection and kernel, providing new tools for quaternionic analysis.

Findings

Weighted $L^2$ estimates for the $k$-Cauchy-Fueter operator.

Introduction of the $k$-Bergman orthogonal projection and kernel.

Asymptotic decay properties of the kernel function.

Abstract

The $k$-Cauchy-Fueter operators, $k=0,1,\ldots$, are quaternionic counterparts of the Cauchy-Riemann operator in the theory of several complex variables. The weighted $L^2$ method to solve Cauchy-Riemann equation is applied to find the canonical solution to the non-homogeneous $k$-Cauchy-Fueter equation in a weighted $L^2$-space, by establishing the weighted $L^2$ estimate. The weighted $k$-Bergman space is the space of weighted $L^2$ integrable functions annihilated by the $k$-Cauchy-Fueter operator, as the counterpart of the Fock space of weighted $L^2$-holomorphic functions on $\mathbb{C}^n$. We introduce the $k$-Bergman orthogonal projection to this closed subspace, which can be nicely expressed in terms of the canonical solution operator, and its matrix kernel function. We also find the asymptotic decay for this matrix kernel function.

Equations

$$\begin{array}{rl}{\mathcal{D}}_0^{(k)}:C^{\infty }\left({\mathbb{R}}^{4n},{\odot }^k{\mathbb{C}}^2\right)& ⟶C^{\infty }\left({\mathbb{R}}^{4n},{\odot }^{k-1}{\mathbb{C}}^2\otimes {\mathbb{C}}^{2n}\right),\end{array}$$

$$\mathscr{D}:C^{1}(\mathbb{H}^{n},\mathbb{H})\rightarrow C(\mathbb{H}^{n},\mathbb{H}^{n}),\qquad\qquad\mathscr{D}f=\left(\begin{array}[]{c}\overline{\partial}_{{\mathbf{q}}_{0}}f\\ \vdots\\ \overline{\partial}_{{\mathbf{q}}_{n-1}}f\end{array}\right),$$

$${\mathcal{D}}_0^{(k)}u=f,$$

$${\odot }^k{\mathbb{C}}^2\simeq {\mathbb{C}}^{k+1},{\odot }^{k-1}{\mathbb{C}}^2\otimes {\mathbb{C}}^{2n}\simeq {\mathbb{C}}^{2kn},$$

$${\mathcal{D}}_1^{(k)}f=0,$$

$$\begin{array}{rl}0\to C^{\infty }\left({\mathbb{R}}^{4n},{\mathcal{V}}_0\right)& \stackrel{{\mathcal{D}}_0^{(k)}}{\to }{C}^{\infty }\left({\mathbb{R}}^{4n},{\mathcal{V}}_1\right)\stackrel{{\mathcal{D}}_1^{(k)}}{\to }{C}^{\infty }\left({\mathbb{R}}^{4n},{\mathcal{V}}_2\right)\stackrel{{\mathcal{D}}_2^{(k)}}{\to }\cdots ,\end{array}$$

$$\begin{array}{r}{\mathcal{V}}_0:={\odot }^k{\mathbb{C}}^2,{\mathcal{V}}_1:={\odot }^{k-1}{\mathbb{C}}^2\otimes {\mathbb{C}}^{2n},{\mathcal{V}}_2:={\odot }^{k-2}{\mathbb{C}}^2\otimes {\wedge }^2{\mathbb{C}}^{2n}.\end{array}$$

$$(u,v{)}_{\phi }:={\int }_{{\mathbb{R}}^{4n}}u\overline{v}{e}^{-2\phi }dV,$$

$$⟨f,g{⟩}_{\phi }:={\int }_{{\mathbb{R}}^{4n}}⟨f,g⟩e^{-2\phi }dV,$$

$$A_{(k)}^2({\mathbb{R}}^{4n},\phi ):=\left\{f\in L_{\phi }^2({\mathbb{R}}^{4n},{\odot }^k{\mathbb{C}}^2);{\mathcal{D}}_0^{(k)}f=0\right\}.$$

$$L_{\phi }^2({\mathbb{R}}^{4n},{\mathcal{V}}_0)\stackrel{{\mathcal{D}}_0}{\to }{L}_{\phi }^2({\mathbb{R}}^{4n},{\mathcal{V}}_1)\stackrel{{\mathcal{D}}_1}{\to }{L}_{\phi }^2({\mathbb{R}}^{4n},{\mathcal{V}}_2)$$

$${\mathcal{D}}_{0}u\in \mathrm{Dom}({\mathcal{D}}_1)\mathrm{and}{\mathcal{D}}_1{\mathcal{D}}_{0}u=0.$$

$${\square }_{\phi }:={\mathcal{D}}_0{\mathcal{D}}_0^{\ast }+{\mathcal{D}}_1^{\ast }{\mathcal{D}}_1.$$

$$\Vert N_{\phi }f{\Vert }_{\phi }\le \frac{1}{4}\Vert f{\Vert }_{\phi },\mathrm{for}\mathrm{any}f\in L_{\phi }^2({\mathbb{R}}^{4n},{\mathcal{V}}_1).$$

$${\mathcal{D}}_0{\mathcal{D}}_0^{\ast }{N}_{\phi }f=f$$

$$\Vert {\mathcal{D}}_0^{\ast }{N}_{\phi }f{\Vert }_{\phi }\le \frac{1}{2}\Vert f{\Vert }_{\phi },\Vert {\mathcal{D}}_1{N}_{\phi }f{\Vert }_{\phi }\le \frac{1}{2}\Vert f{\Vert }_{\phi }.$$

$$4{\Vert f\Vert }_{\phi }^2\le {\Vert {\mathcal{D}}_0^{\ast }f\Vert }_{\phi }^2+{\Vert {\mathcal{D}}_{1}f\Vert }_{\phi }^2$$

$$Pf=f-{\mathcal{D}}_0^{\ast }{N}_{\phi }{\mathcal{D}}_{0}f$$

$$f(x)={\int }_{{\mathbb{R}}^{4n}}K(x,y)f(y)e^{-2\phi }dV$$

$$|K(x,y)|\le C{e}^{|x{|}^2+|y{|}^2+\frac{\epsilon }{2}(|x|+|y|)-\epsilon |x-y|}$$

$$(Z_{AA^{\prime}}):=\left(\begin{array}[]{ll}Z_{00^{\prime}}&{Z}_{01^{\prime}}\\ Z_{10^{\prime}}&{Z}_{11^{\prime}}\\ \quad\vdots&\quad\vdots\\ Z_{(2l)0^{\prime}}&{Z}_{(2l)1^{\prime}}\\ Z_{(2l+1)0^{\prime}}&{Z}_{(2l+1)1^{\prime}}\\ \quad\vdots&\quad\vdots\\ \end{array}\right):=\left(\begin{array}[]{ll}\partial_{x_{1}}+\textbf{i}\partial_{x_{2}}&-\partial_{x_{3}}-\textbf{i}\partial_{x_{4}}\\ \partial_{x_{3}}-\textbf{i}\partial_{x_{4}}&\hskip 8.53581pt\partial_{x_{1}}-\textbf{i}\partial_{x_{2}}\\ \qquad\vdots&\qquad\vdots\\ \partial_{x_{4l+1}}+\textbf{i}\partial_{x_{4l+2}}&-\partial_{x_{4l+3}}-\textbf{i}\partial_{x_{4l+4}}\\ \partial_{x_{4l+3}}-\textbf{i}\partial_{x_{4l+4}}&\hskip 8.53581pt\partial_{x_{4l+1}}-\textbf{i}\partial_{x_{4l+2}}\\ \qquad\vdots&\qquad\vdots\\ \end{array}\right),$$

$$x_{1}+\mathbf{i}x_{2}+\mathbf{j}x_{3}+\mathbf{k}x_{4}\longmapsto\left(\begin{array}[]{rr}x_{1}+\textbf{i}x_{2}&-x_{3}-\textbf{i}x_{4}\\ x_{3}-\textbf{i}x_{4}&\hskip 8.53581ptx_{1}-\textbf{i}x_{2}\end{array}\right).$$

$$(\varepsilon_{A^{\prime}B^{\prime}})=\left(\begin{array}[]{cc}0&1\\ -1&0\end{array}\right),\qquad(\varepsilon^{A^{\prime}B^{\prime}})=\left(\begin{array}[]{cc}0&-1\\ 1&0\end{array}\right)$$

