Abstract
Many fundamental results of pluripotential theory on the quaternionic space \({\mathbb {H}}^n\) are extended to the Heisenberg group. We introduce notions of a plurisubharmonic function, the quaternionic Monge–Ampère operator, differential operators \(d_0\) and \(d_1\) and a closed positive current on the Heisenberg group. The quaternionic Monge–Ampère operator is the coefficient of \( (d_0d_1u)^n\). We establish the Chern–Levine–Nirenberg type estimate, the existence of quaternionic Monge–Ampère measure for a continuous quaternionic plurisubharmonic function and the minimum principle for the quaternionic Monge–Ampère operator. Unlike the tangential Cauchy–Riemann operator \( {\overline{\partial }}_b \) on the Heisenberg group which behaves badly as \( \partial _b{\overline{\partial }}_b\ne -{\overline{\partial }}_b\partial _b \), the quaternionic counterpart \(d_0\) and \(d_1\) satisfy \( d_0d_1=-d_1d_0 \). This is the main reason that we have a good theory for the quaternionic Monge–Ampère operator than \( (\partial _b{\overline{\partial }}_b)^n\).
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1 Introduction
The theory of subharmonic functions and potential theory has already been generalized to Carnot groups in terms of SubLaplacians (cf. e.g. [9, 11] and references therein), and the generalized horizontal Monge–Ampère operator and H-convex functions on the Heisenberg group have been studied for more than a decade (cf. [7, 11,12,13, 15, 17, 19, 22] and references therein). For the 3-dimensional Heisenberg group, Gutiérrez and Montanari [15] proved that the Monge–Ampère measure defined by
can be extended to H-convex functions, where \( Hess_X(u)\) is the symmetric \(2\times 2\)-matrix
and \(X_1,X_2, T\) are standard left invariant vector fields on the 3-dimensional Heisenberg group. u is called H-convex on a domain \(\Omega \) if for any \(\xi ,\eta \in \Omega \) such that \(\xi ^{-1}\eta \in H_0\) and \( \xi \delta _r(\xi ^{-1}\eta )\in \Omega \) for \(r\in [0,1]\), the function of one real variable \(r\rightarrow u(\xi \delta _r (\xi ^{-1}\eta )) \) is convex in [0, 1], where \(\delta _r\) is the dilation and \(H_0\) indicates the subset of horizontal directions through the origin. It was generalized to the 5-dimensional Heisenberg group by Garofalo and Tournier [13], and to k-Hessian measures for k-convex functions on any dimensional Heisenberg groups by Trudinger and Zhang [22].
In the theory of several complex variables, we have a powerful pluripotential theory about the complex Monge–Ampère operator \((\partial {\overline{\partial }} )^n\) and closed positive currents, where \( {\overline{\partial }} \) is the Cauchy-Riemann operator (cf. e.g. [18]). It is quite interesting to develop its CR version over the Heisenberg group. A natural CR generalization of the complex Monge–Ampère operator is \((\partial _b{\overline{\partial }}_b)^n\), where \( {\overline{\partial }}_b \) is the tangential Cauchy-Riemann operator. But unlike \({\overline{\partial }}\partial =-\partial {\overline{\partial }}\), it behaves badly as
because of the noncommutativity of horizontal vector fields (cf. Subsection 3.1). So it is very difficult to investigate the operator \((\partial _b{\overline{\partial }}_b)^n\), e.g. its regularity. On the other hand, pluripotential theory has been extended to the quaternionic space \({\mathbb {H}}^n\) (cf. [1,2,3,4,5,6, 10, 14, 23,24,25,26,27,28, 33] and references therein). If we equip the \((4n+1)\)-dimensional Heisenberg group a natural quaternionic strucuture on its horizontal subspace, we can introduce differential operators \(d_0\), \(d_1\) and \(\triangle u=d_0d_1u\) in terms of complex horizontal vector fields, as the quaternionic counterpart of \(\partial _b\), \( {\overline{\partial }}_b \) and \(\partial _b {\overline{\partial }}_b \). They behave so well that we can extend many fundamental results of quaternionic pluripotential theory on \({\mathbb {H}}^n\) to the Heisenberg group.
The \((4n+1)\)-dimensional Heisenberg group \({\mathscr {H}} \) is the vector space \({\mathbb {R}}^{4n+1}\) with the multiplication given by
for \(x,y\in {\mathbb {R}}^{ 4n}\), \({t},{s }\in {\mathbb {R}} \). Here \(\langle \cdot ,\cdot \rangle \) is the standard symplectic form. We introduce a partial quaternionic structure on the Heisenberg group simply by identifying the underlying space of \( {\mathbb {H}}^n \) with \({\mathbb {R}}^{4n}\). For a fixed \( q\in {\mathbb {H}}^n \), consider a 5-dimensional real subspace
which is a subgroup. \({\mathscr {H}}_q\) is nonabelian for all \( q\in {\mathbb {H}}^n \) except for a \({\mathrm{codim}}_{{\mathbb {R}}} 3\) quadratic cone \({\mathfrak {D}}\). For a point \(\eta \in {\mathscr {H}} \), the left translate of the subgroup \({\mathscr {H}}_q\) by \(\eta \),
is a 5-dimensional real hyperplane through \(\eta \), called a (right) quaternionic Heisenberg line. A \([-\infty ,\infty )\)-valued upper semicontinuous function on \({\mathscr {H}} \) is said to be plurisubharmonic if it is \(L^1_{\mathrm{loc}} \) and is subharmonic (in terms of the SubLaplacian) on each quaternionic Heisenberg line \({\mathscr {H}}_{\eta ,q}\) for any \(\eta \in {\mathscr {H}} \), \( q\in {\mathbb {H}}^n{\setminus }{\mathfrak {D}} \).
Let \(X_1,\ldots X_{4n}\) be the standard horizontal left invariant vector fields (2.2) on the Heisenberg group \({\mathscr {H}} \). Denote the tangential Cauchy–Fueter operator on \({\mathscr {H}} \) by
and its conjugate \( {Q_l}=X_{4l+1}-{\mathbf {i}}X_{4l+2}-{\mathbf {j}}X_{4l+3}-{\mathbf {k}}X_{4l+4}, \) \( l =0,\ldots ,n-1\). See [32] for the Cauchy–Fueter operator on other nilpotent groups of step two. Compared to the Cauchy–Fueter operator on \( \mathbb {{H}}^n\), the tangential Cauchy–Fueter operator \(\overline{Q_l}\) is much more complicated because not only \({\mathbf {i}},{\mathbf {j}},{\mathbf {k}}\) are noncommutative, but also \(X_a\)’s are. In particular,
is not real. But for a real \( C^2 \) function u, the \(n\times n\) quaternionic matrix
is hyperhermitian, called the horizontal quaternionic Hessian. It is nonnegative if u is plurisubharmonic. We define the quaternionic Monge–Ampère operator on the Heisenberg group as
where \(\det \) is the Moore determinant.
Alesker obtained Chern–Levine–Nirenberg estimate for the quaternionic Monge–Ampère operator on \({\mathbb {H}}^n\) [4]. We extend this estimate to the Heisenberg group, and obtain the following existence theorem of the quaternionic Monge–Ampère measure for a continuous plurisubharmonic function.
Theorem 1.1
Let \(\{u_j\}\) be a sequence of \(C^2\) plurisubharmonic functions converging to u uniformly on compact subsets of a domain \(\Omega \) in \({\mathscr {H}} \). Then u be a continuous plurisubharmonic function on \(\Omega \). Moreover, \(\det \left( {\overline{Q_l}}Q_mu_j+ 8 \delta _{lm}{\mathbf {i}} \partial _{t }u_j\right) \) is a family of uniformly bounded measures on each compact subset K of \(\Omega \) and weakly converges to a non-negative measure on \(\Omega \). This measure depends only on u and not on the choice of an approximating sequence \(\{u_j\}\).
It is worth mentioning that compared to the real case (1.2), our quaternionic Monge–Ampère operator need not to be symmetrized for off-diagonal entries and the Monge–Ampère measure does not have an extra term \((Tu)^2\) as in (1.1).
As in [20, 27, 29, 30], motivated by the embedding of quaternionic algebra \({\mathbb {H}}\) into \({\mathbb {C}}^{2\times 2}:\)
we consider complex left invariant vector fields
where \(X_a\)’s are the standard horizontal left invariant vector fields (2.2) on \({\mathscr {H}} \). Let \(\wedge ^{p}{\mathbb {C}}^{2n}\) be the complex exterior algebra generated by \({\mathbb {C}}^{2n}\), \( p=0,\ldots ,2n\). Denote by \(\{\omega ^0,\omega ^1,\ldots \), \(\omega ^{2n-1}\}\) the standard basis of \({\mathbb {C}}^{2n}\). For a domain \(\Omega \) in \({\mathscr {H}} \), we define differential operators \(d_0,d_1:C^1(\Omega ,\wedge ^{p}{\mathbb {C}}^{2n})\rightarrow C (\Omega ,\wedge ^{p+1}{\mathbb {C}}^{2n})\) by
for \(F=\sum _{I}f_{I}\omega ^I\in C^1(\Omega ,\wedge ^{p}{\mathbb {C}}^{2n})\), where \(\omega ^I:=\omega ^{i_1}\wedge \cdots \wedge \omega ^{i_{p}}\) for the multi-index \(I=(i_1,\ldots ,i_{p})\). We call a form F closed if \(d_0F=d_1F=0. \)
In contrast to the bad behaviour (1.3) of \( \partial _b{\overline{\partial }}_b\), we have the following nice identities for \(d_0\) and \(d_1\):
which is the main reason that we could have a good theory for the quaternionic Monge–Ampère operator on the Heisenberg group.
Proposition 1.1
-
1.
