Abstract
We construct the tangential k-Cauchy–Fueter complexes on the right quaternionic Heisenberg group, as the quaternionic counterpart of \(\overline{\partial }_b\)-complex on the Heisenberg group in the theory of several complex variables. We can use the \(L^2\) estimate to solve the nonhomogeneous tangential k-Cauchy–Fueter equation under the compatibility condition over this group modulo a lattice. This solution has an important vanishing property when the group is higher dimensional. It allows us to prove the Hartogs’ extension phenomenon for k-CF functions, which are the quaternionic counterpart of CR functions.
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1 Introduction
The \(\overline{\partial }\)-complex plays an important role in the theory of several complex variables since many important results for holomorphic functions can be obtained by solving nonhomogeneous \(\overline{\partial }\)-equation. We obtain \(\overline{\partial }_b\)-complex when it is restricted to a CR submanifold, and many important results for CR functions can also be obtained by solving \(\overline{\partial }_b\)-equation. In general, for a differential complex, there is an abstract way to obtain a boundary complex restricted to a submanifold, which is written down in terms of quotient sheafs (cf., e.g., [3, 4, 25]).
In quaternionic analysis, we now know the k-Cauchy–Fueter complexes explicitly (cf. [1, 5, 8, 9, 11, 30, 35, 41] and references therein), which are used to show several interesting properties of k-regular functions (cf. [12, 35, 40, 42] and references therein). When restricted to a quadratic hypersurface in \(\mathbb {H}^{n+1},\) we have the tangential k-Cauchy–Fueter operators and k-CF functions (cf. [39] for \(k=1,n=2\)), corresponding to \(\overline{\partial }_b\) and CR functions over a CR manifold. In this paper, we will consider their restriction to a model quadratic hypersurface
in \(\mathbb {H}^{n+1},\) where
Here, we write \(q'=(\ldots ,q_l,\ldots ),q_l=x_{4l+1}+\mathbf {i}x_{4l+2}+ \mathbf {j}x_{4l+3}+\mathbf {k}x_{4l+4}.\) This hypersurface has the structure of the right quaternionic Heisenberg group\(\mathscr {H}=\mathbb {H}^n\times {\mathrm{Im}}\ \mathbb {H}\) with the multiplication given by
where \(x,y\in \mathbb {H}^{n}\) and \({t},s\in {\mathrm{Im}}\ \mathbb {H}.\) We construct a family of differential complexes on \( \mathscr {H} \), the tangential k-Cauchy–Fueter complexes, given by
for a domain \(\Omega \) in \(\mathscr {H} \), where
for fixed \(k=0,1,\ldots \), and \(\odot ^{p}\mathbb {C}^{2}\) is the pth symmetric power of \(\mathbb {C}^2.\) They are the quaternionic counterpart of \(\bar{\partial }_b\)-complex over the Heisenberg group in the theory of several complex variables. They have the same form as the k-Cauchy–Fueter complexes on \(\mathbb {H}^n\) (cf. Remark 2.1), but \(\mathscr {D}_{j}\)’s are given in terms of left invariant vector fields (2.23) (2.26) (2.27), which are differential operators of variable coefficients. So we cannot use the computational algebraic method in [12] to construct these complexes. This family of complexes are natural in the sense that they can be viewed as the restriction to the hypersurface \(\mathcal {S}\) of complexes on \(\mathbb {H}^{n+1},\) but not natural in the sense that they are not invariant under the conformal transformation group \(\mathrm{Sp}( n+1,1)\) of \({\mathscr {H}}\) (cf. Sect. 2.5).
\(\mathscr {D}_{0}\) in (1.4) is called the tangential k-Cauchy–Fueter operator. A \(\odot ^{k}\mathbb {C}^{2}\)-valued distribution f on \(\Omega \) is called k-CF if \(\mathscr {D}_{0}f=0\) in the sense of distributions. The space of all k-CF functions on \(\Omega \) is denoted by \(\mathcal {A}_k(\Omega ).\) A 1-CF function is also called anti-CRF function in [18, 19]. Such functions play an important role in the study of pseudo-Einstein equation over the quaternionic Heisenberg group [19].
On the other hand, when the hypersurface is the boundary of the Siegel upper half space, i.e., the defining function in (1.1) is given by
the corresponding group is the left quaternionic Heisenberg group\(\widetilde{{\mathscr {H}}}:=\mathbb {H}^n\times \mathrm{Im}\,\mathbb {H}\) with the multiplication given by
We already know the tangential k-Cauchy–Fueter complex (cf. [37, Theorem 1.0.1]) on the left quaternionic Heisenberg group by using the twistor method (see also [6, 27] for constructing complexes by this method) . But in this case, \(\wedge ^j\mathbb {C}^{2n}\) in (1.5) must be replaced by the irreducible representation of \(\mathfrak {sp}(2n,\mathbb {C})\) with the highest weight to be the jth fundamental weight (cf. Sect. 2.5). Since it is more complicated than the right case, we only consider the right quaternionic Heisenberg group in this paper. We see that when restricted to different submanifolds, we get different differential complexes. This is a new phenomenon compared to several complex variables, where expressions of \(\overline{\partial }_b\)-complex for different CR submanifolds are the same. It is an interesting problem to write down explicitly the tangential k-Cauchy–Fueter complexes for all quadratic hypersurfaces in \(\mathbb {H}^{n+1}\) (cf. [39] for such hypersurfaces).
In this paper, we prove Hartogs’ phenomenon for k-CF functions over right quaternionic Heisenberg group.
Theorem 1.1
Let \(\Omega \) be a bounded open set in the right quaternionic Heisenberg group \(\mathscr {H}\) with \(\mathrm{dim}\ \mathscr {H}\ge 19,\) and let K be a compact subset of \(\Omega \) such that \(\Omega {\setminus } K\) is connected. Then, for each \(u\in \mathcal {A}_k(\Omega {\setminus } K),\)\(k=2,3,\ldots ,\) we can find \(U\in \mathcal {A}_k(\Omega )\) such that \(U=u\) in \(\Omega {\setminus } K.\)
The restriction of \(\mathrm{dim}\ \mathscr {H} \) and k in this theorem comes from the technical difficulty to establish the \(L^2\) estimate in the remaining cases. A form of Hartogs’ phenomenon was proved for many elliptic differential systems (cf. [12, 26] and references therein). Notably, in our case \(\mathscr {D}_0\) as a matrix-valued horizontal vector field is not an elliptic system, and (1.4) is not an elliptic complex. This is because symbols of \(\mathscr {D}_j\)’s vanish at the cotangent vectors annihilating horizontal vector fields.
In the complex case, we have deep Hartogs–Bochner effect for CR functions on CR submanifolds, which are usually proved by using integral representation formulae (cf. [15, 23, 29] and references therein for further development of this effect). But in the quaternionic case, the integral representation formulae are not sufficiently developed, and only Bochner–Martinelli-type formulae are known (cf. [34, 35]). As in the theory of several complex variables, the formulae with Bochner–Martinelli- type kernels are not good enough to prove the extension phenomenon.
Given a differential complex, it is a fundamental problem to investigate its cohomology group or its Poincaré lemma over a domain (cf., e.g., [7, 16]). In particular, we hope to solve the nonhomogeneous tangential k-Cauchy–Fueter equation
for \(f\in L^2(\mathscr {H},\mathscr {V}_1)\), under the compatibility condition
i.e., f is \(\mathscr {D}_1\)-closed. If we can find compactly supported solution of (1.7)–(1.8) when f is compactly supported, it is a standard procedure to derive Hartogs’ phenomenon (cf., e.g., [17, 35]). One way to solve (1.7)–(1.8) is to consider the associated Hodge–Laplacian
By identifying \(\mathscr {V}_1=\odot ^{k-1}\mathbb {C}^2\otimes \mathbb {C}^{2n}\) with \(\mathbb {C}^{2nk},\) we can see that \(\Box _1\) is a \((2kn)\times (2kn)\) matrix-valued differential operator of the second order, which is not diagonal (cf. Appendix for the expression in the case \(n=2,k=2\)). So it is not easy to verify the subellipticity of \(\Box _1\) and find its fundamental solution, while in the complex case, the Hodge–Laplacian associated with \(\overline{\partial }_b\)-complex is diagonal and it is easy to find its fundamental solution (cf. [13]).