$$Z_A^{A^{\prime }}=\sum _{B^{\prime }=0^{\prime },1^{\prime }}{Z}_{A{B}^{\prime }}{\epsilon }^{B^{\prime }{A}^{\prime }}=Z_{A{0}^{\prime }}{\epsilon }^{0^{\prime }{A}^{\prime }}+Z_{A{1}^{\prime }}{\epsilon }^{1^{\prime }{A}^{\prime }}.$$

$${\epsilon }^{1^{\prime }{0}^{\prime }}=-{\epsilon }^{0^{\prime }{1}^{\prime }}=1,{\epsilon }^{0^{\prime }{0}^{\prime }}={\epsilon }^{1^{\prime }{1}^{\prime }}=0$$

$$\left(Z_{A}^{A^{\prime}}\right):=\left(\begin{array}[]{ll}Z_{0}^{0^{\prime}}&{Z}_{0}^{1^{\prime}}\\ Z_{1}^{0^{\prime}}&{Z}_{1}^{1^{\prime}}\\ \quad\vdots&\quad\vdots\\ Z_{(2n-2)}^{0^{\prime}}&{Z}_{(2n-2)}^{1^{\prime}}\\ Z_{(2n-1)}^{0^{\prime}}&{Z}_{(2n-1)}^{1^{\prime}}\\ \end{array}\right):=\left(\begin{array}[]{ll}{Z}_{01^{\prime}}&-Z_{00^{\prime}}\\ {Z}_{11^{\prime}}&-Z_{10^{\prime}}\\ \quad\vdots&\quad\vdots\\ {Z}_{(2n-2)1^{\prime}}&-Z_{(2n-2)0^{\prime}}\\ {Z}_{(2n-1)1^{\prime}}&-Z_{(2n-1)0^{\prime}}\\ \end{array}\right).$$

$$(\epsilon_{AB})=\left(\begin{array}[]{rrrrr}0&1&&&\\ -1&0&&&\\ &&\ddots&&\\ &&&0&1\\ &&&-1&0\end{array}\right)$$

$$Z_{\phi }^{\ast }=-\overline{Z}+2\overline{Z}\phi .$$

$$Z^{A{A}^{\prime }}=\overline{Z_{A{A}^{\prime }}},$$

$${\left(Z_A^{A^{\prime }}\right)}_{\phi }^{\ast }=Z_{A^{\prime }}^A-2Z_{A^{\prime }}^A\phi .$$

Full text

Department of Mathematics, Zhejiang University, Zhejiang 310027, PR China, Email: wwang@zju.edu.cn.

Supported by National Nature Science Foundation in China (No. 11571305)

1. Introduction

The $k$-Cauchy-Fueter operators over $\mathbb{R}^{4n}$

[TABLE]

$k=0,1,\ldots$, are quaternionic counterparts of the Cauchy-Riemann operator $\overline{\partial}$ in the theory of several complex variables, where $\odot^{p}\mathbb{C}^{2}$ is the $p$-th symmetric tensor product of $\mathbb{C}^{2}$. If we write a vector in the quaternionic space $\mathbb{H}^{n}$ as $\mathbf{q}=(\mathbf{q}_{0},\ldots,\mathbf{q}_{n-1})$, the usual Cauchy-Fueter operator is defined as

[TABLE]

for $f\in C^{1}(\mathbb{H}^{n},\mathbb{H})$, where $\overline{\partial}_{\mathbf{q}_{l}}=\partial_{x_{4l+1}}+\mathbf{i}\partial_{x_{4l+2}}+\mathbf{j}\partial_{x_{4l+3}}+\mathbf{k}\partial_{x_{4l+4}},$ if we write $\mathbf{q}_{l}=x_{4l+1}+x_{4l+2}\mathbf{i}+x_{4l+3}\mathbf{j}+x_{4l+4}\mathbf{k}\in\mathbb{H}$, $l=0,1,\ldots,n-1$. It is known that the Cauchy-Fueter operator coincides with the $1$-Cauchy-Fueter operator [13]. In the quaternionic case, we have a family of operators acting on $\odot^{k}\mathbb{C}^{2}$-valued functions, $k=0,1,\ldots$, because SU$(2)$ as the group of unit quaternions has a family of irreducible representations $\odot^{k}\mathbb{C}^{2}$, while $S^{1}$ as the group of unit complex numbers has only one irreducible representation. The $k$-Cauchy-Fueter operators over $\mathbb{R}^{4}$ also have the origin in physics: they are the elliptic version of spin $k/2$ massless field operators over the Minkowski space (cf. e.g. [4] [11] [16] [17]): $\mathscr{D}_{0}^{(1)}\phi=0$ corresponds to the Dirac-Weyl equation whose solutions correspond to neutrinos; $\mathscr{D}_{0}^{(2)}\phi=0$ corresponds to the Maxwell equation whose solutions correspond to photons; $\mathscr{D}_{0}^{(3)}\phi=0$ corresponds to the Rarita-Schwinger equation; $\mathscr{D}_{0}^{(4)}\phi=0$ corresponds to linearized Einstein’s equation whose solutions correspond to weak gravitational fields; etc..

To develop the function theory of several quaternionic variables, we need to solve the non-homogeneous $k$-Cauchy-Fueter equation:

[TABLE]

where $u$ is $\odot^{k}\mathbb{C}^{2}$-valued and $f$ is $\odot^{k-1}\mathbb{C}^{2}\otimes\mathbb{C}^{2n}$-valued. Under the identification

[TABLE]

${\mathscr{D}}_{0}^{(k)}$ is a $2kn\times(k+1)$-matrix valued differential operator of the first order with constant coefficients. The equation (1.1) is overdetermined and its compatibility condition is that $f$ is ${\mathscr{D}}_{1}^{(k)}$-closed, i.e.

[TABLE]

where ${\mathscr{D}}_{1}^{(k)}$ is the second operator in the $k$-Cauchy-Fueter complex:

[TABLE]

and

[TABLE]

Here $\wedge^{2}\mathbb{C}^{2n}$ is the $2$-th exterior product of $\mathbb{C}^{2n}$. These complexes play the role of Dolbeault complex in several complex variables, and are now explicitly known [21] (cf. also [1] [2] [3] [7] [8]).

The author [20] [21] solved the non-homogeneous $k$-Cauchy-Fueter equation in $L^{2}$-space over ${\mathbb{R}}^{4n}$ by using the method of classical harmonic analysis, and deduced Hartogs’ phenomenon and integral representation formulae. In this paper, the weighted $L^{2}$ method to solve the $\overline{\partial}$ equation on $\mathbb{C}^{n}$ (see e.g. [9] [12] [14] and references therein) is extended to solve the non-homogeneous $k$-Cauchy-Fueter equation (1.1). The $L^{2}$ method is a general method to deal with overdetermined systems of linear differential equations when we can establish the necessary $L^{2}$ estimate, e.g. it is applied to the Dirac operator in Clifford analysis [15]. The reason to consider the weighted $L^{2}$-space is as follows. $f$ is called $k$-regular if ${\mathscr{D}}_{0}^{(k)}f=0$ in the sense of distributions. It is known that the space of $k$-regular polynomials are infinite dimensional (cf. [13]), and such functions are $L^{2}$-integrable with Gaussian weight. This is similar to complex analysis, where one consider the space of $L^{2}$-integrable holomorphic functions with Gaussian weight, called Fock space. Without a weight, a $L^{2}$-integrable holomorphic (or $k$-regular) function must vanish. Given a nonnegative function $\varphi$, called a weighted function, consider the Hilbert space $L_{\varphi}^{2}(\mathbb{R}^{4n},\mathbb{C})$ with the weighted inner product

[TABLE]

where $dV$ is the Lebegues measure on $\mathbb{R}^{4n}$. For a complex linear space $\mathscr{V}$ with an inner product $\langle\cdot,\cdot\rangle$ (e.g. $\mathscr{V}=\odot^{k}\mathbb{C}^{2}$ or $\odot^{k-1}\mathbb{C}^{2}\otimes\mathbb{C}^{2n}$), we define $L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V})$ with the weighted inner product

[TABLE]

and the weighted norm $\|f\|_{\varphi}:=\langle f,f\rangle_{\varphi}^{\frac{1}{2}}$. The weighted $k$-Bergman space with respect to weight $\varphi=|x|^{2}$ is then defined as

[TABLE]

It is infinite dimensional [13] because $k$-regular polynomials are integrable with respect to this weight.