\(d_0^2=d_1^2=0\).
-
2.
The identity (1.9) holds.
-
3.
For \(F\in C^1(\Omega ,\wedge ^{p}{\mathbb {C}}^{2n})\), \(G\in C^1(\Omega ,\wedge ^{q}{\mathbb {C}}^{2n})\), we have
$$\begin{aligned} d_\alpha (F\wedge G)=d_\alpha F\wedge G+(-1)^{p}F\wedge d_\alpha G,\qquad \alpha =0,1. \end{aligned}$$
We introduce a second-order differential operator \(\triangle :C^2(\Omega ,\wedge ^{p}{\mathbb {C}}^{2n})\rightarrow C (\Omega ,\wedge ^{p+2}{\mathbb {C}}^{2n})\) by
which behaves nicely as \( \partial {\overline{\partial }}\) as in the following proposition.
Proposition 1.2
For \(u_1,\ldots , u_n\in C^2\),
The quaternionic Monge–Ampère operator can be expressed as the exterior product of \(\triangle u\).
Theorem 1.2
For a real \(C^2\) function u on \({\mathscr {H}} \), we have
where
Theorem 1.3
(The minimum principle) Let \(\Omega \) be a bounded domain with smooth boundary in \({\mathscr {H}} \), and let u and v be continuous plurisubharmonic functions on \(\Omega \). Assume that \( (\triangle u)^n\le (\triangle v)^n. \) Then
An immediate corollary of this theorem is that the uniqueness of continuous solution to the Dirichlet problem for the quaternionic Monge–Ampère equation.
Originally, we define differential operators \(d_0\) and \(d_1\) and the quaternionic Monge–Ampère operator on the right quaternionic Heisenberg group. Later we find that these definitions also work on the Heisenberg group, on which the theory is simplified because its center is only 1-dimensional while the right quaternionic Heisenberg group has a 3-dimensional center. See [20, 21] for the tangential k-Cauchy–Fueter complexes on the Heisenberg group and the right quaternionic Heisenberg group.
This paper is arranged as follows. In Sect. 2, we give preliminaries on the Heisenberg group, the group structure of right quaternionic Heisenberg line \({\mathscr {H}}_q\), the SubLaplacian on \({\mathscr {H}}_q\) and its fundamental solution. After recall fundamental results on subharmonic functions on a Carnot group, we give basic properties of plurisubharmonic functions on the Heisenberg group. In Sect. 3, we discuss operators \(d_0\), \(d_1\) and nice behavior of brackets \( [Z_{AA '},Z_{BB '}] \), by which we can prove Proposition 1.1. Then we show that the horizontal quaternionic Hessian \((\overline{Q_l} {Q_m} u +8 \delta _{lm}{\mathbf {i}}\partial _t u )\) for a real \(C^2 \) function u is hyperhermitian, and prove the expression of the quaternionic Monge–Ampère operator in Theorem 1.2 by using linear algebra we developed before in [33]. In Sect. 4, we recall definitions of real forms and positive forms, and show that \(\triangle u\) for a \(C^2 \) plurisubharmonic function u is a closed strongly positive 2-form. Then we introduce notions of a closed positive current and the “integral” of a positive 2n-form current, and show that for any plurisubharmonic function u, \(\triangle u\) is a closed positive 2-current. In Sect. 5, we give proofs of Chern-Levine-Nirenberg estimate, the existence of the quaternionic Monge–Ampère measure for a continuous plurisubharmonic function and the minimum principle.
2 Plurisubharmonic functions over the Heisenberg group
2.1 The Heisenberg group
We have the following conformal transformations on \({\mathscr {H}} \): (1) dilations: \(\delta _r: (x,{t})\longrightarrow (r x,r^{2}{t}),\) \( r>0; \) (2) left translations: \( \tau _{(y,s)}:(x,{t})\longrightarrow (y,s)\cdot (x,{t}); \) (3) rotations: \({R_U}:(x,{t})\longrightarrow ( Ux,t),\) for \( U\in \mathrm{U}(n),\) where \( \mathrm{U}(n) \) is the unitary group; (4) the inversion: \( R:(x,{t})\longrightarrow \left( \frac{x}{|x|^{2}+{\mathbf {i}}t }, \frac{ {t}}{|x|^{4}+|{t}|^{2}}\right) . \) Define vector fields:
on the Heisenberg group \({\mathscr {H}} \), where \(e_a=(\ldots ,0,1,0,\ldots )\in {\mathbb {R}}^{4n}\) with only the \(a\hbox {th}\) entry nonvanishing, \(a=1,2,\ldots 4n\). It follows from the multiplication law (1.4) that
\(l= 1,\cdots , 2 n \), whose brackets are
\(X_a \) is left invariant in the sense that for any \((y,s)\in {\mathscr {H}}\),
by definition (2.1), which means for fixed \((y,s)\in {\mathscr {H}}\),
where the pull back function \((\tau _{(y,s)}^{*}f)(x,t) :=f((y,s)(x,t)) \). On the left hand side above, \(X_a\) is the differential operator in (2.2) with coefficients at point (x, t), while on the right hand side, \(X_a\) is the differential operator with coefficients at point (y, s)(x, t),
2.2 Right quaternionic Heisenberg lines
For quaternionic numbers \( q,p\in {\mathbb {H}} \), write
Let \( {\widehat{p}}\) be the column vector in \({\mathbb {R}}^4\) represented by p, i.e. \( {\widehat{p}}:= \left( y_1,y_{2},y_3,y_{4 }\right) ^t,\) and let \(q^{{\mathbb {R}}}\) be the \(4\times 4\) matrix representing the transformation of left multiplying by q, i.e.
It is direct to check (cf. [31]) that
and
The multiplication law (1.4) of the Heisenberg group can be written as
with
The multiplication of the subgroup \( {{\mathscr {H}}}_{q }\) is given by
where
for \(\lambda =\lambda _1+{\mathbf {i}} \lambda _2+{\mathbf {j}}\lambda _3+{\mathbf {k}}\lambda _4,\lambda '=\lambda _1'+{\mathbf {i}} \lambda _2'+{\mathbf {j}}\lambda _3'+{\mathbf {k}}\lambda _4'\in {\mathbb {H}}\). \(B^q\) is a \(4\times 4\) skew symmetric matrix. So if we consider the group \(\widetilde{{\mathscr {H}}}_q\) as the vector space \({\mathbb {R}}^5\) with the multiplication given by
we have the isomorphism of groups
\(\widetilde{{\mathscr {H}}}_q\) is different from the 5-dimensional Heisenberg group in general. Note that the subgroup \( {{\mathscr {H}}}_{q }\) of \( {\mathscr {H}} \) is the same if q is replaced by \(q q_0\) for \(0\ne q_0\in {\mathbb {H}}\),
Write \(\mathbf{i }_1:=1\), \(\mathbf{i }_2:=\mathbf{i }\), \(\mathbf{i }_3:=\mathbf{j }\) and \(\mathbf{i }_4:=\mathbf{k }\). Consider left invariant vector fields on \(\widetilde{{\mathscr {H}}_q}\): \( \widetilde{X_j}u( \lambda , t):=\left. \frac{d u}{d\varsigma }( ( \lambda , t)(\varsigma {\mathbf {i}}_j,0))\right| _{\varsigma =0} \) for \(( \lambda , t)\in \widetilde{{\mathscr {H}}_q} \). Since
we get
Define the SubLaplacian on the right quaternionic Heisenberg line \(\widetilde{{\mathscr {H}}_{ q}}\) as
Note that for \( q\in {\mathbb {H}} \),
by using (2.8) and \({\mathbf {i}}^{{\mathbb {R}}}=-J\) by (2.7). Then
by (2.8) again. If write \(q_l = x_{4l+1}+ {\mathbf {i}}x_{4l+2}+ {\mathbf {j}}x_{4l+3}+ {\mathbf {k}}x_{4l+4 }\), we have \( \Lambda _q^2:= S_1^2+S_2^2+S_3^2, \) where \( S_1:= \sum _{l=0}^{n-1} (x_{4l+1}^2+x_{4l+2}^2-x_{4l+3}^2-x_{4l+4}^2) ,\) \(S_2:= 2\sum _{l=0}^{n-1} (-x_{4l+1} x_{4l+4} + x_{4l+2} x_{4l+3}) ,\) \( S_3: = 2 \sum _{l=0}^{n-1} (x_{4l+1} x_{4l+3} +x_{4l+2} x_{4l+4}), \) The degenerate locus \({\mathfrak {D}}:=\{q\in {\mathbb {H}}^n;\Lambda _q=0\}\) is the intersection of three quadratic hypersurfaces in \({\mathbb {R}}^{4n}\) given by \( S_1= S_2= S_3=0\). Thus \({{\mathscr {H}}}_{q }\) is abelian if and only if \(q\in {\mathfrak {D}}\).