By using the \(L^2\) method, we establish the following estimate: when \(\mathrm{dim}\ \mathscr {H}\ge 19,\) there exists some constant \(c>0\) such that
for \(f\in C^2\left( \mathscr {H},\mathscr {V}_1\right) \cap L^2\left( \mathscr {H},\mathscr {V}_1\right) ,\) where \(\Delta _b\) is the SubLaplacian on the right quaternionic Heisenberg group. But \(\langle \Delta _bf,f\rangle \) does not control the \(L^2\) norm of f. It only controls \(\Vert f\Vert ^2_{L^{\frac{Q+2}{Q-2}}}\) by the well-known Sobolev inequality [19], where \(Q=4n+6\) is the homogeneous dimension of \(\mathscr {H}.\) To avoid this difficulty, we consider the locally flat compact manifold \(\mathscr {H}/\mathscr {H}_{\mathbb {Z}},\) where
is a lattice of \(\mathscr {H}.\) It is a spherical qc manifold (cf. [31]). Because the self-adjoint subelliptic operator \(\Delta _b\) over a compact manifold has discrete spectra, \(\langle \Delta _bf,f\rangle \) controls the \(L^2\) norm of f for \(f\perp \mathrm{ker}\,\Delta _b.\) Moreover, by the Poincaré-type inequality we can show \(\mathrm{ker}\,\Delta _b\) consisting of constant vectors. Namely, there exists some \(c''>0\) such that
for \(f\in C^2(\mathscr {H}/\mathscr {H}_{\mathbb {Z}},\mathscr {V}_1)\) and \(f\perp \) constant vectors. It is a standard way to use the \(L^2\) estimate to solve the nonhomogeneous tangential k-Cauchy–Fueter equation (1.7)–(1.8) on \(\mathscr {H}/\mathscr {H}_\mathbb {Z}\). The solution has an important vanishing property which allows us to prove Hartogs’ phenomenon. See also [13] for the existence theorem for \(\overline{\partial }_b\)-equation over compact CR manifolds by establishing a priori estimate.
In Sect. 2, we give preliminaries on the right quaternionic Heisenberg group, the horizontal complex vector fields \(Z_A^{A'}\)’s and nice behavior of their commutators. We also give the definition of the tangential k-Cauchy–Fueter operators and their basic properties. It is checked directly that (1.4)–(1.5) is a complex. We compare the complexes on the left and right quaternionic Heisenberg groups. In Sect. 3, we use integration by part and Poincaré-type inequality to show the \(L^2\) estimate (1.10) (1.12) for the tangential k-Cauchy–Fueter operator. In Sect. 4, we use the \(L^2\) estimate to solve the nonhomogeneous tangential k-Cauchy–Fueter equation (1.7)–(1.8) over the quotient manifold \(\mathscr {H}/\mathscr {H}_{\mathbb {Z}} \) and derive the Hartogs’ phenomenon. In Sect. 5, we construct the nilpotent Lie groups of step two associated with quadratic hypersurfaces. By constructing a diffeomorphism from the group \(\mathscr {H}\) to the hypersurface \(\mathcal {S}\) in (1.1), we show that the pushforward of the tangential k-Cauchy–Fueter operator on the group \(\mathscr {H} \) coincides with the restriction of the k-Cauchy–Fueter operator on \(\mathbb {H}^{n+1}\) to this hypersurface. Therefore, the restriction of a k-regular functions to \(\mathcal {S}\) is k-CF on \(\mathscr {H}.\)k-CF functions are abundant because so are k-regular functions on \(\mathbb {H}^{n+1}\) [21]. In Appendix, we give the expression of \(\Box _1\) for \(n=2,k=2.\)
2 The tangential k-Cauchy–Fueter complexes
2.1 The right quaternionic Heisenberg group \(\mathscr {H}\) and the locally flat compact manifold \(\mathscr {H}/\mathscr {H}_\mathbb {Z}\)
The multiplication of the right quaternionic Heisenberg group \(\mathscr {H}\) can be written in terms of real variables (cf. [36, (2.13)]) as
for \(x,y\in \mathbb {R}^{4n},\ t,s\in \mathbb {R}^{3},\ \beta =1,2,3,\) where \(B_{kj}^{\beta }\) is the (k, j)th entry of the following matrices
satisfying the commutating relation of quaternions \((B^{1})^{2}=(B^{2})^{2}=(B^{3})^{2}=-id,\ B^{1}B^{2}=B^{3}.\) This is because for \(x=x_{1}+x_{2}{} \mathbf i +x_{3}{} \mathbf j +x_{4}{} \mathbf k \) and \(x'=x'_{1}+x'_{2}{} \mathbf i +x'_{3}{} \mathbf j +x'_{4}{} \mathbf k ,\) we have
where \(\mathbf i _0=1,\mathbf i _1=\mathbf i , \mathbf i _2=\mathbf j ,\mathbf i _3=\mathbf k \). For fixed point \((y,s)\in \mathscr {H},\) the left translate\(\tau _{(y,s)}:\mathscr {H}\longrightarrow \mathscr {H},\)\( (x,t)\longmapsto (y,s)\cdot (x,t),\) is an affine transformation given by a lower triangular matrix by (2.1). So the Lebesgue measure on \(\mathbb {R}^{4n+3}\) is an invariant measure under the left translation of \(\mathscr {H}.\) Recall that we have the following left invariant vector fields on \(\mathscr {H} \):
where \(e_{a}\) is \((0,\ldots ,1,\ldots ,0)\) with only the ath entry equal to 1. Then,
whose brackets are
where \(l,l'=0,1,\ldots ,n-1,\)\(j,k=1,\ldots ,4.\) The SubLaplacian is defined as
The norm of the right quaternionic Heisenberg group \(\mathscr {H}\) is defined by
Define balls \(B(\xi ,r):=\{\eta \in \mathscr {H} ;\Vert \xi ^{-1}\cdot \eta \Vert <r\}\) for \(\xi \in \mathscr {H}, r>0.\) The fundamental set of \(\mathscr {H}\) under the action of the lattice \(\mathscr {H}_{\mathbb {Z}}\) in (1.11) is
\(\mathscr {H}/\mathscr {H}_{\mathbb {Z}}\) is equivalent to \(\mathscr {F}\) as a set.
Proposition 2.1
\(\mathscr {H}\) is the disjoint union of \(\tau _{(n,m)}\mathscr {F}\) with \((n,m)\in \mathscr {H}_{\mathbb {Z}}.\)
Proof
We need to prove that for any \((y,s)\in \mathscr {H},\) there exist unique \((y',s')\in \mathscr {F}\) and \((n,m)\in \mathscr {H}_{\mathbb {Z}}\) such that \((y,s)=(n,m)\cdot (y',s').\) Let \( (n_a,m_a)\in \mathscr {H}_{\mathbb {Z}},a=1,2.\) By the multiplication law (2.1), we have
If \(n_1\ne n_2,\) the y-coordinates of \((n_1,m_1)\cdot ({y},{s})\) and \((n_2,m_2)\cdot ({y},{s})\) are \(n_1+y\) and \(n_2+y,\) respectively, which are different. If \(n_1= n_2,m_1\ne m_2,\) we see that their s-coordinates in (2.9) must be different. This proves the uniqueness.
For \((y,s)=(y_1,\ldots ,y_{4n},s_1,s_2,s_3),\) we can choose \(y'\in \mathbb {R}^{4n}\) with \(0\le y_j'<1 \) and \(n\in \mathbb {Z}^{4n}\) such that \(y_j={n}_j+y_j' \). Then, we can determine \(s'\in \mathbb {R}^{3}\) and \(m\in \mathbb {Z}^{3}\) satisfying
for \(\beta =1,2,3.\) So \(\mathscr {H}\) is the disjoint union of \(\tau _{(n,m)}\mathscr {F}.\) The proposition is proved. \(\square \)
\(\mathscr {H}/\mathscr {H}_{\mathbb {Z}}\) has the structure of a locally flat manifold as follows (cf. [22, p. 238]). Let \(\pi :\mathscr {H}\rightarrow \mathscr {H}/\mathscr {H}_{\mathbb {Z}}\) be the projection. We can find a finite number of balls \(B(\xi _j,r),\ j=1,\ldots ,N,\) covering \(\mathscr {F}\) with r sufficiently small so that \(\tau _{(n,m)} B(\xi _j,r)\cap B(\xi _j,r)=\emptyset \) for any \((0,0)\ne (n,m)\in \mathscr {H}_{\mathbb {Z}}.\) Note that \(\pi B(\xi _i,r)\cap \pi B(\xi _j,r)\ne \emptyset \) for \(i\ne j\) if and only if there exist unique \({(n,m)}\in \mathscr {H}_{\mathbb {Z}},\) such that
Then, we can construct coordinates charts \((\pi B(\xi _j,r),\phi _j),\) where \(\phi _j:\pi B(\xi _j,r)\rightarrow B(\xi _j,r)\) and the transition function \(\phi _j\circ \phi _i^{-1}\) is given by \(\tau _{(n,m)}\) for some \((n,m)\in \mathscr {H}_{\mathbb {Z}}\) such that (2.10) holds.