In the sequel, we will drop the superscript for fixed $k$ for simplicity.

[TABLE]

is a complex, i.e. for any $u\in{\rm Dom}({\mathscr{D}}_{0})$,

[TABLE]

Then if $f\in L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{1})$ is ${\mathscr{D}}_{1}$-closed, the nonhomogeneous $k$-Cauchy-Fueter equation (1.1) has at most one solution $u\in{\rm Dom}({\mathscr{D}}_{0})$ orthogonal to $A^{2}_{(k)}(\mathbb{R}^{4n},\varphi)$. If it exists, it is called the canonical solution to the nonhomogeneous $k$-Cauchy-Fueter equation (1.1). Consider the associated Laplacian operator $\Box_{\varphi}:L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{1})\longrightarrow L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{1})$ given by

[TABLE]

Theorem 1.1**.**

Suppose that $\varphi(x)=|x|^{2}$ and $k=2,3,\ldots$. Then

$\Box_{\varphi}$* has a bounded, self-adjoint and non-negative inverse $N_{\varphi}$ such that*

[TABLE]

${\mathscr{D}}_{0}^{*}N_{\varphi}f$* is the canonical solution operator to the nonhomogeneous $k$-Cauchy-Fueter equation (1.1), i.e. if $f\in{\rm Dom}({\mathscr{D}}_{1})$ is ${\mathscr{D}}_{1}$-closed, then*

[TABLE]

and ${\mathscr{D}}_{0}^{*}N_{\varphi}f$ orthogonal to $A^{2}_{(k)}(\mathbb{R}^{4n},\varphi)$. Moreover,

[TABLE]

The key step to prove this theorem is to establish the following weighted $L^{2}$ estimate.

Theorem 1.2**.**

Suppose that $\varphi(x)=|x|^{2}$ and $k=2,3,\ldots$. Then

[TABLE]

for any $f\in{\rm Dom}({\mathscr{D}}_{0}^{*})\cap{\rm Dom}({\mathscr{D}}_{1})$.

The reason we only consider the weight $\varphi(x)=|x|^{2}$ is that the weighted $L^{2}$ estimate in this case is relatively easier. On $\mathbb{R}^{4n}$ for $n>1$, the operators ${\mathscr{D}}_{0}^{(0)}$ and ${\mathscr{D}}_{1}^{(1)}$ are differential operators of the second order, and the weighted $L^{2}$ estimate is more difficult in these cases. While on $\mathbb{R}^{4}$, the $k$-Cauchy-Fueter complexes for $k=0,1$ are trivial. So we restrict to the case $k\geq 2$.

The weighted $k$-Bergman space $A^{2}_{(k)}(\mathbb{R}^{4n},\varphi)$ is a closed Hilbert subspace. We call the orthogonal projection $P:L_{\varphi}^{2}(\mathbb{R}^{4n},\odot^{k}\mathbb{C}^{2})\longrightarrow A^{2}_{(k)}(\mathbb{R}^{4n},\varphi)$ the weighted $k$-Bergman projection. It can be nicely expressed in terms of the the canonical solution operator as

[TABLE]

for $f\in{\rm Dom}({\mathscr{D}}_{0})$, as in the theory of several complex variables (cf. theorem 4.4.5 in [5]).

If we use the first isomorphism in (1.2), a function in $L_{\varphi}^{2}(\mathbb{R}^{4n},\odot^{k}\mathbb{C}^{2})$ is $\mathbb{C}^{k+1}$-valued. The weighted $k$-Bergman projection $P$ has a kernel $K(x,y)$ such that the following integral formula holds

[TABLE]

for any $f\in A^{2}_{(k)}(\mathbb{R}^{4n},\varphi)$. The kernel $K(x,y)$ is a $(k+1)\times(k+1)$-matrix valued function, which is $k$-regular in variables $x$ and anti-$k$-regular in variables $y$.

The main difference between the $k$-Cauchy-Fueter complexes and Dolbeault complex in the theory of several complex variables is that there exist symmetric forms except for the exterior forms. The analysis of exterior forms is classical, while the analysis of symmetric forms is relatively new. We can handle components of a $\odot^{k}\mathbb{C}^{2}$- or $\odot^{k-1}\mathbb{C}^{2}\otimes\mathbb{C}^{2n}$-valued function. Such notations are used by physicists as two-spinor notations for the massless field operators (cf. e.g. [16] [17] and references therein). They also appear in studying of quaternionic manifolds (cf. e.g. [22] and references therein).

The weighted $L^{2}$ estimate for the model case: $n=1$ and $k=2$, is obtain in section 2. The general case is proved in section 3. Based on the weighted $L^{2}$ estimate, Theorem 1.1 is proved in section 3. In section 4, we establish a localized a priori estimate for $\Box_{\varphi}$ and the Caccioppoli-type estimate, which hold for many systems of PDEs of the divergence form. From these estimates and the weighted $L^{2}$ estimate, we derive the asymptotic decay of the canonical solution ${\mathscr{D}}_{0}^{*}N_{\varphi}f$ to the nonhomogeneous $k$-Cauchy-Fueter equation (1.1) when $f$ is compactly supported. Then by choosing suitable $f$ in (1.9), we find the asymptotic estimate for the weighted $k$-Bergman kernel from the asymptotic behavior of the canonical solution.

Theorem 1.3**.**

Suppose that $\varphi(x)=|x|^{2}$ and $k=2,3,\ldots$. Then we have the following pointwise estimate for the weighted $k$-Bergman kernel: there exists $\varepsilon>0$ only depending on $k,n$ such that

[TABLE]

for any $x,y\in\mathbb{R}^{4n}$ with $|x-y|>3$, and some constant $C>0$ only depending on $k,n,\varepsilon$.

The first estimate for the Bergman kernel of the weighted $L^{2}$-holomorphic functions over the complex plane $\mathbb{C}$ is due to Christ [6]. The result of Christ was extended by Delin [10] to several complex variables for strict plurisubharmonic weights. See also [9] [14] and references therein for recent results. Our estimate is a little bit weaker than the complex case because we have an extra factor $e^{\frac{\varepsilon}{2}(|x|+|y|)}$. But the estimate is the same when $|y|$ is larger compared to $|x|$ (cf. Remark 4.1).

I would like to thank the referee for valuable suggestions.

2. The weighted $L^{2}$ estimate in the model case: $n=1$ and $k=2$

2.1. The complex vector fields $Z_{AA^{\prime}}$’s on $\mathbb{R}^{4n}$ and their formal adjoints

To give the definition of the $k$-Cauchy-Fueter operator, we need the following complex vector fields

[TABLE]

where $A=0,\ldots,2n-1$, $A^{\prime}=0^{\prime},1^{\prime}$. This is motivated by the embedding of the quaternion algebra into the space of complex $2\times 2$-matrices:

[TABLE]

We will use

[TABLE]

to raise or lower primed indices, where $(\varepsilon^{A^{\prime}B^{\prime}})$ is the inverse of $(\varepsilon_{A^{\prime}B^{\prime}})$, i.e., $\sum_{B^{\prime}=0^{\prime},1^{\prime}}\varepsilon_{A^{\prime}B^{\prime}}\varepsilon^{B^{\prime}C^{\prime}}=\delta_{A^{\prime}}^{C^{\prime}}=\sum_{B^{\prime}=0^{\prime},1^{\prime}}\varepsilon^{C^{\prime}B^{\prime}}\varepsilon_{B^{\prime}A^{\prime}}.$ For example,

[TABLE]

In particular, we have $Z_{A}^{0^{\prime}}=Z_{A1^{\prime}},Z_{A}^{1^{\prime}}=-Z_{A0^{\prime}}$ by

[TABLE]

in (2.2). Then

[TABLE]

We also use

[TABLE]

and $(\epsilon^{AB})$, the inverse of $(\epsilon_{AB})$, to raise or lower unprimed indices, e.g. $Z_{A^{\prime}}^{A}=\sum_{B=0}^{2n-1}Z_{A^{\prime}B}\epsilon^{BA}.$ The advantage of using raising indices is that the adjoint of $Z_{A}^{A^{\prime}}$ can be written in a very simple form.

Proposition 2.1**.**

(1) The formal adjoint operator $Z_{\varphi}^{*}$ of a complex vector field $Z$ is

[TABLE]

(2) We have

[TABLE]

and the formal adjoint operator of of $Z_{A}^{A^{\prime}}$ is

[TABLE]

Proof.