Proposition 2.1
For \( q\in {\mathbb {H}}^n {\setminus } {\mathfrak {D}}\), the fundamental solution of \(\widetilde{\triangle _q}\) on \(\widetilde{{\mathscr {H}}}_q\) is
where
Proof
Note that for \(\varepsilon >0\), we have
It follows from the expression (2.15) of \(\widetilde{ X_j}\) that
and
by skew symmetry of \(B^q\) and using (2.16). On the other hand, we have
by \(\sum _{ j, k =1}^4 B^q_{k j}\lambda _k\lambda _{j }=0\). Substituting (2.19)–(2.20) into (2.18) to get
Then \(\int \varphi {\widetilde{\triangle }}_b (\frac{-1}{\rho _q+\varepsilon } )\rightarrow C_q^{-1}\varphi (0,0) \) for \(\varphi \in C_0^\infty (\widetilde{{\mathscr {H}}}_q) \) by recaling and letting \(\varepsilon \rightarrow 0+\). We get the result.\(\square \)
2.3 Subharmonic functions on Carnot groups
A Carnot group \( {\mathbb {G}}\) of step \(r \ge 1\) is a simply connected nilpotent Lie group whose Lie algebra \({\mathfrak {g}}\) is stratified, i.e. \({\mathfrak {g}}={\mathfrak {g}}_1\oplus \cdots \oplus {\mathfrak {g}}_r\) and \([{\mathfrak {g}}_1, {\mathfrak {g}}_j]={\mathfrak {g}}_{j+1}\). Let \(Y_1, \cdots ,Y_p\) are smooth left invariant vector fields on a Carnot group \( {\mathbb {G}}\) and homogeneous of degree one with respect to the dilation group of \( {\mathbb {G}}\), such that \(\{Y_1, \cdots ,Y_p\}\) is a basis of \({\mathfrak {g}}_1\). There exists a homogeneous norm \(\Vert \cdot \Vert \) on a Carnot group \( {\mathbb {G}} \) [9] such that
for some \(Q>0\) is a fundamental solution for the SubLaplacian \(\triangle _{{\mathbb {G}}}\) given by \( \triangle _{{\mathbb {G}}}=\sum _{j=1}^p Y_j^2, \) (the fundamental solution used in [9] is different from the usual one (2.21) by a minus sign). \( \triangle _{{\mathbb {G}}} \) is not elliptic except for \({\mathbb {G}}\) abelian. But it is hypoelliptic since vector fields \(\{Y_1, \cdots ,Y_p\}\) satisfy Hörmander’s hypoellipticity condition.
We denote by \(D(\xi , r)\) the ball of center \(\xi \) and radius r, i.e.
Recall the representation formulae [9] for any smooth function u on \( {\mathbb {G}}\):
for every \(\xi \in \Omega \) and \(r > 0 \) such that \(D(\xi , r)\subset \Omega \), where
and
for some positive constants \(m_Q,n_Q\), and
Here \(\nabla _{{\mathbb {G}}}\) the vector valued differential operator \((Y_1, \ldots ,Y_p)\) and \(\nabla \) is the usual gradient, \(d(\xi )=\Vert \xi \Vert \), dV is the volume element and dS is the surface measure on \(\partial D(\xi , r)\). Integrals \( {{M}}_r^{{\mathbb {G}}}(u)\) and \(\mathscr {{M}}_r^{{\mathbb {G}}}(u)\) are related by the coarea formula.
A function u on a domain \(\Omega \subset \triangle _{{\mathbb {G}}}\) is called harmonic if \(\triangle _{{\mathbb {G}}}u=0\) in the sense of distributions. Then a harmonic function u in an open set \(\Omega \) satisfies the mean-value formula
by (2.23). For an open set \( \Omega \subset {\mathbb {G}}\), we say that an upper semicontinuous function function \(u : \Omega \rightarrow [-\infty ,\infty )\) is \(\triangle _{{\mathbb {G}}}\) -subharmonic if for every \(\xi \in \Omega \) there exists \(r_\xi > 0\) such that
Proposition 2.2
(The maximum principle for the SubLaplacian [9]) If \(\Omega \subseteq {\mathbb {G}}\) is a bounded open set, for every \(u \in C^2(\Omega )\) satisfying \(\Delta _{{\mathbb {G}}} u \ge 0\) in \(\Omega \) and \(lim sup_{\xi \rightarrow \eta } u(\xi ) \le 0\) for any \(\eta \in \partial \Omega \), we have \(u \le 0\) in \(\Omega \).
Theorem 2.1
(Theorem 4.3 in [9]) Let \(\Omega \) be an open set in \({\mathbb {G}}\) and \(u : \Omega \rightarrow [-\infty ,+\infty )\) be an upper semicontinuous function. Then, the following statements are equivalent:
-
(i)
u is subharmonic;
-
(ii)
\(u \in L^1_{loc}(\Omega )\), \(u(\xi ) = lim_{r\rightarrow 0+} M_r^{{\mathbb {G}}}(u)(\xi )\) for every \(\xi \in \Omega \) and \(\triangle _{{\mathbb {G}}} u \ge 0 \) in \(\Omega \) in the sense of distributions.
When \({\mathbb {G}}\) is the Heisenberg group \({\mathscr {H}} \) in (1.4), the SubLaplacian is
where \(X_a\)’s are given by (2.2). It is not elliptic, but is subelliptic. It is known that the fundamental solution of \( {\triangle }_b\) is \(-C_Q\Vert \cdot \Vert ^{-Q+2}\) for some constant \(C_Q>0\) as in Proposition 2.1, with the norm given by
The invariant Haar measure on \({\mathscr {H}} \) is the usual Lebesgue measure dxdt on \({\mathbb {R}}^{4n+3}\).
in the mean-value formula, where \(Q:=4n+2\), the homogeneous dimension of the \((4n+1)\)-dimensional Heisenberg group \({\mathscr {H}} \).
When \({\mathbb {G}}\) is the group \(\widetilde{{\mathscr {H}}}_q\) in (2.13), the SubLaplacian is \( \widetilde{{\triangle }}_b\). Because of the fundamental solution of \( \widetilde{{\triangle }}_b\) given in Proposition 2.1, its norm is given by
The invariant Haar measure on \(\widetilde{{\mathscr {H}}}_q\) is the usual Lebesgue measure \(d\lambda dt \) on \({\mathbb {R}}^{5}\). Its homogeneous dimension is 6, and the mean-value formulae becomes
where \(D_q(0,r)\) is the ball of radius r and centered at the origin in \(\widetilde{{\mathscr {H}}}_q\) in terms of the norm \(\Vert \cdot \Vert _q\), \(m_q\) is the constant in the representation formula (2.24) for the group \(\widetilde{{\mathscr {H}}}_q\), and
by (2.20), which is homogeneous of degree 0.
2.4 Plurisubharmonic functions on the Heisenberg group
Although \({\mathscr {H}}_{\eta ,q}\) is not a subgroup, by the embedding
we say that u is subharmonic function on \({\mathscr {H}}_{\eta ,q}\) if \(\iota _{\eta ,q }^*u\) is \(\widetilde{\triangle _q}\)-subharmonic on \(\widetilde{{\mathscr {H}}_{ q}}\). Thus, a \([-\infty ,\infty )\)-valued upper semicontinuous function u on a domain \(\Omega \subset {\mathscr {H}} \) is plurisubharmonic if u is \(L^1_{\mathrm{loc}}(\Omega )\) and \(\iota _{\eta ,q }^*u\) is \(\widetilde{\triangle _q}\)-subharmonic on \(\iota _{\eta ,q }^*\Omega \cap \widetilde{{\mathscr {H}}_{ q}}\) for any \( q\in {\mathbb {H}}^n {\setminus } {\mathfrak {D}} \) and \(\eta \in {\mathscr {H}}^n\). Denote by \(PSH(\Omega )\) the class of all plurisubharmonic functions on \(\Omega \).
Recall that the convolution of two functions u and v over \({\mathscr {H}}\) is defined as
Then
for any left invariant vector field Y by (2.5), and
by taking transformation \( (y,s)^{-1}(x,t)\rightarrow (y ,s ) \) for fixed (x, t), whose Jacobian can be easily checked to be identity. By the non-commutativity
in general, we have \(u*v\ne v *u\), and
Consider the standard regularization given by the convolution \(\chi _\varepsilon * u\) with
where \(0\le \chi \in C_0^\infty (D(0,1))\), \(\int _{{\mathscr {H}} }\chi (\xi ) dV(\xi )=1\). Then \(\chi _\varepsilon * u\) subharmonic if u is (cf. Proposition 2.3 (6)), but we do not know whether \(\chi _\varepsilon * u\) is decreasing as \(\varepsilon \) decreasing to 0, which we could not prove as in the Euclidean case, because of the non-commutativity.
Remark 2.1
-
(1)
It is a consequence of Theorem 2.1 that a function is in \(L^1_{\mathrm{loc}}(\Omega )\) if it is \(\triangle _b\)-subharmonic on \( \Omega \subset {\mathscr {H}} \). But since \( {{\mathscr {H}}_{ q}}\) is different in general for different \( q\in {\mathbb {H}}^n{\setminus } {\mathfrak {D}} \), we do not know wether a \(PSH(\Omega )\) function is \(\triangle _b\)-subharmonic on \(\Omega \). So we require it as a condition in the definition.
-
(2)
In the characterization of subharmonicity in Theorem 2.1 there is an additional condition \(u(\xi ) = lim_{r\rightarrow 0+} M_r^{{\mathbb {G}}}(u)(\xi )\). We know that \( M_r^{{\mathbb {G}}}(u)(\xi )\) is increasing in r if \(\triangle _{{\mathbb {G}}} u \ge 0 \).
The following basic properties of PSH functions also hold on the Heisenberg group.
Proposition 2.3
Assume that \(\Omega \) is a bounded domain in \({\mathscr {H}}\). Then we have that
-
(1)
If \( u, v \in PSH(\Omega )\), then \(au + bv \in PSH(\Omega )\), for positive constants a, b;
-
(2)
If \( u, v \in PSH(\Omega )\), then \(\max \{ u ,v\} \in PSH(\Omega )\);
-
(3)
If \(\{u_\alpha \}\) is a family of locally uniformly bounded functions in \(PSH(\Omega )\), then the upper semicontinuous regularization \((\sup _\alpha u_\alpha )^*\) is a PSH function;
-
(4)
If \(\{u_n\}\) is a sequence of functions in \(PSH(\Omega )\) such that \(u_n \) is decreasing to \(u\in L^1_{\mathrm{loc}}(\Omega ) \), then \( u \in PSH(\Omega )\);
-
(5)
If \( u \in PSH(\Omega )\) and \(\gamma : {\mathbb {R}} \rightarrow {\mathbb {R}}\) is convex and nondecreasing, then \(\gamma \circ u \in PSH(\Omega )\);
-
(6)
If \( u \in PSH(\Omega )\), then the regularization \( \chi _\varepsilon * u(\xi ) \) is also PSH on \(\Omega '\subset {\mathscr {H}} \), where \(\Omega '\) is subdomain such that \(\Omega 'D(0,\varepsilon )\subset \Omega \). Moreover, if u is also continuous, then \(\chi _\varepsilon * u\) converges to u uniformly on any compact subset.