A function is called periodic on \(\mathscr {H}\) if
for any \((n,m)\in \mathscr {H}_{\mathbb {Z}}.\) A function over \(\mathscr {H}/\mathscr {H}_{\mathbb {Z}}\) can be viewed as a function on \(\mathscr {F}\) and be extended to a periodic function on \(\mathscr {H}\) by
for \((y,s)=(n,m)\cdot (y',s')\) and \((y',s')\in \mathscr {F}.\) If f is periodic, then so is \(Y_af\) for any a. This is because
for \(e_{a} \) as in (2.3). Thus, the action of \(Y_a\) on functions over \(\mathscr {H}/\mathscr {H}_{\mathbb {Z}}\) is well-defined, i.e., it is a vector field over \(\mathscr {H}/\mathscr {H}_{\mathbb {Z}}.\)
2.2 Complex horizontal vector fields \(Z_A^{A'}\)’s and the tangential k-Cauchy–Fueter operator
We consider the following complex horizontal left invariant vector fields on \(\mathscr {H} \):
where \(A=0,1,\ldots ,2n-1,\)\(A'=0',1'.\) It is motivated by the embedding \(\tau \) of quaternionic algebra \(\mathbb {H}\) into \(\mathfrak {gl}(2,\mathbb {C}):\)
and vector fields
to construct the k-Cauchy–Fueter operators on \(\mathbb {H}^{n+1}\) in [35]. We will use matrices
to raise or lower primed indices, e.g., \(Z_{A}^{A'}=\sum _{B'=0',1'} Z_{AB'}\varepsilon ^{B'A'}.\) Here, \((\varepsilon ^{A'B'})\) is the inverse of \((\varepsilon _{A'B'}).\) Then,
and
An element of \(\mathbb {C}^{2}\) is denoted by \((f_{A'})\) with \(A'=0',1'.\) The symmetric power \(\odot ^{p}\mathbb {C}^{2}\) is a subspace of \(\otimes ^{p}\mathbb {C}^2,\) whose element is a \(2^p\)-tuple \((f_{A'_{1}A'_2\ldots A'_{p}})\) with \(A'_{1},A'_2,\ldots ,A'_{p}=0',1',\) such that \(f_{A'_{1}A'_2\ldots A'_{p}}\in \mathbb {C}\) are invariant under permutations of subscripts, i.e.,
for any \(\sigma \) in the group \(S_p\) of permutations of p letters. An element of \(\odot ^{p}\mathbb {C}^2\otimes \wedge ^q\mathbb {C}^{2n}\) is given by a tuple \((f_{A_1'\ldots A_p'A_1 \ldots A_q})\in (\otimes ^{p}\mathbb {C}^2)\otimes (\otimes ^q\mathbb {C}^{2n}),\) which is invariant under permutations of subscripts of \(A_1',\ldots ,A_p',\) and antisymmetric under permutations of subscripts of \(A_1,\ldots ,A_q =0,1,\ldots 2n-1.\) In the sequel, we will write \(f_{A A_2'A_3'\ldots A_k'}:=f_{A_2'A_3'\ldots A_k'A}\) and \(f_{A_3'\ldots A_k'AB}:=f_{ABA_3'\ldots A_k'}\) for convenience. We will use symmetrization of primed indices
The tangential k-Cauchy–Fueter operator in (1.4) is given by
for \(f\in C^1(\Omega ,\mathscr {V}_0) \). The k-Cauchy–Fueter operator on \(\mathbb {H}^{n+1}\) [35] is \(\widehat{\mathscr {D}}_0: C^1(\mathbb {H}^{n+1},\mathscr {V}_0)\rightarrow C^1(\mathbb {H}^{n+1},\mathscr {V}_1)\) with
where \(\nabla \) is given by (2.14). A \(\mathscr {V}_0\)-valued distribution f is called k-regular on \(\Omega \in \mathbb {H}^{n+1}\) if \(\widehat{\mathscr {D}}_0f=0\) on \(\Omega \) in the sense of distributions.
2.3 Commutators of complex horizontal vector fields
The following nice behavior of commutators of \(Z_A^{A'}\)’s plays a very important role to show that (1.4) is a complex and to establish the \(L^2\) estimate. It is also the reason why the tangential k-Cauchy–Fueter complex on the right Heisenberg group is simpler than that on the left one.
Lemma 2.1
-
(1)
Vector fields in each column in (2.16) are commutative, i.e., for fixed \(A'=0'\ \mathrm{or}\ 1',\)
$$\begin{aligned}{}[Z_A^{A'},Z_B^{A'}]=0, \end{aligned}$$(2.19)for any \(A,B=0,\ldots ,2n-1.\)
-
(2)
We have
$$\begin{aligned} \begin{aligned}&[Z_{2l}^{0'},Z_{2l}^{1'}]= \overline{[Z_{2l+1}^{0'},Z_{2l+1}^{1'}]} =8\left( \partial _{s_2}+\mathbf {i}\partial _{s_3}\right) ,\\&[Z_{2l}^{0'},Z_{2l+1}^{1'}] =[Z_{2l+1}^{0'},Z_{2l}^{1'}]=8\mathbf {i}\partial _{s_1}, \end{aligned} \end{aligned}$$(2.20)\(l=0,\ldots ,n-1,\) and any other bracket vanishes.
Proof
- (1):
-
If \(\{A,B\}\ne \{2l,2l+1\}\) for any integer l, we have
$$\begin{aligned}{}[Z_A^{A'},Z_B^{B'}]=0,\quad \mathrm{for}\ A',B'=0',1', \end{aligned}$$by using (2.5) because \(Z_A^{A'}\) and \(Z_B^{B'}\) only involve \(Y_{4l+j}\)’s for different l. It follows from (2.2) (2.5) that
$$\begin{aligned} \begin{aligned} \ [Y_{4l+1},Y_{4l+2}]&=\ \ [Y_{4l+3},Y_{4l+4}]=-\,4\partial _{s_1},\\ [Y_{4l+1},Y_{4l+3}]&=-\,[Y_{4l+2},Y_{4l+4}]=-\,4\partial _{s_2},\\ [Y_{4l+1},Y_{4l+4}]&=\ \ [Y_{4l+2},Y_{4l+3}]=-\,4\partial _{s_3}. \end{aligned} \end{aligned}$$(2.21)Then, for \(\{A,B\}=\{2l,2l+1\},\) we have
$$\begin{aligned} \begin{aligned} \ [Z_{2l}^{0'},Z_{2l+1}^{0'}]&=[-Y_{4l+3}- \mathbf {i}Y_{4l+4},Y_{4l+1}-\mathbf {i}Y_{4l+2}]\\&=[Y_{4l+1},Y_{4l+3}]+[Y_{4l+2},Y_{4l+4}]-\mathbf {i}[Y_{4l+2}, Y_{4l+3}]+\mathbf {i}[Y_{4l+1},Y_{4l+4}]=0,\\ \ [Z_{2l}^{1'},Z_{2l+1}^{1'}]&=[-Y_{4l+1}-\mathbf {i}Y_{4l+2},-Y_{4l+3} +\mathbf {i}Y_{4l+4}]\\&=[Y_{4l+1},Y_{4l+3}]+[Y_{4l+2},Y_{4l+4}]+\mathbf {i}[Y_{4l+2}, Y_{4l+3}]-\mathbf {i}[Y_{4l+1},Y_{4l+4}]=0, \end{aligned} \end{aligned}$$ - (2):
-
Similarly, we have
$$\begin{aligned} \begin{aligned} \ [Z_{2l}^{0'},Z_{2l}^{1'}]&=[-Y_{4l+3}-\mathbf {i} Y_{4l+4},-Y_{4l+1}-\mathbf {i}Y_{4l+2}]\\&=-\,[Y_{4l+1},Y_{4l+3}]+[Y_{4l+2},Y_{4l+4}]\\&\quad -\,\mathbf {i}[Y_{4l+2}, Y_{4l+3}]-\mathbf {i}[Y_{4l+1},Y_{4l+4}]= 8(\partial _{s_{2}}+\mathbf {i}\partial _{s_{3}}),\\ \ [Z_{2l+1}^{0'},Z_{2l+1}^{1'}]&=\overline{[Z_{2l}^{0'},Z_{2l}^{1'}]} =8(\partial _{s_{2}}-\mathbf {i}\partial _{s_{3}}),\\ \ [Z_{2l}^{0'},Z_{2l+1}^{1'}]&=[-Y_{4l+3}- \mathbf {i}Y_{4l+4},-Y_{4l+3}+\mathbf {i}Y_{4l+4}] =-\,2\mathbf {i}[Y_{4l+3},Y_{4l+4}] =8\mathbf {i}\partial _{s_{1}},\\ \ [Z_{2l+1}^{0'},Z_{2l}^{1'}]&=[Y_{4l+1}-\mathbf {i}Y_{4l+2},-Y_{4l+1}- \mathbf {i}Y_{4l+2}] =-\,2\mathbf {i}[Y_{4l+1},Y_{4l+2}] =8\mathbf {i}\partial _{s_{1}}, \end{aligned} \end{aligned}$$by (2.21). The lemma is proved. \(\square \)
On the left quaternionic Heisenberg group, vector fields in each column are not commutative (2.43)–(2.44). We have the following corollary directly by Lemma 2.1 (2).