(1) For a complex vector field $Z$, we have

[TABLE]

for $u,v\in C_{0}^{\infty}(\Omega,\mathbb{C})$. This is because

[TABLE]

(2) By raising indices, $Z^{AA^{\prime}}=\sum_{B=0}^{2n-1}\sum_{B^{\prime}=0^{\prime},1^{\prime}}Z_{BB^{\prime}}\epsilon^{BA}\varepsilon^{B^{\prime}A^{\prime}}$. It is direct from definition of $Z_{AA^{\prime}}$’s in (2.1) to see that

[TABLE]

by (2.3) and similar relations for $\epsilon^{AB}$. Then $\overline{Z_{AA^{\prime}}}=Z^{AA^{\prime}}$. Since $(Z_{A}^{A^{\prime}})_{\varphi}^{*}=-\overline{Z_{A}^{A^{\prime}}}+2\overline{Z_{A}^{A^{\prime}}\varphi}$ by (1), and

[TABLE]

we get (2.7). Here $\varepsilon^{B^{\prime}A^{\prime}}=-\varepsilon_{B^{\prime}A^{\prime}}$ by (2.2). ∎

We will use the notations of the following complex differential operators:

[TABLE]

for $A=0,\ldots,2n-1$, $A^{\prime}=0^{\prime},1^{\prime}$. Then we have $(Z_{A}^{A^{\prime}})_{\varphi}^{*}=\delta_{A^{\prime}}^{A}$ and

[TABLE]

for $u,v\in C_{0}^{1}(\Omega,\mathbb{C})$. By taking conjugate, we also have

[TABLE]

2.2. The weighted $L^{2}$ estimate in the model case $n=1$ and $k=2$

In this case,

[TABLE]

By definition, $\odot^{2}\mathbb{C}^{2}$ is a subspace of $\otimes^{2}\mathbb{C}^{2}$, and an element $f$ of $L_{\varphi}^{2}(\mathbb{R}^{4},\odot^{2}\mathbb{C}^{2})$ has $4$ components $f_{0^{\prime}0^{\prime}},f_{1^{\prime}0^{\prime}},f_{0^{\prime}1^{\prime}}$ and $f_{1^{\prime}1^{\prime}}$ such that $f_{1^{\prime}0^{\prime}}=f_{0^{\prime}1^{\prime}}$ . Its $L^{2}$ inner product is induced from that of $L_{\varphi}^{2}(\mathbb{R}^{4},\otimes^{2}\mathbb{C}^{2})$ by

[TABLE]

$f\in L_{\varphi}^{2}(\mathbb{R}^{4},\mathbb{C}^{2}\otimes\mathbb{C}^{2})$ has $4$ components $f_{A^{\prime}A}$, $A=0,1,A^{\prime}=0^{\prime},1^{\prime}$, and

[TABLE]

while $f\in L_{\varphi}^{2}(\mathbb{R}^{4},\wedge^{2}\mathbb{C}^{2})$ has components $f_{AB}$ with $f_{AB}=-f_{BA}$, among which there is only one nontrivial (i.e. $f_{00}=f_{11}=0$, $f_{01}=-f_{10}$), and

[TABLE]

The operators in the $2$-Cauchy-Fueter complex over $\mathbb{R}^{4}$ are given by

[TABLE]

for $\phi\in C^{1}(\mathbb{R}^{4},\odot^{2}\mathbb{C}^{2})$ where $A=0,1,A^{\prime}=0^{\prime},1^{\prime}$, and

[TABLE]

for $\psi\in C^{1}(\mathbb{R}^{4},\mathbb{C}^{2}\otimes\mathbb{C}^{2})$, where

[TABLE]

is the antisymmetrisation. Here and in the sequel, we write $\psi_{AA^{\prime}}:=\psi_{A^{\prime}A}$ for convenience. It is direct to see that

[TABLE]

by relabeling indices, $\phi_{C^{\prime}A^{\prime}}=\phi_{A^{\prime}C^{\prime}}$ and the commutativity $\nabla^{A^{\prime}}_{B}\nabla^{C^{\prime}}_{A}=\nabla^{C^{\prime}}_{A}\nabla^{A^{\prime}}_{B}$, as scalar differential operators of constant complex coefficients (cf. (2.11) in [4]).

Lemma 2.1**.**

(1) For any $h\in L_{\varphi}^{2}(\mathbb{R}^{4},\odot^{2}\mathbb{C}^{2})$ and $H\in L_{\varphi}^{2}(\mathbb{R}^{4},\otimes^{2}\mathbb{C}^{2})$, we have

[TABLE]

where

[TABLE]

is the symmetrisation, i.e. $({H_{(A^{\prime}B^{\prime})}})\in L_{\varphi}^{2}(\mathbb{R}^{4},\odot^{2}\mathbb{C}^{2})$.

For any $h\in L_{\varphi}^{2}(\mathbb{R}^{4},\wedge^{2}\mathbb{C}^{2})$ and $H\in L_{\varphi}^{2}(\mathbb{R}^{4},\otimes^{2}\mathbb{C}^{2})$, we have

[TABLE]

For any $h,H\in L_{\varphi}^{2}(\mathbb{R}^{4},\otimes^{2}\mathbb{C}^{2})$, we have

[TABLE]

Proof.

(1) This is because

[TABLE]

by changing indices and $h_{A^{\prime}B^{\prime}}=h_{B^{\prime}A^{\prime}}$.

(2) This is because

[TABLE]

by changing indices and $h_{BA}=-h_{AB}$.

(3) This is because

[TABLE]

and the second term in the right hand side is $-2\sum_{A,B}h_{[AB]}\overline{H_{AB}}=-2\sum_{A,B}h_{[AB]}\overline{H_{[AB]}}$ by the identity (2.19). ∎

Lemma 2.2**.**

For $f\in C_{0}^{\infty}(\mathbb{R}^{4},\mathbb{C}^{2}\otimes\mathbb{C}^{2})$, we have

[TABLE]

Proof.

For any $g\in C_{0}^{\infty}(\mathbb{R}^{4},\odot^{2}\mathbb{C}^{2})$, we have

[TABLE]

by using (2.10) and Lemma 2.1 (1). Here we have to symmetrise $(A^{\prime}B^{\prime})$ in $\sum_{A}\delta_{A^{\prime}}^{A}{f_{B^{\prime}A}}$ since only after symmetrisation it becomes an element of $C_{0}^{\infty}(\mathbb{R}^{4},\odot^{2}\mathbb{C}^{2})$, i.e. a $\odot^{2}\mathbb{C}^{2}$-valued function. ∎

Theorem 2.1**.**

Suppose that there exist a constant $c>0$ such that the weight $\varphi$ satisfies

[TABLE]

for any $x\in\mathbb{R}^{4n}$ and $(\xi_{A^{\prime}A})\in\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. Then we have the weighted $L^{2}$ estimate

[TABLE]

for any $f\in{\rm Dom}({\mathscr{D}}_{0}^{*})\cap{\rm Dom}({\mathscr{D}}_{1})$.

Proof.

By definition, we have ${\rm Dom}({\mathscr{D}}_{1}):=\left\{f\in L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{1});{\mathscr{D}}_{1}f\in L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{2})\right\}$. Then ${\mathscr{D}}_{1}$ is densely-defined since $C_{0}^{\infty}(\mathbb{R}^{4n},\mathscr{V}_{1})$ is contained in its domain. It is also closed since differentiation is continuous on distributions. So is ${\mathscr{D}}_{0}^{*}$ as a differential operator given by (2.21). Therefore it is sufficient to show (2.23) for $f\in C_{0}^{\infty}(\mathbb{R}^{4n},\mathscr{V}_{1})$. It follows from the definition of ${\mathscr{D}}_{0}$ in (2.13), ${\mathscr{D}}_{0}^{*}$ in Lemma 2.2 and the definition of symmetrisation that

[TABLE]

where

[TABLE]

and

[TABLE]

by using the formal adjoint operator (2.11), relabeling indices and using the commutator

[TABLE]

which follows from (2.8)-(2.9) and the commutativity $Z^{A^{\prime}}_{B}Z_{B^{\prime}}^{A}=Z_{B^{\prime}}^{A}Z^{A^{\prime}}_{B}$ as scalar differential operators of constant coefficients. The first summation in the right hand side of (2.26) is equal to

[TABLE]

by applying (2.18) with $h_{BA}=\sum_{A^{\prime}}Z^{A^{\prime}}_{B}f_{AA^{\prime}}$, $H_{AB}=\sum_{B^{\prime}}Z^{B^{\prime}}_{A}f_{BB^{\prime}}$. Now substituting (2.25)-(2.26) into (2.24) and using the above identity, we get

[TABLE]

Now the resulting estimate follows from the assumption (2.22) for $\varphi$. ∎

Remark 2.1**.**

(1) We do not handle the term $\Sigma_{0}$ in (2.25) by using commutators. Because if we do so

[TABLE]

the first term in the right hand side above is quite difficult to control. But over $\mathbb{R}^{4}$ it can be controlled in terms of ${\mathscr{D}}_{0}^{*}f$ and ${\mathscr{D}}_{1}f$. Based on such estimates, we can solve the Neumann problem for the $k$-Cauchy-Fueter complexes over $k$-pseudoconvex domains in $\mathbb{R}^{4}$ (cf. [23]).