-
(7)
If \(\omega \Subset \Omega \), \( u \in PSH(\Omega )\), \( v \in PSH(\omega )\), and \(\limsup _{\xi \rightarrow \eta } v(\xi ) \le u(\eta )\) for all \(\eta \in \partial \omega \), then the function defined by
$$\begin{aligned} \phi =\left\{ \begin{array}{l} u, \qquad \qquad \qquad {\mathrm{on}}\quad \Omega {\setminus }\omega ,\\ \max \{u, v\}, \quad {\mathrm{on}}\quad \omega , \end{array} \right. \end{aligned}$$is PSH on \(\Omega \).
Proof
(1)–(3) follows from definition trivially.
(4) It holds since for any fixed \( q\in {\mathbb {H}}^n{\setminus } {\mathfrak {D}} \), \(\eta \in \Omega \) and small \(r>0\),
by the monotone convergence theorem.
(5) It holds since
by Jensen’s inequality for nondecreasing convex function \(\gamma \), since \(\frac{m_q}{r^6} K_{ q} (\lambda ,t) \) is nonnegative and its integral over \( D_q(0,r)\) is 1. The latter fact follows from the mean value formula for the harmonic function \(\equiv 1\).
(6) For fixed \( q\in {\mathbb {H}}^n{\setminus } {\mathfrak {D}} \) and \(\eta \in \Omega \), \(\chi _\varepsilon * u\) is PSH since it is smooth and
by Fubini’s theorem and subharmonicity of u on the open subset \(\Omega \cap {\mathscr {H}}_{(y,s)^{-1}\eta ,q}\). The uniform convergence is trivial.
(7) \(\phi \) is obviously in \(L^1_{\mathrm{loc}}(\Omega )\), and is PSH on \(\mathring{\omega }\) by (2). For \(\eta \in \partial \omega \),
for small \( r>0\). \(\square \)
Remark 2.2
Our notion of plurisubharmonic functions is different from that introduced by Harvey and Lawson [16] for calibrated geometries, i.e. an upper semicontinuous function u satisfies \(\triangle u\ge 0\) on each calibrated submanifold in \({\mathbb {R}}^N\), where \(\triangle \) is the Laplacian associated to the induced Riemannian metric on the calibrated submanifold. In our definition we require \(\triangle _q u\ge 0\) for SubLaplacian \(\triangle _q \), which is subelliptic, on each 5-dimensional real hyperplane \( {{\mathscr {H}}_{\eta , q}}\) for \( q\in {\mathbb {H}}^n{\setminus } {\mathfrak {D}} \).
3 Differential operators \(d_0\), \(d_1\), \(\triangle \) and the quaternionic Monge–Ampère operator on the Heisenberg group
3.1 Differential operators \(d_0\) and \(d_1\)
Denote \( {\overline{W}}_j:=X_{2j-1}+{\mathbf {i}}X_{2j}\), \( {W}_j:=X_{2j-1}-{\mathbf {i}}X_{2j}, \) \( j=1,\ldots 2n.\) Then
and all other brackets vanish by (2.2). Let \(\{ \ldots , \theta ^{{\overline{j}}}, \theta ^{ {j}},\ldots ,\theta \}\) be the basis dual to \(\{\ldots ,{\overline{W}}_j, {W}_j\), \(\ldots , \partial _t\}\). The tangential Cauchy-Riemann operator is defined as \({\overline{\partial }}_bu=\sum _{j=1}^{2n} {\overline{W}}_ju\,\theta ^{{\overline{j}}}\) for a function u and
where \(\theta ^{ {J}} =\theta ^{ {j_1}}\wedge \cdots \wedge \theta ^{ {j_l}} \), \(\theta ^{ {\overline{K}}} =\theta ^{ {{\overline{k}}_1}}\wedge \cdots \wedge \theta ^{ {{\overline{k}}_m}} \) for multi-indices \(J =(j_1,\ldots ,j_l)\), \({\overline{K}} =({\overline{k}}_1,\ldots ,{\overline{k}}_m)\). Similarly, \( {\partial }_bu=\sum _{j=1}^{2n} {W}_ju\,\theta ^{ {j}}\) for a function u and is extended to forms as in (3.1). Then
Thus \(\partial _b{\overline{\partial }}_b\ne -{{\overline{\partial }}}_b{\partial }_b\).
By the definition of the operator \(\triangle \) in (1.10), we have
for \(F= \sum _{ I} f_{I}~ \omega ^I\). Now for a function \(u\in C^2\) we define
\(2\triangle _{AB}\) is the determinant of \((2\times 2)\)-submatrix of Ath and Bth rows in (1.7). Note that \(Z_{B0' }Z_{A1' }u\) in the above definition could not be replaced by \(Z_{A1' }Z_{B0' }u\) in general because of noncommutativity. Then we can write
When \(u_1=\ldots =u_n=u\), \(\triangle u_1\wedge \cdots \wedge \triangle u_n\) coincides with \((\triangle u)^n:=\wedge ^n\triangle u\).
The following nice behavior of brackets plays a key role in the proof of properties of \(d_0, d_1\).
Proposition 3.1
-
(1)
For fixed \(A'=0'\) or \(1'\), we have \( [Z_{AA '},Z_{BA '}]=0\) for any \(A,B=0,\ldots 2n-1\), i.e. each column \(\{Z_{0A '},\ldots , Z_{(2n-1)A '}\}\) in (1.7) spans an abelian subalgebra.
-
(2)
If \(|A-B|\ne 0,n\), we have
$$\begin{aligned}{}[Z_{A0 '},Z_{B1 '}]=0, \end{aligned}$$(3.5)and
$$\begin{aligned}{}[ Z_{l 0'}, Z_{ (n+l ) 1' } ] = [ Z_{(n+l )0'}, Z_{ l 1' } ] = - 8 {\mathbf {i}} \partial _{t }, \end{aligned}$$(3.6)for \(l=0,,\ldots n-1\), and
$$\begin{aligned} \begin{aligned}&2\triangle _{l(n+l)}=X_{4l+1}^2+ X_{4l+2}^2+X_{4l+3}^2+ X_{4l+4}^2. \end{aligned} \end{aligned}$$(3.7)
Proof
Noting that by (1.7), \(Z_{AA'}\) and \(Z_{BB'}\) for \(|A-B|\ne 0\) or n are linear combinations of \(X_{2l+j}\)’s, \(j=1,2\), with different l, and so their bracket vanishes by (2.3). Thus (1) and (3.5) hold. (3.6) follows from brackets in (2.3) and the expression of \(Z_{AA'}\)’s in (1.7). (3.7) holds by
and using (2.3). \(\square \)
Proof of Proposition 1.1
-
(1)
For any \(F=\sum _If_I\omega ^I\), note that we have \(Z_{A0'} Z_{B0'}f_I=Z_{B0'}Z_{A0'}f_I \) by Proposition 3.1 (1). So we have
$$\begin{aligned} d_0^2F=\sum _{ I}\sum _{A,B=0}^{2n-1}Z_{A0'}Z_{B0'}f_I~\omega ^A\wedge \omega ^B\wedge \omega ^I=0, \end{aligned}$$by \(\omega ^A\wedge \omega ^B =-\omega ^B\wedge \omega ^A.\) It is similar for \(d_1^2 =0\).
-
(2)
For any \(F=\sum _If_I\omega ^I\), we have
$$\begin{aligned} \begin{aligned} d_0d_1F&=\sum _{ I}\sum _{A , B }Z_{A0' }Z_{B1' }f_{I} \omega ^A\wedge \omega ^B\wedge \omega ^I=\quad \sum _I\sum _{|A-B|\ne 0, n} Z_{A0' }Z_{B1' }f_{I} \omega ^A\wedge \omega ^B\wedge \omega ^I\\ {}&\quad +\sum _{ I}\sum _{l=0}^{n-1}\left( Z_{ l 0' }Z_{(n+l )1'}-Z_{(n+l )0'} Z_{l1' }\right) f_{I} \omega ^{ l}\wedge \omega ^{ n+l }\wedge \omega ^I \\&=-\sum _I\sum _{|A-B|\ne 0,n} Z_{B1' }Z_{A0' }f_{I}\omega ^B\wedge \omega ^A\wedge \omega ^I\\ {}&\quad -\sum _{ I}\sum _{l=0}^{n-1}(Z_{ l 1' }Z_{(n+l )0'}-Z_{(n+l )1'} Z_{l0' }) f_{I} \omega ^{ l}\wedge \omega ^{ n+l }\wedge \omega ^I \\ {}&=-\sum _{A,B,I}Z_{A1'}Z_{B 0' }f_{I} \omega ^A\wedge \omega ^B\wedge \omega ^I =-d_1d_0F, \end{aligned} \end{aligned}$$by using commutators (3.5)–(3.6) in Proposition 3.1 in the third identity.