Corollary 2.1
for any \(A,B=0,\ldots ,2n-1.\)
2.4 The tangential k-Cauchy–Fueter complex
Differential operators in the complex (1.4) are as follows. For \(j=0,1,\ldots ,k-1,\)\(\mathscr {D}_j:C^{\infty }(\Omega ,\mathscr {V}_j)\rightarrow C^{\infty }(\Omega ,\mathscr {V}_{j+1})\) with \(\mathscr {V}_j=\odot ^{k-j}\mathbb {C}^2\otimes \wedge ^j\mathbb {C}^{2n} \) is a differential operator of the first order given by
where \([A_0A_1\ldots A_j]\) is the antisymmetrization of unprimed indices given by
In particular, \(h_{[AB]}:=\frac{1}{2}(h_{AB}-h_{BA})\). By definition, we have
\(\mathscr {D}_k:C^{\infty }(\Omega ,\mathscr {V}_k)\rightarrow C^{\infty }(\Omega ,\mathscr {V}_{k+1})\) with \(\mathscr {V}_k=\wedge ^k\mathbb {C}^{2n}\) and \(\mathscr {V}_{k+1}=\wedge ^{k+2}\mathbb {C}^{2n}\) is a differential operator of the second order given by
For \(j=k+1,\ldots ,2n-2,\)\(\mathscr {D}_j:C^{\infty }(\Omega ,\mathscr {V}_j)\rightarrow C^{\infty }(\Omega ,\mathscr {V}_{j+1})\) with \(\mathscr {V}_j=\odot ^{j-k-1}\mathbb {C}^2\otimes \wedge ^{j+1}\mathbb {C}^{2n} \) is a differential operator of the first order given by
Remark 2.1
The k-Cauchy–Fueter complex on \(\mathbb {H}^n\) [35, 41] is the same as (1.4)–(1.5) with \({\mathscr {H}}\) replaced by \(\mathbb {H}^n\) and \(Z_{A}^{A'}\) in definition of \(\mathscr {D}_{j}\)’s in (2.23), (2.26) and (2.27) replaced by \(\nabla _{A}^{A'}\) in (2.14).
Lemma 2.2
for any \(A,B=0,\ldots ,2n-1\) and \(A',B'=0',1'.\)
Proof
Note that
by (2.19), and
by Corollary 2.1. The lemma is proved. \(\square \)
Now, let us check (1.4) to be a complex by direct calculation as in [41, Section 3.1].
Theorem 2.1
(1.4) is a complex, i.e.,
for each j.
Proof
For \(A,B=0,\ldots , 2n-1\) and \(A_3',\ldots , A_k'=0',1',\) we have
by Lemma 2.2 and \(f_{C'A'A_3'\ldots A_k'}=f_{A'C'A_3'\ldots A_k'}\). For general \(j=1,\ldots ,k-2,\) we have
by using (2.25) repeatedly, Lemma 2.2 and f symmetric in the primed indices again.
For \(j=k-1,\) we have
This is because if \(A'=1',\)\(Z_{[A_1}^{0'}Z_{[A_2}^{1'}Z_{A_3]}^{1'}f_{A_4\ldots A_{k+2}]1'}=0\) by using (2.29), and if \(A'=0',\)
by using (2.25) repeatedly and Corollary 2.1.
For \(j=k,\) we have
This is because if \(A'=0',\)\(Z_{[[A_1}^{0'}Z_{A_2]}^{0'}Z_{A_3}^{1'}f_{A_4\ldots A_{k+3}] }=0\) by using (2.29), and if \(A'=1',\)
by using (2.25) repeatedly and Corollary 2.1.
For \(j=k+1,\ldots ,2n-2,\) we have
by Lemma 2.2. The theorem is proved. \(\square \)
2.5 Comparison with the left case
Recall that a transformation T on \(\mathscr {H}\) is called conformal if \(\Vert T_*W_1\Vert =\Vert T_*W_2\Vert \) for any two horizontal vector fields \(W_1\) and \(W_2\) with \(\Vert W_1\Vert =\Vert W_2\Vert ,\) where \(\Vert W\Vert ^2:= \sum _{j=1}^{4n}a_j^2 \) if we write \(W=\sum _{j=1}^{4n}a_jY_j.\) It is known that the group of conformal transformations on \(\mathscr {H}\) is the real semisimple Lie group \(\mathrm{Sp}(n+1,1)\) of rank one (cf., e.g., [18]) generated by the following transformations:
- (1)
Dilations:
$$\begin{aligned} D_{\delta }:(y,s)\longrightarrow (\delta y,\delta ^{2}s),\ \delta >0; \end{aligned}$$(2.32) - (2)
Left translations:
$$\begin{aligned} \tau _{(x,{t})}:(y,{s})\longrightarrow (x,{t})\cdot (y,{s}); \end{aligned}$$(2.33) - (3)
Rotations:
$$\begin{aligned} R_{\mathbf {a}}:(y,s)\longrightarrow (y{\mathbf {a}},s),\ \mathrm{for} \ {\mathbf {a}}\in \mathrm{Sp}(n), \end{aligned}$$(2.34)where
$$\begin{aligned} \mathrm{Sp}(n)=\{{\mathbf {a}}\in \mathrm{GL}(n,\mathbb {H})|{{\mathbf {a}}\bar{{\mathbf {a}}}^{t}}=I_{n}\}; \end{aligned}$$ - (4)
The inversion:
$$\begin{aligned} R:(y,s)\longrightarrow \left( -\,(|y|^{2}-s)^{-1}y, \frac{-s}{|y|^{4}+|s|^{2}}\right) ; \end{aligned}$$(2.35) - (5)
\(\mathrm{Sp}(1)\) acts on \(\mathscr {H}\) as
$$\begin{aligned} \sigma :(y,s)\longrightarrow (\sigma y,\sigma {s}\sigma ^{-1}), \end{aligned}$$(2.36)where the action on the first factor is left multiplication by \(\sigma \in \mathbb {H}\) with \(|\sigma |=1,\) while the action on the second factor is isomorphism with \(\mathrm{SO}(3)\).
It is known that \(\mathrm{Sp}(n+1,1)\) is a real form of \(\mathrm{Sp}(2(n+2),\mathbb {C})\), whose Lie algebra \(\mathfrak {g}=\mathfrak {sp}(2(n+2),\mathbb {C})\) has the decomposition \(\mathfrak {g}=\mathfrak {g}_{-2}\oplus \mathfrak {g}_{-1}\oplus \mathfrak {g}_{0} \oplus \mathfrak {g}_{1}\oplus \mathfrak {g}_{2},\) where \(\mathfrak {g}_{-2}\) is an complex abelian subalgebra generated by \(T_1,T_2,T_3,\) and \(\mathfrak {g}_{-1}\) is generated by \(\{Y_{AA'}\},A=0,1,\ldots ,2n-1,A'=0',1'\) with
and any other bracket vanishes (cf. [37, (2.10)]). \({\mathfrak {p}}:=\mathfrak {g}_{0} \oplus \mathfrak {g}_{1}\oplus \mathfrak {g}_{2}\) is a parabolic subgroup. \(\mathfrak {u}_-:=\mathfrak {g}_{-2}\oplus \mathfrak {g}_{-1}.\) Then,
Let \(\mathrm{U}_-\) be the complex Lie group with Lie algebra \(\mathfrak {u}_-\). There exist exact sequences [37, Theorem 3.2.1] on \(\mathrm{U}_-\)
for \(0\le k\le n-2,\) where operators \(Q_{j}^{(k)}\)’s are defined in terms of \(Y_{AA'},T_\beta \) (cf. [37, Theorem 1.0.1]). Here, \(V^{(j)}\) is the irreducible representation of \(\mathfrak {sp}(2n,\mathbb {C})\) with the highest weight to be the jth fundamental weight \(\omega _j\) and \(\mathcal {R}( \mathrm{U}_-,V)\) is the space of V-valued polynomials over \(\mathrm{U}_-.\) These complexes are constructed by twistor method, and operators \(Q_{j}^{(k)}\)’s are invariant under \(\mathrm {Sp}(2(n+2),\mathbb {C}).\)
The multiplication (1.6) of the left quaternionic Heisenberg group \(\widetilde{{\mathscr {H}}}\) can be written as
for \(x,y\in \mathbb {R}^{4n},\ t,s\in \mathbb {R}^{3},\, \beta =1,2,3,\) where \(I_{kj}^{\beta }\) is the (k, j)th entry of the following matrices
satisfying the commutating relation of quaternions. Note that
is standard left invariant vector field on \(\widetilde{\mathscr {H}}.\) Denote
where \(A=0,1,\ldots ,2n-1,\)\(A'=0',1'.\) They satisfy the following commutating relations:
\(l=0,\ldots ,n-1,\) and any other bracket vanishes. So by embedding the real Lie algebra of \(\widetilde{{\mathscr {H}}}\) into the complex Lie algebra \(\mathfrak {u}_-\) by \(\widetilde{Z}_{(2l)A'}\mapsto Y_{lA'},\widetilde{Z}_{(2l+1)A'}\mapsto Y_{(n+l)A'}\) we get tangential k-Cauchy–Fueter complexes on \(\widetilde{{\mathscr {H}}}\) (cf. [37, Theorem 1.0.1]), on which \(G=\mathrm{Sp}(2(n+2),\mathbb {C})\) acts naturally.
Now, consider complexes on the right quaternionic Heisenberg group. We can show the following proposition as [38, Proposition 3.1].