(2) $\varphi=|x|^{2}$ satisfies the assumption (2.22) for $\varphi$ with $c=4$ by the following Lemma 3.2.

3. The canonical solution operator to the nonhomogeneous $k$-Cauchy-Fueter equation

3.1. The weighted $L^{2}$ estimate in the general case

Recall that the symmetric power $\odot^{k}\mathbb{C}^{2}$ is a subspace of $\otimes^{k}\mathbb{C}^{2}$, and an element of $\odot^{k}\mathbb{C}^{2}$ is given by a $2^{k}$-tuple $(f_{A_{1}^{\prime}\ldots A_{k}^{\prime}})\in\otimes^{k}\mathbb{C}^{2}$ with $A_{1}^{\prime}\ldots A_{k}^{\prime}=0^{\prime},1^{\prime}$, where $f_{A_{1}^{\prime}\ldots A_{k}^{\prime}}$ is invariant under permutations of subscripts, i.e.

[TABLE]

for any $\sigma\in S_{k}$, the group of permutations of $k$ letters. Note that $\dim(\odot^{k}\mathbb{C}^{2})=k+1$ (cf. (4.1)) while $\dim(\otimes^{k}\mathbb{C}^{2})=2^{k}$. An element of the exterior power $\wedge^{2}\mathbb{C}^{2n}$ is given by a tuple $(f_{AB})$ with $f_{AB}=-f_{BA}$, $A,B=0,\ldots,2n-1$. An element of $\odot^{k-1}\mathbb{C}^{2}\otimes\mathbb{C}^{2n}$ is given by a tuple $(f_{A_{2}^{\prime}\ldots A_{k}^{\prime}A})\in\otimes^{k-1}\mathbb{C}^{2}\otimes\mathbb{C}^{2n}$, which is invariant under permutations of $A_{2}^{\prime},\ldots,A_{k}^{\prime}$. We will use symmetrisation of primed indices

[TABLE]

The first two operators in $k$-Cauchy-Fueter complex (1.4)-(1.5) over $\mathbb{R}^{4n}$ are given by

[TABLE]

for $f\in C^{1}(\mathbb{R}^{4n},\mathscr{V}_{0})$, $h\in C^{1}(\mathbb{R}^{4n},\mathscr{V}_{1})$, where $A,B=0,1,\ldots,2n-1$, $A_{2}^{\prime},\ldots,A_{k}^{\prime}=0^{\prime},1^{\prime}$. Here and in the sequel, we write $h_{AA_{2}^{\prime}A_{3}^{\prime}\ldots A_{k}^{\prime}}:=h_{A_{2}^{\prime}A_{3}^{\prime}\ldots A_{k}^{\prime}A}$ for convenience. It is direct to check that ${\mathscr{D}}_{1}\circ{\mathscr{D}}_{0}=0$ as (2.15).

The weighted inner product of $L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{0})$ is induced from that of $L_{\varphi}^{2}(\mathbb{R}^{4n},\otimes^{k}\mathbb{C}^{2})$. Namely we define

[TABLE]

for $f,h\in L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{0})$, and $\|f\|_{\varphi}=\left\langle f,{f}\right\rangle_{\varphi}^{\frac{1}{2}}$. We define the weighted induced inner products of $L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{1})$ and $L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{2})$ similarly.

Lemma 3.1**.**

For $f\in C_{0}^{\infty}(\mathbb{R}^{4n},\mathscr{V}_{1})$, we have

[TABLE]

Proof.

For any $g\in C_{0}^{\infty}(\mathbb{R}^{4n},\mathscr{V}_{0})$ we have

[TABLE]

by using (2.10) and symmetrisation

[TABLE]

for any $g\in L_{\varphi}^{2}(\mathbb{R}^{4n},\odot^{k}\mathbb{C}^{2})$, $G\in L_{\varphi}^{2}(\mathbb{R}^{4n},\otimes^{k}\mathbb{C}^{2})$. Here we have to symmetrise indices $(A_{1}^{\prime}\ldots A_{k}^{\prime})$ in $\sum_{A}\delta_{A_{1}^{\prime}}^{A}{f_{A_{2}^{\prime}\ldots A_{k}^{\prime}A}}$ since only after symmetrisation it becomes an element of $C_{0}^{\infty}(\mathbb{R}^{4n},\mathscr{V}_{0})$, i.e. a $\odot^{k}\mathbb{C}^{2}$-valued function. (3.4) is a generalization of Lemma 2.1 (1). It holds because

[TABLE]

by relabeling indices, which equals to L.H.S. by $g$ symmetric in the indices, i.e. $g_{A_{\sigma^{-1}(1)}^{\prime}\ldots A_{\sigma^{-1}(k)}^{\prime}}=g_{A_{1}^{\prime}\ldots A_{k}^{\prime}}$ for any permutation $\sigma$. ∎

Proof of Theorem 1.2. As in the model case $n=1$, $k=2$, it is sufficient to show the weighted $L^{2}$-estimate (1.8) for $f\in C_{0}^{\infty}(\mathbb{R}^{4n},\odot^{k-1}\mathbb{C}^{2}\otimes\mathbb{C}^{2n})$. Recall that if $(F_{A_{1}^{\prime}\ldots A_{k}^{\prime}})\in\otimes^{k}\mathbb{C}^{2}$ is symmetric in $A_{2}^{\prime}\ldots A_{k}^{\prime}$, then we have

[TABLE]

by definition of symmetrisation (3.1). Now we expand the symmetrisation to get

[TABLE]

by the adjoint operator ${\mathscr{D}}_{0}^{*}$ in Lemma 3.1. Here we split the sum into the cases $s=1$ and $s\geq 2$ as in the model case (cf. Remark 2.1). Note that

[TABLE]

by using (2.10), and

[TABLE]

by using commutators. For the second sum,

[TABLE]

for $\varphi(x)=|x|^{2}$ by the following Lemma 3.2 and $f$ symmetric in the primed indices. On the other hand,

[TABLE]

by $f$ symmetric in the primed indices and relabelling indices. Then applying Lemma 2.1 (3) ((2.20) holds for $A,B=0,\ldots,2n-1$) to the right hand side with $h_{BA}=\sum_{A^{\prime}}Z^{A^{\prime}}_{B}f_{AA^{\prime}B_{3}^{\prime}\ldots B_{k}^{\prime}}$ and $H_{AB}=\sum_{A^{\prime}}Z^{A^{\prime}}_{A}f_{BA^{\prime}B_{3}^{\prime}\ldots B_{k}^{\prime}}$ for fixed $B_{3}^{\prime},\ldots,B_{k}^{\prime}$, we get

[TABLE]

Now we get

[TABLE]

The estimate (1.8) follows.                                                                                                    $\Box$

Lemma 3.2**.**

$Z^{A^{\prime}}_{B}\overline{{Z^{B^{\prime}}_{A}}}|x|^{2}=4\delta_{AB}\delta_{A^{\prime}B^{\prime}}$. In particular, $\varphi=|x|^{2}$ satisfies the assumption (2.22) for $\varphi$ with $c=4$.

To prove this lemma, we introduce complex linear functions

[TABLE]

where $A=0,\ldots,2n-1$, $A^{\prime}=0^{\prime},1^{\prime}$. $z_{AA^{\prime}}$ is obtained by replacing $\partial_{x_{j}}$ in $\overline{Z_{AA^{\prime}}}$ in (2.1) by $x_{j}$. By the following lemma, $z_{AA^{\prime}}$’s can be viewed as independent variables and $Z_{AA^{\prime}}$’s are derivatives with respect to these variables formally.