-
(3)
Write \(G=\sum _Jg_J\omega ^J\). We have
$$\begin{aligned} \begin{aligned}d_\alpha (F\wedge G)&=\sum _{A,I,J}[Z_{A\alpha '}(f_I)g_J+f_IZ_{A\alpha '}(g_J)]~\omega ^A\wedge \omega ^I\wedge \omega ^J\\&= \sum _{A,I }Z_{A\alpha '}(f_I)~\omega ^A\wedge \omega ^I\wedge \sum _{ J} g_J\omega ^J+(-1)^{p}\sum _{A,I }f_I\omega ^I\wedge \sum _{ J}Z_{A\alpha ' }(g_J)\omega ^A\wedge \omega ^J\\&=d_\alpha F\wedge G+(-1)^{p}F\wedge d_\alpha G. \end{aligned} \end{aligned}$$by \(\omega ^A\wedge \omega ^I =(-1)^{p}\omega ^I\wedge \omega ^A \). \(\square \)
Corollary 3.1
For \(u_1,\ldots , u_n\in C^3\), \(\triangle u_1\wedge \cdots \wedge \triangle u_k\) is closed, \(k=1,\ldots ,n,\).
Proof
By Proposition 1.1 (3), we have
for \(\alpha =0,1\). Note that \(d_0\triangle =d_0^2d_1=0\) and \( d_1\triangle =-d_1^2d_0=0 \) by using Proposition 1.1 (1)–(2). It follows that \(d_\alpha (\triangle u_1\wedge \cdots \wedge \triangle u_k)=0\). \(\square \)
Proof of Proposition 1.2
It follows from Corollary 3.1 that
By Proposition 1.1 (3),
So we have
\(\square \)
3.2 The quaternionic Monge–Ampère operator on the Heisenberg groups
A quaternionic \((n\times n)\)-matrix \(({\mathcal {M}}_{jk})\) is called hyperhermitian if \({{\mathcal {M}}}_{jk}=\overline{{\mathcal {M}}_{kj}}\).
Proposition 3.2
(Claim 1.1.4, 1.1.7 in [1]) For a hyperhermitian \((n\times n)\)-matrix \({\mathcal {M}}\), there exists a unitary matrix \({\mathcal {U}}\) such that \({\mathcal {U}}^*{\mathcal {M}}{\mathcal {U}}\) is diagonal and real.
Proposition 3.3
(Theorem 1.1.9 in [1])
-
(1)
The Moore determinant of any complex hermitian matrix considered as a quaternionic hyperhermitian matrix is equal to its usual determinant.
-
(2)
For any quaternionic hyperhermitian \((n\times n)\)-matrix \({\mathcal {M}}\) and any quaternionic \((n\times n)\)-matrix \({\mathcal {C}}\)
$$\begin{aligned} \det ({\mathcal {C}}^*{\mathcal {M}}{\mathcal {C}})=\det ({\mathcal {A}} )\det ({\mathcal {C}}^* {\mathcal {C}}). \end{aligned}$$
Proposition 3.4
For a real \(C^2 \) function u, the horizontal quaternionic Hessian \((\overline{Q_l} {Q_m} u +8 \delta _{lm}{\mathbf {i}}\partial _t u )\) is hyperhermitian.
Proof
It follows from definition (1.7) of \(Z_{AA'}\)’s that
and so
When \(l=m\), it follows that
by using (3.5)–(3.6). Thus \(\overline{Q_l} {Q_l} u + 8 {\mathbf {i}} \partial _{t }u\) is real by (3.7). If \(l\ne m\), we have
by using commutators
by Proposition 3.1.
To see the horizontal quaternionic Hessian to be hyperhermitian, note that for \(l\ne m\)
and
by the conjugate of \(Z_{AA'}\)’s in (1.7) and (3.12). Similarly, for any l, m, we have
and so
by using (3.8) and (3.12). Now substitute (3.14) and (3.16) into (3.13) to get
for \(l\ne m\). This together with the reality of \(\overline{Q_l} {Q_l} u + 8 {\mathbf {i}} \partial _{ t }u\) implies that the quaternionic Hessian \((\overline{Q_l} {Q_m} u + 8\delta _{lm} {\mathbf {i}} \partial _{ t }u )\) is hyperhermitian. \(\square \)
As in [33], denote by \(M_{{\mathbb {F}}}(p,m)\) the space of \({\mathbb {F}}\)-valued \((p\times m)\)-matrices, where \({\mathbb {F}}={\mathbb {R}},{\mathbb {C}},{\mathbb {H}}\). For a quaternionic \(p\times m\)-matrix \({\mathcal {M}}\), write \({\mathcal {M}}=a+b{\mathbf {j}}\) for some complex matrices \(a, b\in M_{{\mathbb {C}}}( p, m)\). Then we define the \( {\tau }({\mathcal {M}})\) as the complex \((2p\times 2m)\)-matrix
Recall that for skew symmetric matrices \(M_\alpha =(M_{\alpha ;AB})\in M_{{\mathbb {C}}}(2n, 2n )\), \(\alpha =1,\ldots , n\), such that 2-forms \(\omega _\alpha =\sum _{i,j} M_{\alpha ;AB}\omega ^A\wedge \omega ^B \) are real, define
Consider the homogeneous polynomial \(\text {det}(\lambda _1{\mathcal {M}}_1+\ldots +\lambda _n{\mathcal {M}}_n)\) in real variables \(\lambda _1,\ldots ,\lambda _n\) of degree n. The coefficient of the monomial \(\lambda _1\ldots \lambda _n\) divided by n! is called the mixed discriminant of the matrices \({\mathcal {M}}_1,\ldots ,{\mathcal {M}}_n\), and it is denoted by \(\text {det}({\mathcal {M}}_1,\ldots ,{\mathcal {M}}_n)\). In particular, when \({\mathcal {M}}_1=\ldots ={\mathcal {M}}_n={\mathcal {M}}\), \(\text {det}({\mathcal {M}}_1,\ldots ,{\mathcal {M}}_n)=\text {det}({\mathcal {M}})\).
Theorem 3.1
(Theorem 1.2 in [33]) For hyperhermitian matrices \({\mathcal {M}}_1,\ldots , {\mathcal {M}}_n\in M_{{\mathbb {H}}} (n,n)\), we have
where
Proof of Theorem 1.2
The proof is similar to that of Theorem 1.3 in [33] except that \(Z_{AA'}\)’s are noncommutative. (3.10)–(3.11) implies that the quaternionic Hessian can be written as
with \(n\times n\) complex matrices
Thus
Note that \(\triangle _{ l l }u= \triangle _{ (n+l ) (n+l) }{u}=0\) by definition. For \(l\ne m\),
by (3.14), while for \(l= m\) we also have
by using Proposition 3.1 (2). Moreover,
which follows from (3.15). Therefore we have
Then the result follows from applying Theorem 3.1 to matrices \({\mathcal {M}}_j=2\left( \overline{Q_l} {Q_m} u_j + 8\delta _{lm}{\mathbf {i}}\partial _tu_j\right) \). \(\square \)
4 Closed positive currents on the quaternionic Heisenberg group
4.1 Positive 2k-forms
Now let us recall definitions of real forms and positive 2k-forms (cf. [4, 27, 33] and references therein). Let \(\{\omega ^0,\omega ^1,\ldots ,\omega ^{2n-1}\}\) be the standard basis of \({\mathbb {C}}^{2n}\) and
Then \( \beta _n^n=\wedge ^n\beta _n=n!~\Omega _{2n},\) where \(\Omega _{2n}\) is given by (1.12). For \({\mathcal {A}}\in \text {GL}_{{\mathbb {H}}}(n)\), define the induced \({\mathbb {C}}\)-linear transformation of \({\mathcal {A}}\) on \({\mathbb {C}}^{2n}\) as \({\mathcal {A}}.\omega ^p=\tau ({\mathcal {A}}). \omega ^p\) with
for \(M\in M_{{\mathbb {C}}}(2n,2n)\), and define the induced \({\mathbb {C}}\)-linear transformation of \({\mathcal {A}}\) on \(\wedge ^{2k}{\mathbb {C}}^{2n}\) as
Therefore for \(A\in \text {U}_{{\mathbb {H}}}(n)\), \({\mathcal {A}}.\beta _n =\beta _n, \). Consequently \({\mathcal {A}}.(\wedge ^n \beta _n)=\wedge ^n \beta _n\), i.e., \({\mathcal {A}}.\Omega _{2n}=\Omega _{2n}\).
\(\mathbf{j }\) defines a real linear map
which is not \({\mathbb {C}}\)-linear, where \({\mathbb {J}}\) is given by (3.20). Also the right multiplying of \(\mathbf{i }\): \((q_1,\ldots ,q_n)\mapsto (q_1\mathbf{i },\ldots ,q_n\mathbf{i })\) induces
Thus \(\rho \) defines \(GL_{{\mathbb {H}}}(1)\)-action on \({\mathbb {C}}^{2n}\). The actions of \(GL_{{\mathbb {H}}}(1)\) and \(GL_{{\mathbb {H}}}(n)\) on \({\mathbb {C}}^{2n}\) are commutative, and equip \({\mathbb {C}}^{2n}\) a structure of \(GL_{{\mathbb {H}}}(n)GL_{{\mathbb {H}}}(1)\)-module. This is because \((MN).\omega ^p=M.(N.\omega ^p)\) by definition and \(\rho (\mathbf{j })\rho (\mathbf{i })=-\rho (\mathbf{i })\rho (\mathbf{j })\). This action extends to \(\wedge ^{2k}{\mathbb {C}}^{2n}\) naturally.
The real action (4.3) of \( \rho (\mathbf{j })\) on \({\mathbb {C}}^{2n}\) naturally induces an action on \(\wedge ^{2k}{\mathbb {C}}^{2n}\). An element \(\varphi \) of \(\wedge ^{2k}{\mathbb {C}}^{2n}\) is called real if \(\rho (\mathbf{j })\varphi =\varphi \). Denote by \(\wedge ^{2k}_{{\mathbb {R}}}{\mathbb {C}}^{2n}\) the subspace of all real elements in \(\wedge ^{2k}{\mathbb {C}}^{2n}\). These forms are counterparts of \((k,k)-\)forms in complex analysis.