Proposition 2.2
Under the transformation \(M_{\mathbf {a}}:{\mathbb {H}}^n\rightarrow {\mathbb {H}}^n,\)\(q\mapsto q'=q{\mathbf {a}}\) with \(\mathbf {a}=(a_{jk})\in GL(n,\mathbb {H}),\) where \(q=(q_1,q_2,\ldots ,q_{n})\) with \(q_{l+1}={x_{4l+1}}+\mathbf {i}{x_{4l+2}} +{\mathbf {j}}{x_{4l+3}}+{\mathbf {k}}{x_{4l+4}},\) we have
where \(\overline{\partial }_{q_{l+1}}=\partial _{x_{4l+1}}+\mathbf {i}\partial _{x_{4l+2}} +\mathbf {j}\partial _{x_{4l+3}}+\mathbf {k}\partial _{x_{4l+4}}.\)
Proof
Denote \({\widehat{q}}=(x_1,\ldots ,x_{4n}) \). Since \(M_{\mathbf {a}}\) defines a real linear transformation on the underlying vector space \(\mathbb {R}^{4n}\), we have \(\widehat{q{\mathbf {a}}}={\widehat{q}}{\mathbf {a}}^\mathbb {R} \) for some \((4n)\times (4n)\) real matrix \({\mathbf {a}}^\mathbb {R}\) associated with \({\mathbf {a}}.\) As the bth element of \(\widehat{q{\mathbf {a}}}\) is \(\sum _{a=1}^{4n}x_a{\mathbf {a}}^\mathbb {R}_{a b},\) we have
Note that we can write \(q_{l+1} = \sum _{j=1}^4{\mathbf {i}}_{j-1} x_{4l+j} \). Therefore,
by \(\left( {{\mathbf {a}}}^\mathbb {R}\right) ^t=\left( \overline{{{\mathbf {a}}}}^t\right) ^\mathbb {R}\), which can be proved as [38, Lemma 2.1 (1)]. The proposition is proved. \(\square \)
Corollary 2.2
Let \(\overline{Q}_{l+1}:=X_{4l+1}+\mathbf iX_{4l+2}+\mathbf jX_{4l+3}+\mathbf kX_{4l+4}.\) Then, \({R_\mathbf {a}}_*\left( \overline{Q}_1,\ldots ,\overline{Q}_n\right) =\left( \overline{Q}_1,\ldots ,\overline{Q}_n\right) \bar{{\mathbf {a}}}^t,\) for rotation \({R_\mathbf {a}}\) in (2.34) with \({\mathbf {a}}\in \mathrm{Sp}(n) \).
Since \(\overline{Q}_l=\overline{\partial }_{q_l}\) at the origin of \({\mathscr {H}},\) the above identity holds at the origin by Proposition 2.2. It holds at other places by the left invariance. By applying the representation \(\tau \) in (2.13), i.e., \(\tau (q_1q_2)=\tau (q_1)\tau (q_2)\) for any \(q_1,q_2\in \mathbb {H} \) (cf. [33, Proposition 2.1]), we get
where \(\tau (\bar{{\mathbf {a}}}^t)\) is a \((2n)\times (2n)\) complex matrix with \(\overline{\mathbf {a}}_{jk}\) replaced by the \(2\times 2\) matrix \(\tau (\overline{\mathbf {a}}_{jk}).\) (2.46) implies that each column in (2.14) is not preserved under rotations (2.34) of \(\mathrm{Sp}(n).\) The commutativity (2.19) of each column that plays a very important role in the construction of our complexes (1.4) is destroyed. So by definition (2.23), (2.26) and (2.27), the differential operators \({\mathscr {D}}_j\)’s in the complex (1.4) in terms of \(Z_{A}^{A'}\)’s are not invariant under \(\mathrm{Sp}(n).\) Therefore, they are not invariant under \(\mathrm{Sp}(2(n+2),\mathbb {C}).\)
Another difference is that the kernel of the tangential k-Cauchy–Fueter in space of \(L^2\) integrable function on the left quaternionic Heisenberg group \(\widetilde{{\mathscr {H}}}\) is infinite dimensional [32], while it is trivial on the right quaternionic Heisenberg group \( {{\mathscr {H}}}\) , since such a function satisfies \(\Delta _b f=0\) by Proposition 2.4 and \(\ker \Delta _b=\{0\}\) in the \(L^2\) space.
On the other hand, if we choose the complex horizontal fields on \({\mathscr {H}}\)
with \(Y_{4l+1}\) replaced by \(-Y_{4l+1},\) then \(\widehat{Z}_{AA'}\)’s satisfy
\(l=0,\ldots ,n-1,\) and any other bracket vanishes, i.e., we can embed the real Lie algebra of \({\mathscr {H}}\) into the complex Lie algebra \({{\mathfrak {u}}}_-.\) Then, the complexes (2.39) on \(\mathrm{U}_-\) induce a family of complexes on \(\mathscr {H}\) invariant under \(\mathrm{Sp}(2(n+2),\mathbb {C}).\) But the first operator is different from the first one in (1.4). Moreover, the \((n-1)\)th operator in the complex induced from \(\mathrm{U}_-\) is a linear combination of \(T_\beta \)’s (cf. [37, Proposition 4.3.3]), while the \((n-1)\)th operator in (1.4) involves only \(Z_{AA'}\)’s. So we get two different families of complexes on \( {{\mathscr {H}}}\). Here, changing \(Y_{4l+1}\) to \(-Y_{4l+1} \) corresponds to changing the sign before \(x_{4l+1}^2\) in the defining function (1.2) of the hypersurface \(\mathcal {S}\). The resulting hypersurface is essentially the boundary of the quaternionic Siegel domain.
On other quadratic hypersurface, there is no reason to expect that the restriction of the k-Cauchy–Fueter operators and complexes is invariant in general under the action of \(\mathrm{Sp}(n).\)
2.6 The adjoint operator
On a domain \(\Omega \subset \mathscr {H},\) denote the inner product
for \(u,v\in L^2(\Omega ,\mathbb {C}),\) where \(\hbox {d}V\) is the Lebesgue measure on \(\mathscr {H}.\) The inner product of \(L^2(\Omega ,\mathscr {V}_1)\) is defined as
for \(f,h\in L^2(\Omega ,\mathscr {V}_1),\) and \(\Vert f\Vert :=\langle f,f\rangle ^{\frac{1}{2}}.\) We define inner products of \(L^2(\Omega ,\mathscr {V}_0)\) and \(L^2(\Omega ,\mathscr {V}_2)\) similarly. Define the \(L^2\)-norm on \(\mathscr {H}/\mathscr {H}_{\mathbb {Z}}\) by
Proposition 2.3
The formal adjoint operator of \(Z_A^{A'}\) is
Proof
For \(u,v\in C_0^\infty (\mathscr {H},\mathbb {C}),\) we have
by integration by part. So \(((Y_a\pm \mathbf {i}Y_b)u,v)=(u,-(Y_a\mp \mathbf {i}Y_b)v).\) Then, (2.48) holds since \(Z_A^{A'}\) has the form \(Y_a\pm \mathbf {i}Y_b\) for some a and b by (2.16). Thus, we have
over \(\mathscr {H}.\) For (2.49) over \(\mathscr {H}/\mathscr {H}_{\mathbb {Z}},\) by using the unit partition, it is sufficient to show it for \(v\in C_0^\infty (\mathscr {H},\mathbb {C}).\) This case follows from the result over \(\mathscr {H}.\)\(\square \)
Lemma 2.3
For \(f\in C_0^1(\mathscr {H},\mathscr {V}_1)\) or \(C^1(\mathscr {H}/\mathscr {H}_{\mathbb {Z}},\mathscr {V}_1),\) we have
Proof
The proof is similar to that for the k-Cauchy–Fueter operator over \(\mathbb {H}^n\) (cf. [40, Lemma 3.1]). For any \(g\in C^1(\mathscr {H}/\mathscr {H}_{\mathbb {Z}},\mathscr {V}_0),\) we have
by using (2.49) and symmetrization
for any \(g\in L^2(\mathscr {H},\odot ^k\mathbb {C}^2),G\in L^2(\mathscr {H},\otimes ^k\mathbb {C}^2).\) (cf. [40, (3.4)]). Here, we have to symmetrise the primed indices in \(\sum _A\delta ^A_{A_1'}f_{A_2'\ldots A_k'A}\) since only after symmetrization it becomes an element of \(C_0^1(\mathscr {H},\mathscr {V}_0).\)\(\square \)
\(\mathscr {D}_0^*\mathscr {D}_0\) is simple since it is diagonal by the following proposition.