Lemma 3.3**.**

$Z_{AA^{\prime}}z_{BB^{\prime}}=2\delta_{AB}\delta_{A^{\prime}B^{\prime}}$.

Proof.

Assume that $A=2l,A^{\prime}=0^{\prime}$. By (3.6), we have

[TABLE]

Note that $Z_{(2l)0^{\prime}}$ is a differential operator with respect to variables $x_{4l+1}$ and $x_{4l+2}$, while $z_{BB^{\prime}}$ for $BB^{\prime}\neq{(2l)0^{\prime}}$ or ${(2l+1)1^{\prime}}$ is independent of variables $x_{4l+1}$ and $x_{4l+2}$. So we get

[TABLE]

for such $BB^{\prime}$. It is similar to check the result directly for other vectors $Z_{(2l)1^{\prime}}$, $Z_{(2l+1)0^{\prime}}$ and $Z_{(2l+1)1^{\prime}}$. ∎

Proof of Lemma 3.2. Note that $(\partial_{x_{j}}\pm\mathbf{i}\partial_{x_{k}})|x|^{2}=2({x_{j}}\pm\mathbf{i}{x_{k}})$. So $\overline{Z_{AC^{\prime}}}|x|^{2}=2z_{AC^{\prime}}$ by definitions of $\overline{Z_{AC^{\prime}}}$’s and $z_{AC^{\prime}}$’s in (3.6). Then we have

[TABLE]

by Lemma 3.3. Here by (2.3), $\varepsilon^{C^{\prime}B^{\prime}}\varepsilon^{C^{\prime}A^{\prime}}=1$ only if $A^{\prime}=B^{\prime}$ and $C^{\prime}$ is different from them. Otherwise, it vanishes. So for any $(\xi_{A^{\prime}A})\in\mathbb{C}^{2}\otimes\mathbb{C}^{2}$,

[TABLE]

3.2. The associated Laplacian operator $\Box_{\varphi}$

By definition,

[TABLE]

We introduce

[TABLE]

for any $f,g\in{\rm Dom}(\mathcal{E}_{\varphi}):={\rm Dom}({\mathscr{D}}_{1})\cap{\rm Dom}({\mathscr{D}}_{0}^{*})$. By definition of adjoint operators, we have

[TABLE]

for any $f\in{\rm Dom}(\Box_{\varphi}),g\in{\rm Dom}(\mathcal{E}_{\varphi})$.

Note that for any $F\in{\rm Dom}({\mathscr{D}}_{0})$, we have ${\mathscr{D}}_{0}F\in{\rm Dom}({\mathscr{D}}_{1})$ and

[TABLE]

This is because ${\mathscr{D}}_{1}{\mathscr{D}}_{0}F=0$ for smooth $F$ and the general result follows from the closedness of ${\mathscr{D}}_{0}$ and ${\mathscr{D}}_{1}$ as differential operators.

Proposition 3.1**.**

The associated Laplacian operator $\Box_{\varphi}$ is a densely-defined, closed, self-adjoint and non-negative operator on $L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{1})$.

Proof.

It is similar to the proof of proposition 4.2.3 of [5] for $\overline{\partial}$-complex. We give the proof here for completeness.

As we mentioned before, ${\mathscr{D}}_{0}$ and ${\mathscr{D}}_{0}^{*}$ as differential operators are both densely-defined and closed. $\Box_{\varphi}$ is densely-defined in the same way. For closedness of $\Box_{\varphi}$, we need to show that for any $f_{n}\in{\rm Dom}(\Box_{\varphi})$ such that $f_{n}\longrightarrow f$ in $L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{1})$ and $\Box_{\varphi}f_{n}$ converges, we have $f\in{\rm Dom}(\Box_{\varphi})$ and $\Box_{\varphi}f_{n}\longrightarrow\Box_{\varphi}f$. Because $f_{n}\in{\rm Dom}(\Box_{\varphi})$, we have

[TABLE]

and so ${\mathscr{D}}_{0}^{*}f_{n}$ and ${\mathscr{D}}_{1}f_{n}$ converge in $L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{0})$ and $L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{1})$, respectively. It follows from the closedness of ${\mathscr{D}}_{0}^{*}$ and ${\mathscr{D}}_{1}$ that $f\in{\rm Dom}({\mathscr{D}}_{0}^{*})\cap{\rm Dom}({\mathscr{D}}_{1})$ and

[TABLE]

Note that ${\mathscr{D}}_{0}{\mathscr{D}}_{0}^{*}f_{n}$ and ${\mathscr{D}}_{1}^{*}{\mathscr{D}}_{1}f_{n}$ are orthogonal to each other by

[TABLE]

by (3.8). So $\Box_{\varphi}f_{n}={\mathscr{D}}_{0}{\mathscr{D}}_{0}^{*}f_{n}+{\mathscr{D}}_{1}^{*}{\mathscr{D}}_{1}f_{n}$ converges implies that both ${\mathscr{D}}_{0}{\mathscr{D}}_{0}^{*}f_{n}$ and ${\mathscr{D}}_{1}^{*}{\mathscr{D}}_{1}f_{n}$ converge. It follows from the closedness of ${\mathscr{D}}_{0}$ and ${\mathscr{D}}_{1}^{*}$ again that ${\mathscr{D}}_{0}^{*}f\in{\rm Dom}({\mathscr{D}}_{0})$, ${\mathscr{D}}_{1}f\in({\mathscr{D}}_{1}^{*})$ and

[TABLE]

Therefore $f\in{\rm Dom}(\Box_{\varphi})$ and $\Box_{\varphi}f_{n}\longrightarrow\Box_{\varphi}f$. So $\Box_{\varphi}$ is a closed operator.

Define

[TABLE]

It is sufficient to show that $L_{1}^{-1}$ is self-adjoint. By a theorem of Von Neumann (cf. §1 in Chapter 8 in [18]), $(I+{\mathscr{D}}_{0}{\mathscr{D}}_{0}^{*})^{-1}$ and $(1+{\mathscr{D}}_{1}^{*}{\mathscr{D}}_{1})^{-1}$ are automatically both bounded and self-adjoint, and so is

[TABLE]

We claim that $Q_{1}=L_{1}^{-1}$. Since

[TABLE]

we see that $\mathcal{R}(I+{\mathscr{D}}_{0}{\mathscr{D}}_{0}^{*})^{-1}\subset{\rm Dom}({\mathscr{D}}_{0}{\mathscr{D}}_{0}^{*})$. Similarly, $\mathcal{R}(I+{\mathscr{D}}_{1}^{*}{\mathscr{D}}_{1})^{-1}\subset{\rm Dom}({\mathscr{D}}_{1}^{*}{\mathscr{D}}_{1})$, and so

[TABLE]

Since ${\mathscr{D}}_{1}{\mathscr{D}}_{0}=0$ by (3.8), we have $\mathcal{R}(Q_{1})\subset{\rm Dom}({\mathscr{D}}_{1}^{*}{\mathscr{D}}_{1})$ and ${\mathscr{D}}_{1}^{*}{\mathscr{D}}_{1}Q_{1}={\mathscr{D}}_{1}^{*}{\mathscr{D}}_{1}(I+{\mathscr{D}}_{1}^{*}{\mathscr{D}}_{1})^{-1}$. Similarly $\mathcal{R}(Q_{1})\subset{\rm Dom}({\mathscr{D}}_{0}{\mathscr{D}}_{0}^{*})$ and ${\mathscr{D}}_{0}{\mathscr{D}}_{0}^{*}Q_{1}={\mathscr{D}}_{0}{\mathscr{D}}_{0}^{*}(I+{\mathscr{D}}_{0}{\mathscr{D}}_{0}^{*})^{-1}$. Consequently, $\mathcal{R}(Q_{1})\subset{\rm Dom}(L_{1})$ and

[TABLE]

by (3.10). This together with the injectivity of $L_{1}$ implies that $L_{1}^{-1}=Q_{1}$. Thus $L_{1}^{-1}$ is self-adjoint. So is its inverse $L_{1}$ (cf. §2 in Chapter 8 in [18] for this general property). ∎

3.3. The canonical solution operator

Proof of Theorem 1.1. (1) The weighted $L^{2}$-estimate (1.8) implies that

[TABLE]

for $g\in{\rm Dom}(\Box_{\varphi})$, by (3.7), i.e.