A right \({\mathbb {H}}\)-linear map \(g:{\mathbb {H}}^{k}\rightarrow {\mathbb {H}}^{m}\) induces a \({\mathbb {C}}\)-linear map \(\tau (g):{\mathbb {C}}^{2k}\rightarrow {\mathbb {C}}^{2m}\). If we write \(g=(g_{jl})_{m\times k}\) with \(g_{jl}\in {\mathbb {H}}\), then \(\tau (g)\) is the complex \((2m\times 2k)\)-matrix given by (3.17). The induced \({\mathbb {C}}\)-linear pulling back transformation of \(g^*:{\mathbb {C}}^{2m}\rightarrow {\mathbb {C}}^{2k}\) is defined as:
where \(\{{\widetilde{\omega }}^0,\ldots ,{\widetilde{\omega }}^{2m-1}\}\) is the standard basis of \({\mathbb {C}}^{2m}\) and \(\{\omega ^0,\ldots ,\omega ^{2k-1}\}\) is the standard basis of \({\mathbb {C}}^{2k}\). It induces a \({\mathbb {C}}\)-linear pulling back transformation on \(\wedge ^{2k}{\mathbb {C}}^{2m}\) given by \( g^*(\alpha \wedge \beta )=g^*\alpha \wedge g^*\beta \) inductively.
An element \(\omega \in \wedge _{{\mathbb {R}}}^{2k}{\mathbb {C}}^{2n}\) is said to be elementary strongly positive if there exist linearly independent right \({\mathbb {H}}\)-linear mappings \(\eta _j:{\mathbb {H}}^n\rightarrow {\mathbb {H}}\) , \(j=1,\ldots ,k\), such that
where \(\{{\widetilde{\omega }}^0,{\widetilde{\omega }}^1\}\) is a basis of \({\mathbb {C}}^{2}\) and \(\eta _j^*:~{\mathbb {C}}^{2}\rightarrow {\mathbb {C}}^{2n}\) is the induced \({\mathbb {C}}\)-linear pulling back transformation of \(\eta _j\). The definition in the case \(k=0\) is obvious: \(\wedge ^{0}_{{\mathbb {R}}}{\mathbb {C}}^{2n}={\mathbb {R}}\) and the positive elements are the usual ones. For \(k=n\), dim\( _{{\mathbb {C}}}\wedge ^{2n}{\mathbb {C}}^{2n}=1\), \(\Omega _{2n}\) defined by (1.12) is an element of \(\wedge _{{\mathbb {R}}}^{2n}{\mathbb {C}}^{2n}\) (\(\rho (\mathbf{j })\beta _n=\beta _n\)) and spans it. An element \(\eta \in \wedge _{{\mathbb {R}}}^{2n}{\mathbb {C}}^{2n}\) is called positive if \(\eta =\kappa ~\Omega _{2n}\) for some non-negative number \(\kappa \). By definition, \(\omega \in \wedge _{{\mathbb {R}}}^{2k}{\mathbb {C}}^{2n}\) is elementary strongly positive if and only if
for some quaternionic matrix \({\mathcal {M}}\in M_{{\mathbb {H}}}(n ,k )\) of rank k.
An element \(\omega \in \wedge _{{\mathbb {R}}}^{2k}{\mathbb {C}}^{2n}\) is called strongly positive if it belongs to the convex cone \(\text {SP}^{2k}{\mathbb {C}}^{2n}\) in \(\wedge _{{\mathbb {R}}}^{2k}{\mathbb {C}}^{2n}\) generated by elementary strongly positive 2k-elements; that is, \(\omega =\sum _{l=1}^m\lambda _l\xi _l\) for some non-negative numbers \(\lambda _1,\ldots ,\lambda _m\) and some elementary strongly positive elements \(\xi _1,\ldots ,\xi _m\). An 2k-element \(\omega \) is said to be positive if for any strongly positive element \(\eta \in \text {SP}^{2n-2k}{\mathbb {C}}^{2n}\), \(\omega \wedge \eta \) is positive. We will denote the set of all positive 2k-elements by \(\wedge ^{2k}_{{\mathbb {R}}+}{\mathbb {C}}^{2n}\). Any 2k element is a \({\mathbb {C}}\)-linear combination of strongly positive 2k elements by Proposition 5.2 in [4], i.e. \({\mathrm{span}}_{\mathbb {C}}\{\varphi ;~\varphi \in \wedge ^{2k}_{{\mathbb {R}}+}{\mathbb {C}}^{2n}\}=span_{\mathbb {C}}\{\varphi ;~\varphi \in \text {SP}^{2k}{\mathbb {C}}^{2n}\}=\wedge ^{2k}{\mathbb {C}}^{2n}\). By definition, \( \beta _n\) is a strongly positive 2-form, and \( \beta _n^n=\wedge ^n\beta _n=n!~\Omega _{2n}\) is a positive 2n-form.
For a domain \(\Omega \) in \({\mathscr {H}} \), let \({\mathcal {D}}_0^{p}(\Omega )=C_0(\Omega ,\wedge ^{p}{\mathbb {C}}^{2n})\) and \( {\mathcal {D}}^{p}(\Omega )=C_0^\infty (\Omega ,\wedge ^{p}{\mathbb {C}}^{2n}). \) An element of the latter one is called a test p-form. An element \(\eta \in {\mathcal {D}}_0^{2k}(\Omega )\) is called a positive 2k-form (respectively, strongly positive 2k-form) if for any \(q\in \Omega \), \(\eta (q)\) is a positive (respectively, strongly positive) element.
Theorem 1.1 in [33] and its proof implies the following result.
Proposition 4.1
For a hyperhermitian \(n\times n\)-matrix \({\mathcal {M}}=({{\mathcal {M}}}_{jk})\), there exists a quaternionic unitary matrix \({\mathcal {E}}\in \text {U}_{{\mathbb {H}}}(n)\) such that \({\mathcal {E}}^*{\mathcal {M}}{\mathcal {E}}=\mathrm{diag} (\nu _0,\ldots ,\nu _{n-1})\). Then the 2-form
with \( M=\tau ({\mathcal {M}}){\mathbb {J}}, \) can be normalized as
with \({\widetilde{\omega }}^A= \mathcal {E^*}.{\omega }^A\). In particular, \(\omega \) is strongly positive if and only if \({\mathcal {M}} \) is nonnegative.
Proposition 4.2
For any \(C^1\) real function u, \(d_0u\wedge d_1u\) is elementary strongly positive if grad \(u\ne 0\).
Proof
Let \(p:=(p_1,\ldots ,p_n)\in {\mathbb {H}}^n\) with \( p_l= X_{4l+1}u +{\mathbf {i}}X_{4l+2}u +{\mathbf {j}}X_{4l+3}u +{\mathbf {k}}X_{4l+4}u .\) Then as (3.9), we have
where
Denote \(n\times n\) quaternionic matrix \(\widetilde{{\mathcal {M}}}:=(\overline{p_l} {p_m})\). Then \(\widetilde{{\mathcal {M}}} =a+b{\mathbf {j}} \) with \(n\times n\) complex matrices \( a=({\widetilde{\triangle }}_{ l(n+m )}u),\) \( b=({\widetilde{\triangle }}_{ l m }u). \) Thus
since we can easily check
Since \({\mathcal {M}}\) has eigenvalues \(|p|^2, 0,\ldots ,0\), we see that
is elementary strongly positive by Proposition 4.1. \(\square \)
See [28, Proposition 3.3] for this proposition for \({\mathbb {H}}^n\) with a different proof.
4.2 The closed strongly positive 2-form given by a smooth PSH
Proposition 4.3
For \(u\in C^2(\Omega )\), u is PSH if and only if the hyperhermitian matrix \((\overline{Q_l} {Q_m} u - 8\delta _{lm}{\mathbf {i}}\partial _tu )\) is nonnegative.
The tangential mapping \(\iota _{\eta ,q *} \) maps horizontal left invariant vector fields on \(\widetilde{{\mathscr {H}}}_q \) to that on the quaternionic Heisenberg line \({\mathscr {H}}_{\eta ,q } \). In particular, we have
Proposition 4.4
For \(q\in {\mathbb {H}}^n{\setminus } {\mathfrak {D}}\),
Proof
Since \(\iota _{\eta ,q}=\tau _\eta \circ \iota _{q}\) and \(X_j\)’s are invariant under \(\tau _\eta \), it sufficient to prove (4.10) for \(\eta =0\). For fixed \(j=1,2,3,4\) and \(l=1,\ldots , n\), note that
by (2.6). Thus for \(q=(q_1,\ldots ,q_n)\in {\mathbb {H}}^n\) and \(\varsigma \in {\mathbb {R}}\), if we write \(\iota _{q } (\lambda ,t)=(q\lambda ,t)=(x,t)\), we get
by the multiplication (2.13) of the group \(\widetilde{{\mathscr {H}}}_q\) and \(B^q\) in (2.12). So
by (2.8). \(\square \)
Proof of Proposition 4.3
Denote \( \overline{{\widetilde{Q}}}:= \widetilde{X_1} +{\mathbf {i}}\widetilde{X_{ 2}}+{\mathbf {j}}\widetilde{X_{ 3 }}+{\mathbf {k}}\widetilde{X_{ 4}} . \) Then we have
by Proposition 4.4, (2.6) and definition of \({q }^{{\mathbb {R}}}\) in (2.7), and \( \iota _{\eta ,q*} {{\widetilde{Q}}}= \sum _{l=0 }^{n-1} {Q_l} {q}_l \) by taking conjugate. Therefore for real u, we have
On the other hand, we have
Since the horizontal quaternionic Hessian \((\overline{Q_l} {Q_m} u +8 \delta _{lm}{\mathbf {i}}\partial _t u )\) is hyperhermitian by Proposition 3.4, we see that the above quadratic form is real for any q. Note that \( {\overline{p}} {\mathbf {i}}p\in {\mathrm{Im}}\,{\mathbb {H}}\) for any \(0\ne p\in {\mathbb {H}}\). Therefore, we get
for \(q\in {\mathbb {H}}^n{\setminus } {\mathfrak {D}}\) by (4.12).