Proposition 2.4
For \(f\in C^2(\Omega ,\mathscr {V}_0),\) we have
Proof
Recall that for a \(\otimes ^k\mathbb {C}^2\)-valued function \(F_{A_1'\ldots A_k'}\) symmetric in \(A_2'\ldots A_k',\) we have
by the definition of symmetrization (2.17). As usual, a hat means omittance of the corresponding index. Then, for fixed \(A'_1,\ldots ,A'_k=0',1',\)
by using the following Lemma 2.4 and f symmetric in the primed indices, where \(\mathscr {D}_0^*\) is given by (2.50). The proposition is proved. \(\square \)
Lemma 2.4
For \(A',B'=0',1',\) we have
Proof
Note that
by (2.21), whose summation over l gives us (2.53) for \(A'=B'=0'.\) Similarly, we have
by (2.21), whose summation over l gives us (2.53) for \(A'=0,B'=1'.\) Similarly, (2.53) holds for \(A'=1,B'=0'\) and \(A'=B'=1'\) by
Then, (2.53) follows. \(\square \)
3 The \(L^2\) estimate
We begin with the following Poincaré-type inequality, which was proved for general vector fields satisfying Hörmander’s condition (cf. [20, Theorem 2.1]). So it holds over \(\mathscr {H}.\)
Proposition 3.1
(Poincaré-type inequality) For each f with \(\sum _{a=1}^{4n}|Y_af|^2\in L^1(\mathscr {H}),\) we have
where \(B_r\) is a ball of radius r and \(f_{B_r}={\int _{B_r}f\mathrm{d}V}/{\int _{B_r}\mathrm{d}V}.\)
We say \(f\in L^2(\mathscr {H}/\mathscr {H}_{\mathbb {Z}},\mathscr {V}_1)\) satisfies \(f\perp {\,constant}\)vectors if \(\langle f,C\rangle =0\) for any constant vector \(C\in \mathscr {V}_1.\)
Lemma 3.1
There exists some \(c>0\) such that
for \(f\in C^2\left( \mathscr {H}/\mathscr {H}_{\mathbb {Z}},\mathscr {V}_1\right) \) and \(f\perp \,\)constant vectors.
Proof
As \(\bigcup \limits _{(n,m)\in \mathscr {H}_{\mathbb {Z}}} \tau _{(n,m)}\mathscr {F}=\mathscr {H}\) by Proposition 2.1, we can choose some \(r>0\) and a finite number of elements \((n_i,m_i)\in \mathscr {H}_\mathbb {Z},i=1,\ldots ,N,\) such that
Recall that if we identify \(f\in C^2(\mathscr {H}/\mathscr {H}_{\mathbb {Z}},\mathscr {V}_1)\) with a periodic function on \(\mathscr {H},\) so is \(Y_af.\) Then, the Poincaré-type inequality (3.1) implies that
Since \(f\bot \) constant vectors, we have
Thus, we find that
for constant \(c=\frac{1}{NCr^2}.\)\(\square \)
Lemma 3.2
(cf. [40, Lemma 2.1]) For any \(h,H\in \mathbb {C}^{2n}\otimes \mathbb {C}^{2n},\) we have
We have the following \(L^2\) estimate.
Theorem 3.1
For \(n>3,k\ge 2,\) there exists some \(c_{n,k}>0\) such that
for \(f\in Dom(\mathscr {D}_1)\cap Dom(\mathscr {D}_0^*)\) and \(f\perp \,\)constant vectors over \(\mathscr {H}/\mathscr {H}_{\mathbb {Z}}\).
Proof
We use the \(L^2\) method for the k-Cauchy–Fueter operator on \(\mathbb {H}^n\) in [40]. Since \(C^2\) functions are dense in \( Dom(\mathscr {D}_1)\cap Dom(\mathscr {D}_0^*)\) for the compact manifold \(\mathscr {H}/\mathscr {H}_{\mathbb {Z}}\), it is sufficient to prove (3.2) for \(f\in C^2(\mathscr {H}/\mathscr {H}_{\mathbb {Z}},\odot ^{k-1}\mathbb {C}^2\otimes \mathbb {C}^{2n}).\) We have
by using (2.51) to expand the symmetrization. Note that
and
by using commutators. For the first sum, we have
by relabeling indices and f symmetric in the primed indices. Then, by applying Lemma 3.2 with \(h_{BA}=\sum _{A'}Z_B^{A'}f_{AA'B_3'\ldots B_k'}\) and \(H_{A B}=\sum _{A'}Z_A^{A'}f_{BA'B_3'\ldots B_k'}\) for fixed \(B_3',\ldots ,B_k',\) we get
where
by Lemma 2.4. Thus, by substituting (3.4)–(3.5) and (3.7)–(3.8) to (3.3), we get
To control the commutator term \({\mathscr {C}}\) in (3.5), note that
by (2.16). Then, it follows from Lemma 2.1 that (1)
\(A,B=0,\ldots ,2n-1;\) (2) for \(A'=B',\) we have
(3) if \(\{A,B\}\ne \{2l,2l+1\}\) for any l, then \(\left[ Z_A^{A'},\overline{Z_B^{B'}}\right] =0\) for any \(A',B'.\) Thus, we have
by using (1) and (3) above, relabeling indices and f symmetric in the primed indices. Apply (3.11) to \({\mathscr {C}}\) above to get
For any \(u,v\in C^1({\mathscr {H}}/\mathscr {H}_\mathbb {Z},\mathbb {C}),\) we have
by (2.21). As
for \(a,b=1,\ldots ,4n,\) we get
Similarly, we have
Then, apply (3.13)–(3.14) to the right-hand side of (3.12) to get
So it follows from estimate (3.9) that
4 Hartogs’ phenomenon
4.1 The nonhomogeneous tangential k-Cauchy–Fueter equation over \(\mathscr {H}/\mathscr {H}_{\mathbb {Z}}\)
Consider the Hilbert subspace \(\mathcal {L}\) consisting of \(f\in L^2\left( \mathscr {H}/\mathscr {H}_{\mathbb {Z}},\mathscr {V}_1\right) \) and \(f\perp \) constant vectors. The domain of \(\Box _1\) over \(\mathcal {L}\) is
Proposition 4.1
The associated Hodge–Laplacian \(\Box _1\) is densely defined, closed, self-adjoint and nonnegative operator on \(\mathcal {L}.\)
The proof is exactly the same as that of Proposition 3.1 in [40] since \(\mathcal {L}\oplus \{const.\}=L^2(\mathscr {H}/\mathscr {H}_\mathbb {Z}, \mathscr {V}_1)\), and the action of \(\Box _1\) on \(\{const.\}\) is trivial. We omit the detail. Now, we can find solution to (1.7)–(1.8), whose proof is similar to that of Theorem 1.2 in [40] for the k-Cauchy–Fueter operator on \(\mathbb {H}^n.\)
Theorem 4.1
Suppose that \(\mathrm{dim}\ \mathscr {H}\ge 19 \) and \(k=2,3,\ldots \). If \(f\in \mathrm{Dom}(\mathscr {D}_1)\) is \(\mathscr {D}_1\)-closed and \(f\perp \) constant vectors, then there exist \(u\in L^2\left( \mathscr {H}/\mathscr {H}_{\mathbb {Z}},\mathscr {V}_0\right) \) such that
Proof
The \(L^2\) estimate (3.2) implies
for \(g\in \mathrm{Dom}(\Box _1),\) i.e.,
Thus, \(\Box _1:\mathrm{Dom}(\Box _1)\rightarrow \mathcal {L}\) is injective. This together with the self-adjointness of \(\Box _1\) by Proposition 4.1 implies the density of the range. For fixed \(f\in \mathcal {L},\) the complex anti-linear functional
is then well-defined on a dense subspace of \(\mathcal {L}.\) It is finite since
for any \(g\in \mathrm{Dom}(\Box _1),\) by (4.1). So \(l_f\) can be uniquely extended to a continuous anti-linear functional on \(\mathcal {L}.\) By the Riesz representation theorem, there exists a unique element \(h\in \mathcal {L}\) such that \(l_f(F)=\langle h,F\rangle \) for any \(F\in \mathcal {L},\) and \(\Vert h\Vert =\Vert l_f\Vert \le \frac{1}{c_{n,k}}\Vert f\Vert .\) Then, we have
for any \(g\in \mathrm{Dom}(\Box _1).\) This implies that \(h\in \mathrm{Dom}(\Box _1^*)\) and \(\Box _1^*h=f,\) and so \(h\in \mathrm{Dom}(\Box _1)\) and \(\Box _1h=f\) by self-adjointness of \(\Box _1.\) We write \(h=Nf.\) Then, \(\Vert Nf\Vert \le \frac{1}{c_{n,k}}\Vert f\Vert .\)
Since \(Nf\in \mathrm{Dom}(\Box _1),\) we have \(\mathscr {D}_0^*Nf\in \mathrm{Dom}(\mathscr {D}_0),\)\(\mathscr {D}_1Nf\in \mathrm{Dom}(\mathscr {D}_1^*),\) and
by \(\Box _1Nf=f.\) Because f and \(\mathscr {D}_0F\) for any \(F\in \mathrm{Dom}(\mathscr {D}_0)\) are both \(\mathscr {D}_1\)-closed, the above identity implies \(\mathscr {D}_1^*\mathscr {D}_1Nf\in \mathrm{Dom}(\mathscr {D}_1)\) and so \(\mathscr {D}_1\mathscr {D}_1^*\mathscr {D}_1Nf=0.\) Then,
i.e., \(\mathscr {D}_1^*\mathscr {D}_1Nf=0.\) Hence, \(\mathscr {D}_0\mathscr {D}_0^*Nf=f\) by (4.2). \(\square \)
4.2 Proof of Hartogs’ phenomenon
We need the analytic hypoellipticity of \(\Delta _b\). Let G be a nilpotent Lie group of step 2, and its Lie algebra \(\mathfrak {g}\) has decomposition: \(\mathfrak {g}=\mathfrak {g}_1\oplus \mathfrak {g}_2\) satisfying \([\mathfrak {g}_1,\mathfrak {g}_1]\subset \mathfrak {g}_2,\ [\mathfrak {g},\mathfrak {g}_2]=0.\) Consider the condition (H): For any \(\lambda \in \mathfrak {g}_2^*{\setminus }\{0\},\) the antisymmetric bilinear form
for \(Y,Y'\in \mathfrak {g}_1\) is nondegenerate. Métivier proved the following theorem for analytic hypoellipticity.