[TABLE]

Thus $\Box_{\varphi}$ is injective. This together with the self-adjointness of $\Box_{\varphi}$ by Proposition 3.1 implies the density of the range (cf. §2 in Chapter 8 in [18] for this general property). For fixed $f\in L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{1})$, the complex anti-linear functional

[TABLE]

is then well-defined on a dense subset $\mathcal{R}(\Box_{\varphi})$ of $L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{1})$. It is finite since

[TABLE]

for any $g\in{\rm Dom}(\Box_{\varphi})$, by (3.11). So $\lambda_{f}$ can be uniquely extended a continuous anti-linear functional on $L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{1})$. By the Riesz representation theorem, there exists a unique element $h\in L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{1})$ such that $\lambda_{f}(F)=\langle h,F\rangle_{\varphi}$ for any $F\in L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{1})$, and $\|h\|_{\varphi}=|\lambda_{f}|\leq\frac{1}{4}\|f\|_{\varphi}$. In particular, we have

[TABLE]

for any $g\in{\rm Dom}(\Box_{\varphi})$. This implies that $h\in{\rm Dom}(\Box_{\varphi}^{*})$ and $\Box_{\varphi}^{*}h=f$, and so $h\in{\rm Dom}(\Box_{\varphi})$ and $\Box_{\varphi}h=f$ by self-adjointness of $\Box_{\varphi}$. We write $h=N_{\varphi}f$. Then $\left\|N_{\varphi}f\right\|_{\varphi}\leq\frac{1}{4}\left\|f\right\|_{\varphi}$.

(2) Since $N_{\varphi}f\in{\rm Dom}(\Box_{\varphi})$, we have ${\mathscr{D}}_{0}^{*}N_{\varphi}f\in{\rm Dom}({\mathscr{D}}_{0})$, ${\mathscr{D}}_{1}N_{\varphi}f\in{\rm Dom}({\mathscr{D}}_{1}^{*})$, and

[TABLE]

by $\Box_{\varphi}N_{\varphi}f=f$. Because $f$ and ${\mathscr{D}}_{0}F$ for any $F\in{\rm Dom}({\mathscr{D}}_{0})$ are both ${\mathscr{D}}_{1}$-closed, the above identity implies ${\mathscr{D}}_{1}^{*}{\mathscr{D}}_{1}N_{\varphi}f\in{\rm Dom}({\mathscr{D}}_{1})$ and so ${\mathscr{D}}_{1}{\mathscr{D}}_{1}^{*}{\mathscr{D}}_{1}N_{\varphi}f=0$ by ${\mathscr{D}}_{1}$ acting in both sides. Then

[TABLE]

i.e. ${\mathscr{D}}_{1}^{*}{\mathscr{D}}_{1}N_{\varphi}f=0.$ Hence ${\mathscr{D}}_{0}{\mathscr{D}}_{0}^{*}N_{\varphi}f=f$ by (3.12). Moreover, we have ${\mathscr{D}}_{0}^{*}N_{\varphi}f\perp A^{2}_{(k)}(\mathbb{R}^{4n},\varphi)$ since $(F,{\mathscr{D}}_{0}^{*}N_{\varphi}f)_{\varphi}=({\mathscr{D}}_{0}F,N_{\varphi}f)_{\varphi}=0$ for any $F\in A^{2}_{(k)}(\mathbb{R}^{4n},\varphi)$. The estimate (1.7) follows from

[TABLE]

Corollary 3.1**.**

The weighted $k$-Bergman projection formula (1.9) holds.

Proof.

For $f\in{\rm Dom}({\mathscr{D}}_{0})$, ${\mathscr{D}}_{0}f$ is automatically ${\mathscr{D}}_{1}$-closed. Apply Theorem 1.1 to ${\mathscr{D}}_{0}f$ to get the canonical solution ${\mathscr{D}}_{0}^{*}N_{\varphi}{\mathscr{D}}_{0}f$ orthogonal to $A^{2}_{(k)}(\mathbb{R}^{4n},\varphi)$. So $f-{\mathscr{D}}_{0}^{*}N_{\varphi}{\mathscr{D}}_{0}f\in A^{2}_{(k)}(\mathbb{R}^{4n},\varphi)$ by ${\mathscr{D}}_{0}(f-{\mathscr{D}}_{0}^{*}N_{\varphi}{\mathscr{D}}_{0}f)=0$, and is exactly the projection of $f$ to the weighted $k$-Bergman space. ∎

Remark 3.1**.**

As in [21], we can use Theorem 1.1 to get compactly supported solution to the nonhomogeneous $k$-Cauchy-Fueter equation (1.1) for ${\mathscr{D}}_{1}$-closed $f\in C_{0}^{1}\left(\mathbb{R}^{4n},\mathscr{V}_{1}\right)$, which implies Hartogs’ phenomenon for $k$-regular functions.

4. Decay of canonical solutions and the weighted $k$-Bergman kernel

4.1. The weighted $k$-Bergman projection and kernel

For $f\in L_{\varphi}^{2}(\Omega,\mathscr{V}_{0})$, it has $k+1$ independent components $f_{0^{\prime}0^{\prime}\ldots 0^{\prime}0^{\prime}},f_{1^{\prime}0^{\prime}\ldots 0^{\prime}0^{\prime}},\ldots,f_{1^{\prime}1^{\prime}\ldots 1^{\prime}1^{\prime}}$. We write

[TABLE]

where $f_{j}:=f_{1^{\prime}\ldots 1^{\prime}0^{\prime}\ldots 0^{\prime}}$ with $j$ indices to be $1^{\prime}$.

Note that for a sequence of $k$-regular functions $F_{n}\in L_{\varphi}^{2}(\Omega,\mathscr{V}_{0})$ (i.e. ${\mathscr{D}}_{0}F_{n}=0$), if $F_{n}\longrightarrow F$ in $L_{\varphi}^{2}(\Omega,\mathscr{V}_{0})$, we have ${\mathscr{D}}_{0}F=0$ by the closedness of ${\mathscr{D}}_{0}$. So $A^{2}_{(k)}(\mathbb{R}^{4n},\varphi)$ is a closed subspace of $L_{\varphi}^{2}(\Omega,\mathscr{V}_{0})$. If $\{\psi_{\alpha}\}$ is an orthonormal basis of the space $A^{2}_{(k)}(\mathbb{R}^{4n},\varphi)$, the weighted $k$-Bergman projection $P$ can be write as $Pf=\sum_{\alpha}\langle f,\psi_{\alpha}\rangle_{\varphi}\psi_{\alpha}.$

Proposition 4.1**.**

If $f\in L_{\varphi}^{2}(\Omega,\odot^{k}\mathbb{C}^{2})$ is $k$-regular, then each component of $f$ is harmonic.

Proof.

It follows from

[TABLE]

where $\Delta:=\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2}+\cdots+\partial_{x_{4n}}^{2}$. See lemma 3.3 of [19] for this identity. ∎

By Proposition 4.1, each component of a $k$-regular function is smooth. So for a fixed point $x\in\mathbb{R}^{4n}$, we can define complex linear functionals

[TABLE]

for $f\in A^{2}_{(k)}(\mathbb{R}^{4n},\varphi)$, $j=0,\ldots k$. Since $f_{j}$ is harmonic by Proposition 4.1, we see that

[TABLE]

where $C_{x}$ only depends on $x$, not on $f$. Consequently, linear functionals $l_{j}$ are bounded on $A^{2}_{(k)}(\mathbb{R}^{4n},\varphi)$. By the Riesz representation theorem, there exists $K_{j}(\cdot,x)\in A^{2}_{(k)}(\mathbb{R}^{4n},\varphi)$ such that

[TABLE]

It is obvious that $\langle g,K_{j}(\cdot,x)\rangle_{\varphi}=0$ for any $g\perp A^{2}_{(k)}(\mathbb{R}^{4n},\varphi)$. So $K(x,y)=\left(\overline{K_{jl}(y,x)}\right)$ is the kernel of the weighted $k$-Bergman projection $P$, which is a $(k+1)\times(k+1)$ matrix anti-$k$-regular in $y$. Then the integral formula (1.10) holds. Since an orthogonal projection $P$ is self-adjoint on $L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{0})$, $K$ has the Hermitian property $K(x,y)=\overline{K(y,x)}^{t}$, and so $K(x,y)$ is $k$-regular in $x$.

4.2. A localized a priori estimate and Caccioppoli-type estimate

It is known that the Caccioppoli-type estimate holds for many systems of PDEs of the divergence form by establishing localized a priori estimate of the following type.