Now if u is PSH, then \( \widetilde{\triangle _q} (\iota ^*_{\eta ,q }u)\) is nonnegative by applying Theorem 2.1 to the group \(\widetilde{{\mathscr {H}}}_q\) for \(q\in {\mathbb {H}}^n{\setminus } {\mathfrak {D}}\). Consequently, (4.13) holds for any \(q\in {\mathbb {H}}^n\) by continuity, i.e. the hyperhermitian matrix \((\overline{Q_l} {Q_m} u + 8\delta _{lm}{\mathbf {i}}\partial _tu )\) is nonnegative. Conversely, if the hyperhermitian matrix is nonnegative, we get u is is subharmonic on each quaternionic Heisenberg line \({\mathscr {H}}_{\eta ,q}\) for any \(q\in {\mathbb {H}}^n{\setminus } {\mathfrak {D}}\) and \(\eta \in {\mathscr {H}}^n\) by applying Theorem 2.1 again. \(\square \)
Corollary 4.1
For \(u\in PSH\cap C^2(\Omega )\), \(\triangle u\) is a closed strongly positive 2-form.
Proof
It follows from applying Proposition 4.1 to nonnegative \({\mathcal {M}}=(\overline{Q_l} {Q_m} u - 8\delta _{lm}{\mathbf {i}}\partial _tu )\) and using (3.23). \(\square \)
Corollary 4.2
A \(C^2\) function u is pluriharmonic if and only if \(\triangle u=0\).
Proof
u is pluriharmonic means that \(\widetilde{\triangle _q}\iota ^*_{\eta ,q}u=0\) on the quaternionic Heisenberg line \(\widetilde{{\mathscr {H}}}_{q}\) for any \(\eta \in {\mathscr {H}} \) and \( q\in {\mathbb {H}}^n{\setminus } {\mathfrak {D}} \). It holds if and only if
for any \(q\in {\mathbb {H}}^n\) by (4.13), i.e. \((\overline{Q_l} {Q_m} u + 8\delta _{lm}{\mathbf {i}}\partial _tu ) =0 \), which equivalent to \(\triangle u=0\) by (3.23). \(\square \)
Recall that the tangential 1-Cauchy–Fueter operator on a domain \(\Omega \) in the Heisenberg group \({\mathscr {H}} \) is \({\mathscr {D}}: C^1(\Omega ,{\mathbb {C}}^2)\rightarrow C^0(\Omega ,{\mathbb {C}}^{2n})\) [20] given by
where \(Z_A^{0'}=Z_{A1'}\) and \(Z_A^{1'}=-Z_{A0'}\). A \({\mathbb {C}}^2\)-valued function \(f=(f_{0'},f_{1'})=(f_1 +{\mathbf {i}}f_2,f_3 +{\mathbf {i}}f_4)\) is called 1-CF if \({\mathscr {D}} f=0\).
Proposition 4.5
Each real component of a 1-CF function \(f: \mathscr {H}^n\rightarrow {\mathbb {C}}^2\) is pluriharmonic.
Proof
Note that \(\sum _{A'=0',1'}Z_A^{A'}f_{A'}=0\) is equivalent to \(\sum _A\sum _{A'=0',1'}Z_A^{A'}f_{A'}\omega ^A=0\), which can be written as
Apply \( d_0\) on both sides to get \(d_0 d_{1}f_{0'}=0\) since \(d_0^2=0\). Similarly, we get \(d_0 d_{1}f_{1'}=0\). Writing \(f_{0'}=f_1+{\mathbf {i}}f_2\) for some real functions \(f_1\) and \(f_2\), we have
Note that for a real valued function u, \(\triangle u\) is a real 2-form by (3.23) and Proposition 4.1, i.e. \(\rho (\mathbf{j })\triangle u=\triangle u\). We get
Thus \( \triangle f_1=0=\triangle f_2\). Similarly, we have \( \triangle f_3=0=\triangle f_4\). \(\square \)
See Corollary 2.1 in [31] for this Proposition on the quaternionic space \({\mathbb {H}}^n\). Since 1-regular functions are abundant, so are pluriharmonic functions on the Heisenberg group.
4.3 Closed positive currents
An element of the dual space \(({\mathcal {D}}^{2n-p}(\Omega ))' \) is called a p-current. A 2k-current T is said to be positive if we have \(T(\eta )\ge 0\) for any strongly positive form \(\eta \in {\mathcal {D}}^{2n-2k}(\Omega )\). Although a 2n-form is not an authentic differential form and we cannot integrate it, we can define
if we write \(F=f~\Omega _{2n}\in L^1(\Omega ,\wedge ^{2n}{\mathbb {C}}^{2n})\), where dV is the Lebesgue measure. In general, for a 2n-current \(F=\mu ~\Omega _{2n}\) with the coefficient to be a measure \(\mu \), define
Now for the p-current F, we define a \((p+1)\)-current \(d_\alpha F\) as
for any test \((2n-p-1)\)-form \(\eta \). We say a current F is closed if \(d_0F=d_1F=0 .\)
An element of the dual space \(({\mathcal {D}}_0^{2n-p}(\Omega ))' \) are called a p-current of order zero. Obviously, a 2n-current is just a distribution on \(\Omega \), whereas a 2n-current of order zero is a Radon measure on \(\Omega \). Let \(\psi \) be a p-form whose coefficients are locally integrable in \(\Omega \). One can associate with \(\psi \) the p-current \(T_{\psi }\) defined by
If T is a 2k-current on \(\Omega \), \(\psi \) is a 2l-form on \(\Omega \) with coefficients in \(C^{\infty }(\Omega )\), and \(k+l\le n\), then the formula
defines a \((2k+2l)\)-current. In particular, if \(\psi \) is a smooth function, \(\psi T(\varphi )=T(\psi \varphi )\).
A 2k-current T is said to be positive if we have \(T(\eta )\ge 0\) for any \(\eta \in C_0^\infty (\Omega ,SP^{2n-2k}{\mathbb {C}}^{2n})\). Namely, T is positive if for any \(\eta \in C_0^\infty (\Omega ,SP^{2n-2k}{\mathbb {C}}^{2n})\), \(T\wedge \eta =\mu ~\Omega _{2n}\) for some positive distribution \(\mu \) (and hence a measure).
Let \(I=(i_1,\ldots ,i_{2k})\) be a multi-index such that \(1\le i_1<\ldots < i_{2k}\le n\). Denote by \({\widehat{I}}=(l_1,\ldots ,l_{2n-2k})\) the increasing complements to I in the set \(\{0,1,\ldots ,2n-1\}\), i.e., \(\{i_1,\ldots ,i_{2k}\}\cup \{l_1,\ldots ,l_{2n-2k}\}=\{0,1,\ldots ,2n-1\}\). For a 2k-current T in \(\Omega \) and multi-index I, define distributions \(T_I\) by \(T_I(f):=\varepsilon _IT(f\omega ^{{\widehat{I}}})\) for \(f\in C_0^\infty (\Omega )\), where \(\varepsilon _I=\pm 1\) is so chosen that
If T is a current of order 0, the distributions \(T_I\) are Radon measures and
for \(\varphi =\sum _{{\widehat{I}}}\varphi _{{\widehat{I}}}\omega ^{{\widehat{I}}}\in {\mathcal {D}}^{2n-2k}(\Omega )\), where I and \({\widehat{I}}\) are increasing. Namely,
where the summation is taken over increasing multi-indices of length 2k, holds in the sense that if we write \(T\wedge \varphi =\mu ~\Omega _{2n}\) for some Radon measure \(\mu \), then we have
Proposition 4.6
Any positive 2k-current T on \(\Omega \) has measure coefficients (i.e. is of order zero), and we can write \(T=\sum _IT_I\omega ^I\) for some complex Radon measures \(T_I\), where the summation is taken over all increasing multi-indices I.
Proof
By Proposition 5.4 in [4], we can find \(\{\varphi _L\}\subseteq SP^{2n-2k}{\mathbb {C}}^{2n}\) such that any \(\eta \in \wedge ^{2n-2k}{\mathbb {C}}^{2n}\) is a \({\mathbb {C}}\)-linear combination of \(\varphi _L\), i.e., \(\eta =\sum \lambda _L\varphi _L\) for some \(\lambda _L\in {\mathbb {C}}\). Let \(\{\widetilde{\varphi _L}\}\) be a basis of \(\wedge ^{2k}{\mathbb {C}}^{2n}\) which is dual to \(\{\varphi _L\}\). Then \(T=\sum T_L\widetilde{\varphi _L}\) with distributional coefficients \(T_L\) as (4.20). If \(\psi \) is a nonnegative test function, \(\psi \varphi _L\in C_0^\infty (\Omega ,SP^{2n-2k}{\mathbb {C}}^{2n})\). Then \(T_L(\psi )=T(\psi \varphi _L)\ge 0\) by definition. It follows that \(T_L\) is a positive distribution, and so is a nonnegative measure. \(\square \)
The following Proposition is obvious and will be used frequently.