Theorem 4.2
([24, Theorem 0]) Let P be a homogeneous left invariant differential operator on a nilpotent Lie group G satisfies condition (H). Then, the following are equivalent:
- (i)
P is analytic hypoelliptic;
- (ii)
P is \(C^\infty \) hypoelliptic.
Corollary 4.1
\(\Delta _b\) is analytic hypoelliptic on a domain \(\Omega \subset \mathscr {H},\) i.e., for any distribution \(u\in S'(\Omega )\) such that \(\Delta _bu\) is analytic, u must be also analytic.
Proof
It follows from the well-known subellipticity of \(\Delta _b \) that u is locally \(C^{k+1}\) if \(\Delta _bu\) is locally \(C^{k }\). So \(\Delta _b \) is \(C^\infty \) hypoelliptic. To obtain the analytic hypoellipticity of \(\Delta _b \) by applying Theorem 4.2, it is sufficient to check the condition (H) for the right quaternionic Heisenberg group \(\mathscr {H}\). In this case, \(\mathfrak {g}_1=\mathrm{span}\{Y_1,\ldots ,Y_{4n}\},\)\(\mathfrak {g}_2=\mathrm{span}\left\{ \partial _{s_1}, \partial _{s_2}, \partial _{s_3}\right\} ,\) where \(Y_1,\ldots ,Y_{4n}\) is the left invariant vector fields in (2.4). Let \(\lambda \in \mathfrak {g}_2^*{\setminus }\{0\}.\) For \(Y_{4l+j},Y_{4l+j'}\in \mathfrak {g}_1,\) we have
by (2.5), if we write \(\lambda (\partial _{s_{\beta }})=\lambda _{\beta } \). Then, the matrix associated with \(B_\lambda \) is
whose determinant is \( \left( \lambda _1^2+\lambda _2^2+\lambda _3^2\right) ^{2n}\) by direct calculation. So \(B_\lambda \) is nondegenerate for \(\lambda \in \mathfrak {g}_2^*{\setminus }\{0\},\) i.e., \(\mathscr {H}\) satisfies condition (H). \(\square \)
Liouville-type theorems hold for SubLaplacian \(\Delta _b\) on the right quaternionic Heisenberg group by the following general theorem of Geller.
Theorem 4.3
([14, Theorem 2]) Let \(\mathscr {L}\) be a homogeneous hypoelliptic left invariant differential operator on a homogeneous group G. Suppose \(u\in S'(G)\) and \(\mathscr {L}u=0.\) Then, u is a polynomial.
Theorem 4.4
Let \(\widetilde{\Omega }\) be an open set in \(\mathscr {F}\) such that \(\widetilde{{\Omega }}\Subset \mathring{\mathscr {F}}\) and \(\mathscr {F}{\setminus }\widetilde{\Omega }\) are connected. If \(f\in C^1(\mathscr {H}/\mathscr {H}_{\mathbb {Z}},\mathscr {V}_1)\) with \(\mathrm{supp}f\subset \widetilde{\Omega }\) is \(\mathscr {D}_1\)-closed and \(f\perp \) constant vectors, then there exist \(u\in C^2\left( \mathscr {H}/\mathscr {H}_{\mathbb {Z}},\mathscr {V}_0\right) \) such that
with \(\mathrm{supp}\,u\subset \widetilde{\Omega }.\)
Proof
By Theorem 4.1, we can find a solution \(u\in L^2\left( \mathscr {H}/\mathscr {H}_{\mathbb {Z}},\mathscr {V}_0\right) \) to (4.4). For \(c\in \mathbb {{H}}\), denote
We see that \(\mathscr {H}'_c\cap \Omega =\emptyset \) for |c| small by \(\widetilde{\Omega }\Subset \mathring{\mathscr {F}}.\)
Since \(\mathscr {D}_0u=0\) on \(\left( \mathscr {H}/\mathscr {H}_{\mathbb {Z}}\right) {\setminus }\widetilde{\Omega },\) we have \(\mathscr {D}_0^*\mathscr {D}_0u=0,\), and then, by Proposition 2.4\(\Delta _bu_{A'_1\ldots A'_{k }A}=0\) on \(\left( \mathscr {H}/\mathscr {H}_{\mathbb {Z}}\right) {\setminus }\widetilde{\Omega }\) in the sense of distributions for any fixed \(A'_1,\ldots ,A'_{k },A\). So it is real analytic on \(\left( \mathscr {H}/\mathscr {H}_{\mathbb {Z}}\right) {\setminus }\widetilde{\Omega }\) by Corollary 4.1. Moreover, u is \(C^2\) on \(\mathscr {H}/\mathscr {H}_{\mathbb {Z}}\) by subellipticity of \(\Delta _b \). In particular, \(u(q',c,s)\) is well-defined on \(\mathscr {H}'_c/\mathscr {H}'_{\mathbb {Z}}\) as a real analytic function. So it can be extended to a periodic function over \(\mathscr {H}'_c\) by (2.11). Now, let \(\mathscr {D}_0'\) be the tangential k-Cauchy–Fueter operator on \(\mathscr {H}_c',\) i.e., \(\mathscr {D}_0'u\) is a \(\odot ^{k-1}\mathbb {C}^2\otimes \mathbb {C}^{2n-2}\)-valued function with
By applying Proposition 2.4 to \(\mathscr {H}_c',\) we see that \(\Delta _b'u=0,\) where \(\Delta _b'=-\sum _{a=0}^{4n-5}Y_a^2.\) Then, apply Liouville-type Theorem 4.3 to the group \(\mathscr {H}_c'\) and \(\Delta _b'\) to get
which must be a constant by periodicity. Thus, u only depends on the variable \(q_n.\)
Similarly, we can prove u is a constant on the subgroup
Now, if replacing u by \(u-\) const., we see that u vanishes in a neighborhood of \(\mathscr {H}_0''.\) Consequently, by the identity theorem for real analytic functions it vanishes on the connected component \(\mathscr {F}{\setminus }\widetilde{{\Omega }}.\) Thus, \(\mathrm{supp}\,u\subset \widetilde{\Omega }.\)\(\square \)
The solution with supp\(\,u\subset \widetilde{\Omega }\) above plays the role of compactly supported solution to \(\overline{\partial }\) equation or the tangential k-Cauchy–Fueter equations (cf., e.g., [17, 35]). It leads to Hartogs’ extension phenomenon as follows.