Proposition 4.2**.**

There exists an absolute constant $C_{0}>0$ such that for any $f\in{\rm Dom}(\Box_{\varphi})$ and real bounded Lipschitzian function $\eta$, we have estimates

[TABLE]

where $|d\eta|^{2}=\sum_{j=1}^{4n}|\frac{\partial\eta}{\partial x_{j}}|^{2}$.

Proof.

Note that

[TABLE]

by $\delta_{A_{1}^{\prime}}^{A}=Z_{A_{1}^{\prime}}^{A}-2Z_{A_{1}^{\prime}}^{A}\varphi$ in (2.9). Then taking summation over $A$ and symmetrising $(A_{1}^{\prime}\ldots A_{k}^{\prime})$, we get

[TABLE]

On the other hand, for fixed $A_{1}^{\prime}\ldots A_{k}^{\prime}$, we have

[TABLE]

by using (3.5) and Cauchy-Schwarz inequality and $f$ symmetric in the primed indices. Note that it directly follows from definition (2.1) of $Z_{AA^{\prime}}$’s that

[TABLE]

for fixed $A^{\prime}=0^{\prime}$ or $1^{\prime}$. Then by raising indices, we get

[TABLE]

and so is the sum of $|Z_{1^{\prime}}^{A}\eta|^{2}$. Apply these to (4.6) to get

[TABLE]

Thus we get the estimate

[TABLE]

by (4.5), and simultaneously,

[TABLE]

Note that by (4.5) again, we get

[TABLE]

by using estimates (4.6)-(4.7) and the trivial inequality $2|ab|\leq\kappa|a|^{2}+\frac{1}{\kappa}|b|^{2}$ for any $\kappa>0$. Thus if we choose $\kappa=1/2$, we get

[TABLE]

But

[TABLE]

by applying estimates similar to (4.6)-(4.7) in the third inequality. Now Substitute (4.10) to (4.9) and using (4.8) to control the term $\kappa\left\|\eta{\mathscr{D}}_{0}^{*}f\right\|_{\varphi}^{2}$, we find that there exists a constant $C_{0}>0$ such that

[TABLE]

Similarly,

[TABLE]

by definition, and so

[TABLE]

The result follows. ∎

As a corollary, we get Caccioppoli-type estimate.

Proposition 4.3**.**

Suppose that $\varphi(x)=|x|^{2}$. If $\Box_{\varphi}F=0$ on $B(x,R)\subset\mathbb{R}^{4n}$, then for $r<R$, we have

[TABLE]

for some constant $C$ only depending on $n$, $k$, $R$ and $r$.

Proof.

Let $\eta$ be a $C_{0}^{\infty}(B(x,R))$ function such that $\eta\equiv 1$ on $B(x,r)$. By the localized a priori estimate (4.4) in Proposition 4.2, we get

[TABLE]

since $\Box_{\varphi}F=0$ on supp$\,\eta$ and $d\eta$ is supported in $B(x,R)$. The result follows by choosing $\eta$. ∎

4.3. Decay of canonical solutions and the weighted $k$-Bergman kernel

Theorem 4.1**.**

Suppose that $\varphi(x)=|x|^{2}$, $k=2,3,\ldots$, and that $f\in L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{1})$ is compactly supported in $B(y,r_{0})$. Then the canonical solution $u={\mathscr{D}}_{0}^{*}N_{\varphi}f$ has the following pointwise estimate: there exists $\varepsilon>0$ only depending on $r_{0}$ and constant $C>0$ only depending on $n$, $k$ and $\varepsilon$ such that

[TABLE]

for any $x$ such that $|x-y|>r_{0}+2$.

Proof.

For the canonical solution $u={\mathscr{D}}_{0}^{*}N_{\varphi}f$, we have ${\mathscr{D}}_{0}u=f$ vanishing outside of $B(y,r_{0})$. Consequently, each component of $u$ is harmonic outside of $B(y,r_{0})$ by Proposition 4.1. By the mean value formula for harmonic functions, we get

[TABLE]

for some constant $C_{\delta}^{\prime}>0$ only depending on $n$, $\delta<1$ and any $x$ such that $|x-y|>r_{0}+1$. Here in the last inequality we apply Caccioppoli-type estimate in Proposition 4.3 to $F=N_{\varphi}f$ with $\Box_{\varphi}N_{\varphi}f=f=0$ outside of $B(y,r_{0})$, and $e^{|x^{\prime}|^{2}}\leq e^{|x|^{2}+2\delta|x|+\delta^{2}}$ for $x^{\prime}\in B(x,\delta)$. We choose $\delta=\frac{\varepsilon}{4}$ for $\varepsilon$ determined later.

For fixed $x$ outside of $B(y,r_{0})$, consider the Lipschitzian function

[TABLE]

Let $l:[0,\infty)\rightarrow[0,1]$ be the Lipschitzian function vanishing on $[0,r_{0}]$, equal to $1$ on $[r_{0}+1,\infty)$, and affine in between. Set $\eta(x^{\prime})=l(|x^{\prime}-y|)$. Applying weighted $L^{2}$ estimate (1.8) and the localized a priori estimate in Proposition 4.2 to $N_{\varphi}f$ with $\eta$ replaced by $\eta e^{\varepsilon b}$, we get

[TABLE]

since the Lipschitzian constant of $b$ is $1$ and $\Box_{\varphi}N_{\varphi}f=f=0$ on the support of $\eta$ ($=B(y,r_{0})^{c}$). Hence if we choose $\varepsilon$ sufficiently small (e.g. $C_{0}(4n\varepsilon)^{2}\leq\frac{1}{2}$), we get

[TABLE]

for some constant $C^{\prime\prime}>0$, by $d\eta$ supported in $B(y,r_{0}+1)$ and $b$ uniformly bounded on $B(y,r_{0}+1)$ ($|b(x^{\prime})|<r_{0}+1$). But $b(x^{\prime})\geq|x-y|-1$ for $x^{\prime}\in B(x,1)$, and so the above estimate implies that

[TABLE]

Substituting this into (4.12), we get the result by the boundedness of $N_{\varphi}$ on $L_{\varphi}^{2}(\mathbb{R}^{4n},\mathscr{V}_{1})$ by Theorem 1.1 (1). ∎

Proof of Theorem 1.3. For fixed $y\in\mathbb{R}^{4n}$, let $\eta_{y}$ be a smooth radial function supported in the ball $B(y,\delta)$ ($\delta<1$) such that $\int\eta_{y}(y^{\prime})dV(y^{\prime})=1$. Set

[TABLE]

for fixed $j$, where only $j$-th entry is nonvanishing. Note that

[TABLE]

by applying the mean value formula for harmonic functions to each component of $K(x,\cdot)$, since $\eta_{y}(\cdot)$ is constant on each sphere centered at $y$. Hence the $j$-th column of $(k+1)\times(k+1)$-matrix $K$ is

[TABLE]

by the identity (1.9). The exponential decay of the canonical solution in Theorem 4.1 implies that there exists a constant $C>0$ only depending on $\varepsilon,n,k$ such that

[TABLE]

for any $x$ such that $|x-y|>3$, since ${\mathscr{D}}_{0}{\mathscr{D}}_{0}^{*}N_{\varphi}{\mathscr{D}}_{0}f_{y}={\mathscr{D}}_{0}f_{y}$ is supported in $B(y,1)$. Note that $|{\mathscr{D}}_{0}f_{y}(y^{\prime})|\leq C_{3}e^{2|y^{\prime}|^{2}}(|y^{\prime}|+1)\chi_{B(y,\delta)}$ for some constant $C_{3}>0$ depending on $n,\delta$, by direct differentiation (4.13). It is direct to check that $\|{\mathscr{D}}_{0}f_{y}\|_{\varphi}\leq C_{4}e^{|y|^{2}+5\delta|y|}$ for some constant $C_{4}>0$ depending on $n,\delta$. The result follows by choose small $\delta$.

Remark 4.1**.**

Our estimate (1.11) has an extra factor $e^{\frac{\varepsilon}{2}(|x|+|y|)}$ compared to the estimate

[TABLE]

for the Bergmann kernel in complex analysis. But when $|y|$ is large compared to $|x|$, e.g. $|y|\geq 4|x|$,

[TABLE]

which has similar exponential decay with respect to the measure $e^{-|y|^{2}}dV$ as in the complex case.