The proof of Theorem 1.1
Without loss of generality, we can assume \(\Omega \Subset \mathring{\mathscr {F}}\) by dilating if necessary. Let \(\chi \in C_0^{\infty }(\Omega )\) be equal to 1 in a neighborhood of K such that \(\mathscr {F}{\setminus }\mathrm{supp}\,\chi \) is connected. Set
Then, \(\widetilde{u}\in C^{\infty }(\Omega ),\) and \(\widetilde{u}|_{\Omega {\setminus } \mathrm{supp}\,{\chi }}=u|_{\Omega {\setminus } \mathrm{supp}\,\chi }.\) We have
on \(\mathscr {H},\) where \(f_{A_2'\ldots A_k'A}=-\sum _{A_1'}Z_A^{A_1'}\chi \cdot u_{A_1'\ldots A_k'}\) by \(\mathscr {D}_0u=0\) on \(\Omega {\setminus } K.\) Hence, \(f\in C_0^\infty (\mathscr {H},\mathscr {V}_1)\) vanishes in K and outside \(\Omega ,\) satisfying \(\mathscr {D}_1f=\mathscr {D}_1\mathscr {D}_0\tilde{u}=0\) by (2.30). We can extend f to a periodic function and view it as an element of \(C^\infty (\mathscr {H}/\mathscr {H}_\mathbb {Z},\mathscr {V}_1).\)
Denote
Then, we have \((f-c) \perp \,\)constant vectors. It follows from Theorem 4.4 that there exists a solution \(\widetilde{U}\in C^2(\mathscr {H}/\mathscr {H}_\mathbb {Z},\mathscr {V}_0)\) to \(\mathscr {D}_0\widetilde{U}=f-c,\) which vanishes outside \(\widetilde{\Omega }:=\mathrm{supp}\,\chi \). Then, \(\mathscr {D}_0(\widetilde{u}-\widetilde{U})=c\) on \(\mathscr {H}/\mathscr {H}_\mathbb {Z}.\) So \(c=\mathscr {D}_0\widetilde{u}|_{\Omega {\setminus } \widetilde{\Omega }}=\mathscr {D}_0 {u}|_{\Omega {\setminus } \widetilde{\Omega }}=0.\) Therefore, \( U=\widetilde{u}-\widetilde{U} \) is k-CF in \(\Omega \) since \(\mathscr {D}_0(\widetilde{u}-\widetilde{U})=0.\) Note that \(\widetilde{U}\equiv 0\) outside \(\widetilde{\Omega }\) and \( \mathscr {F} {\setminus } \widetilde{\Omega }\) is connected. So \(U=u\) in \(\Omega {\setminus } \widetilde{\Omega }.\) Then, \(U=u\) in \(\Omega {\setminus } K\) by the identity theorem for real analytic functions. The theorem is proved. \(\square \)
5 The restriction of the k-Cauchy–Fueter operator to the hypersurface \(\mathcal {S}\)
5.1 The nilpotent Lie groups of step two associated with quadratic hypersurfaces
Let \((x_1,\ldots ,x_{4n},\)\(t_1,t_2,t_3)\) be coordinates of \(\mathbb {R}^{4n+3}.\) Now, consider general quadratic hypersurfaces \(\widehat{\mathcal {S}}\) defined by
for some symmetric matrix \({\mathbb {S}}.\) Define the projection:
where \(\mathbf {t}=t_1\mathbf i +t_2\mathbf j +t_3\mathbf k ,\)\(q_{l+1}=x_{4l+1}+\mathbf i x_{4l+2}+\mathbf j x_{4l+3}+\mathbf k x_{4l+4},\)\(l=0,\ldots ,n-1\) and \(t_\beta =x_{4n+1+\beta }\) for \(\beta =1,2,3.\) Let \(\psi :\mathbb {H}^{n}\times \mathrm{Im}\,\mathbb {H}\longrightarrow \mathcal {S}\subset \mathbb {H}^{n+1}\) be its inverse. The Cauchy–Fueter operator is
Then, \(\overline{\partial }_{q_{l+1}}+\overline{\partial }_{q_{l+1}}\phi \cdot \overline{\partial }_{q_{n+1}}\) is a vector field tangential to the hypersurface \(\widehat{\mathcal {S}},\) since
\(l=0,1,\ldots ,n-1.\) This vector field is exactly the pushforward vector field \(\psi _*\big (\overline{\partial }_{q_{l+1}}+\overline{\partial }_{q_{l+1}}\phi \cdot \overline{\partial }_{\mathbf {t}}\big )\), where \(\overline{\partial }_{\mathbf {t}}=\mathbf i {\partial }_{t_{1}}+ \mathbf j {\partial }_{t_{2}}+\mathbf k {\partial }_{t_{3}}.\) Because
for \(\beta =1,2,3,j=1,\ldots ,4,l=0,\ldots ,n-1 \), and
Denote
Proposition 5.1
We have
where \({\mathbb {I}}^\beta \) is the \((4n)\times (4n)\) matrix \(\mathrm{diag}\left( I^\beta ,\ldots ,I^\beta \right) .\)
Proof
The proof is similar to that of Proposition 2.1 in [39]. Consider right multiplication by \(\mathbf i _\beta \). Note that
we can write
where \(I^\beta \)’s are given by (2.41). \(B^\beta \) in (2.2) is the matrix associated with left multiplication by \(\mathbf {i}_\beta \) ([39, p. 1358]). Then, we have
Substitute it into (5.3) to get
by the antisymmetry of \(I^\beta .\)\(\square \)
By Proposition 5.1, we get
So \(\mathrm{span}_\mathbb {C}\big \{X_1,\ldots ,X_{4n},\partial _{t_1}, \partial _{t_2},\partial _{t_3}\big \}\) is a nilpotent Lie algebra with center \(\mathrm{span}_\mathbb {C}\big \{\partial _{t_1}, \partial _{t_2},\partial _{t_3}\big \}.\) The corresponding nilpotent Lie group of step two is the group associated with the quadratic hypersurface \(\widehat{\mathcal {S}}\).
Now, if we choose the matrix \({\mathbb {S}}\) so that
then the Lie algebra spanned by \(X_1,\ldots ,X_{4n},\partial _{t_1},\partial _{t_1},\partial _{t_3}\) is isomorphic to the Lie algebra of the right quaternionic Heisenberg group \({\mathscr {H}}.\) It is sufficient to choose \(\mathbb {S}=\mathrm{diag}(S,\ldots ,S)\) such that \(SI^\beta +I^\beta S=2B^\beta ,\) where S is a symmetric \(4\times 4\) matrix. Namely,
for \(C^\beta =SI^\beta .\) Then,
and
satisfy (5.5). Thus, the defining function (5.1) of \(\widehat{{\mathcal {S}}}\) in this case is (1.2) of \(\mathcal {S}\), and so the Lie group associate with \({\mathcal {S}}\) is the right quaternionic Heisenberg group.
5.2 The restriction of the k-Cauchy–Fueter operator
\(X_a\)’s for \(\mathcal {S}\) has the form
Since \(C^\beta \) is not antisymmetric, they are different from the standard left invariant vector fields (2.4) on \(\mathscr {H} \). It is standard that they can be transformed to the standard left invariant vector fields (2.4) on \(\mathscr {H} \) by a simple coordinate transformation \(\mathcal {F}:\mathscr {H}\rightarrow \mathbb {R}^{4n+3},(y,s)\mapsto (x,t)\) given by
(cf. [39, (1.8)]) with \( D^\beta :=C^\beta +\left( C^\beta \right) ^t \) symmetric. It is direct to see that
where \(Y_{4l+j}\) is given by (2.4). Then, we find the relationship between complex horizontal vector fields \(Z_A^{A'}\)’s on \(\mathscr {H} \) and \(\nabla _A^{A'}\)’s on \( {\mathbb {H}}^{n+1}\).
Proposition 5.2
Under the diffeomorphism \(\psi \circ \mathcal {F}:\mathscr {H}\rightarrow \mathcal {S}\), we have
for fixed \(A=0,1,\ldots ,2n-1,A'=0',1' \), where \(\tau \) is the embedding given by (2.13).
Proof
As \(\tau \) is a representation, we have
where \(\varepsilon =\left( \begin{matrix}0&{}\quad -\,1\\ 1&{}\quad 0\end{matrix}\right) \) in (2.15). Then, (5.7) follows. \(\square \)
From this proposition, we can derive the relationship between operators in k-Cauchy–Fueter complex on \(\mathbb {H}^{n+1}\) and that in the tangential k-Cauchy–Fueter complex on \(\mathscr {H} \).
Proposition 5.3
Suppose that f is a k-regular function near \(q_0\in \mathcal {S}.\) Then, \(\left( \psi \circ \mathcal {F}\right) ^*f\) is k-CF on \(\mathscr {H}\) near the point \(\mathcal {F}^{-1}(\pi (q_0)) \).
Proof
As f is a k-regular function near \(q_0\in \mathcal {S}\subset \mathbb {H}^{n+1},\) we have \(\sum _{B'=0',1'}\nabla _{A}^{B'}f_{B' A_2'\ldots A_k'}=0\) for any fixed \(A =0,1,\ldots ,2n+1, \)\(A_2',\ldots , A_k' =0',1'.\) Then, we find that
for any fixed \(A =0,1,\ldots ,2n-1, \)\(A_2',\ldots , A_k' =0',1' \), by Proposition 5.2. The proposition is proved. \(\square \)
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Appendix
Appendix
In the case \(n=2,k=2,\) We have isomorphisms
by identifying \(f\in \odot ^{2}\mathbb {C}^{2}\) and \(F\in \mathbb {C}^{2}\otimes \mathbb {C}^{4}\) with
respectively. The operator \(\mathscr {D}_0\) in (2.18) can be written as a \(8\times 3\) matrix-valued differential operator:
Similarly, the operator \(\mathscr {D}_1\) in (2.23) can be written as a \(6\times 8\) matrix-valued differential operator:
Thus, we have \(\mathscr {D}_{0}^*=-\overline{\mathscr {D}_{0}}^t,\ \mathscr {D}_{1}^*=-\overline{\mathscr {D}_{1}}^t.\) Then, by direct calculation we have
with
where \(\Delta _b=-Y_1^2\cdots -Y_8^2,\Delta _1=-Y_1^2-Y_2^2, \Delta _2=-Y_3^2-Y_4^2,\Delta _3=-Y_5^2-Y_6^2,\Delta _4=-Y_7^2-Y_8^2, L_1=8(\partial _{s_2}+\mathbf {i}\partial _{s_3}).\) Because of the complexity of \(\Box _1\) in (6.3), it is not easy to obtain its fundamental solution.
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Shi, Y., Wang, W. The tangential k-Cauchy–Fueter complexes and Hartogs’ phenomenon over the right quaternionic Heisenberg group. Annali di Matematica 199, 651–680 (2020). https://doi.org/10.1007/s10231-019-00895-0
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DOI: https://doi.org/10.1007/s10231-019-00895-0