Proposition 4.7
-
(1)
(linearity) For 2n-currents \(T_1\) and \(T_2\) with (Radon) measure coefficients, we have
$$\begin{aligned} \int _\Omega \alpha T_1+ \beta T_2=\alpha \int _\Omega T_1+ \beta \int _\Omega T_2. \end{aligned}$$ -
(2)
If \(T_1\le T_2\) as positive 2n-currents (i.e. \(\mu _1\le \mu _2\) if we write \(T_j=\mu _j\Omega _{2n}\), \(j=1,2\)), then \(\int _\Omega T_1\le \int _\Omega T_2\).
Lemma 4.1
(Stokes-type formula) Let \(\Omega \) be a bounded domain with smooth boundary and defining function \(\rho \) (i.e. \(\rho =0\) on \(\partial \Omega \) and \(\rho <0\) in \(\Omega \)) such that \(|{\mathrm{grad}} \rho |=1\). Assume that \(T=\sum _AT_A\omega ^{{\widehat{A}}}\) is a smooth \((2n-1)\)-form in \(\Omega \), where \(\omega ^{{\widehat{A}}}=\omega ^A \rfloor \Omega _{2n}\). Then for \(h\in C^1({\overline{\Omega }})\), we have
where dS denotes the surface measure of \(\partial \Omega \). In particular, if \(h=0\) on \(\partial \Omega \), we have
Proof
Note that
Then
by definition (4.14) and integration by part,
for \(j=1,\ldots 4n\). (4.24) holds because the coefficient of \(\partial _t\) is independent of t. (4.22) follows from the above formula and \(d_\alpha (hT)=d_\alpha h\wedge T+hd_\alpha T\). \(\square \)
Now let us show that \(d_\alpha F\) in the generalized sense (4.16), coincides with the original definition when F is a smooth 2k-form. Let \(\eta \) be arbitrary \((2n-2k-1)\)-test form compactly supported in \(\Omega \). It follows from Lemma 4.1 that \(\int _\Omega d_\alpha (F\wedge \eta )=0.\) By Proposition 1.1 (3), \(d_\alpha (F\wedge \eta )=d_\alpha F\wedge \eta +F\wedge d_\alpha \eta \). We have
We also define \(\triangle F\) in the generalized sense, i.e., for each test \((2n-2k-2)\)-form \(\eta \),
As a corollary, \(\triangle F\) in the generalized sense coincides with the original definition when F is a smooth 2k-form:
Corollary 4.3
For \(u\in PSH(\Omega )\), \(\triangle u\) is a closed positive 2-current.
Proof
If u is smooth, \(\triangle u\) is a closed strongly positive 2-form by Corollary 4.1. When u is not smooth, consider regularization \(u_{\varepsilon }=\chi _{\varepsilon }* u\) as in Proposition 2.3 (6). It suffices to show that coefficients \(\triangle _{AB}u_{\varepsilon }\rightarrow \triangle _{AB}u\) in the sense of weak convergence of distributions. For any \(\varphi \in C_0^{\infty }(\Omega )\),
as \(\varepsilon \rightarrow 0\), by using integration by part (4.24) and the standard fact that \(\chi _{\varepsilon }* u\rightarrow u\) in \(L^1_{\mathrm{loc}}(\Omega )\) if \(u \in L^1_{\mathrm{loc}}(\Omega )\) [19]. It follows that the currents \(\triangle u_{\varepsilon }\) converge to \(\triangle u\), and so the current \(\triangle u\) is positive. For any test form \(\eta \),
\(\alpha =0,1,\) where the last identity follows from Corollary 3.1. Here \(u_\varepsilon \) is smooth, and \(d_\alpha \triangle u_\varepsilon \) coincides with its usual definition.\(\square \)
5 The quaternionic Monge–Ampère measure over the Heisenberg group
For positive \((2n-2p)\)-form T and an arbitrary compact subset K, define \(\Vert T\Vert _K:=\int _K T\wedge \beta _n^p,\) where \(\beta _n\) is given by (4.1). In particular, if T is a positive 2n-current, \(\Vert T\Vert _K\) coincides with \(\int _KT\) defined by (4.15). Let \(\Vert \cdot \Vert \) be a norm on \(\wedge ^{2k}{\mathbb {C}}^{2n}\).
Lemma 5.1
(Lemma 3.3 in [27]) For \(\eta \in \wedge ^{2k}_{{\mathbb {R}}}{\mathbb {C}}^{2n}\) with \(\Vert \eta \Vert \le 1\), \(\beta _n^k\pm \varepsilon \eta \) is a positive 2k-form for some sufficiently small \(\varepsilon >0\).
Proposition 5.1
(Chern–Levine–Nirenberg type estimate) Let \(\Omega \) be a domain in \({\mathscr {H}}^n\). Let K and L be compact subsets of \(\Omega \) such that L is contained in the interior of K. Then there exists a constant C depending only on K, L such that for any \(u_1,\ldots u_k\in PSH(\Omega )\cap C^2(\Omega )\), we have
Proof
By Corollary 4.1, \(\triangle u_1\wedge \cdots \wedge \triangle u_k \) is already closed and strongly positive. Since L is compact, there is a covering of L by a family of balls \(D_j'\Subset D_j\subseteq K\). Let \(\chi \ge 0\) be a smooth function equals to 1 on \(\overline{D_j'}\) with support in \(D_j\). For a closed smooth \((2n-2p)\)-form T, we have
by using Stokes-type formula (4.23) and Proposition 1.2. Then
by using (5.2) and Lemma 5.1. The result follows by repeating this procedure. \(\square \)
Proof of Theorem 1.1
It is sufficient to prove for any compactly supported continuous function \(\chi \), the sequence \( \int _\Omega \chi (\triangle u_j)^n \) is a Cauchy sequence. We can assume \(\chi \in C_0^\infty (\Omega )\). Note the following identity
Then we have
as in the proof of Proposition 5.1, where C depends on the uniform upper bound of \(\left\| u_j \right\| _\infty \). \(\square \)
Proposition 5.2
Let \(u ,v\in C ({\overline{\Omega }})\) be plurisubharmonic functions. Then \( (\triangle (u+v))^n\ge (\triangle u )^n+(\triangle v )^n. \)
Proof
For smooth PSH \(u_{\varepsilon }=\chi _{\varepsilon }* u\), we have
The result follows by taking limit \(\varepsilon \rightarrow 0\) and using the convergence of the quaternionic Monge–Ampère measure in Theorem 1.1. \(\square \)
We need the following proposition to prove the minimum principle.
Proposition 5.3
Let \(\Omega \) be a bounded domain with smooth boundary in \({\mathscr {H}} \), and let \(u ,v\in C^2({\overline{\Omega }})\) be plurisubharmonic functions on \(\Omega \). If \(u=v\) on \(\partial \Omega \) and \(u\ge v\) in \(\Omega \), then
Proof
We have
by using (5.3) and Stokes-type formula (4.22), if we write
where \(\rho \) is a defining function of \(\Omega \) with \(|{\mathrm{grad}} \rho |=1\), and \(\omega ^{{\widehat{A}}}=\omega ^A \rfloor \Omega _{2n}\). Note that we have
Since \(u=v\) on \(\partial \Omega \) and \(u\ge v\) in \(\Omega \), for a point \(\xi \in \partial \Omega \) with \({\mathrm{grad}} (v-u)(\xi )\ne 0\) , we can write \( v-u = h \rho \) in a neighborhood of \(\xi \) for some positive smooth function h. Consequently, we have \({\mathrm{grad}} (v-u)(\xi )= h(\xi ) {\mathrm{grad}} \rho \), and so \(Z_{A1'}(v-u)(\xi )= h(\xi ) Z_{A1'}\rho (\xi )\) on \(\partial \Omega \). Thus,
which is strongly positive by Proposition 4.2. Moreover, both \(\triangle v \) and \(\triangle u\) are strongly positive for \(C^2\) plurisubharmonic functions u and v on \(\Omega \) by Proposition 4.1. We find that the right hand of (5.6) is a positive 2n-form, and so the integrant in the right hand of (5.5) on \(\partial \Omega \) is nonnegative if \({\mathrm{grad}} (v-u)(\xi )\ne 0\), while if \({\mathrm{grad}} (v-u)(\xi )= 0\), the integrant at \(\xi \) in (5.5) vanishes. Therefore the difference in (5.5) is nonnegative. \(\square \)
The proof of the minimum principle is similar to the complex case [8] and the quaternionic case [1], but we need some modifications because we do not know whether the regularization \( \chi _\varepsilon *u \) of a PSH function u on the the Heisenberg group is decreasing as \(\varepsilon \rightarrow 0+\).
Proof of Theorem 1.3
Without loss of generality, we may assume \(\min _{ \partial {\Omega }} \{u-v\}=0\). Suppose that there exists a point \((x_0,t_0)\in \Omega \) such that \(u(x_0,t_0)<v (x_0,t_0)\). Denote \(\eta _0=\frac{1}{2}[v (x_0,t_0)-u(x_0,t_0)]\). Then for each \(0<\eta <\eta _0\), the set \( G(\eta ):=\{(x,t)\in \Omega ; u(x ,t )+\eta <v( x ,t )\} \) is a non-empt, open, relatively compact subset of \(\Omega \). Now consider
There exists an increasing function \(\delta (\eta )\) such that \(G(\eta ,\delta )\) for \(0<\delta <\delta (\eta )\) is a non-empt, open, relatively compact subset of \(\Omega \). On the other hand, there exists small \(\alpha (\eta ,\delta )\) such that for \(0<\alpha <\alpha (\eta ,\delta )\), we have \( \{\xi \in \Omega ; \mathrm{dist} (\xi ,\partial \Omega )>\alpha \}= :\Omega _\alpha \supset G(\eta ,\delta ) \) for \(0<\delta <\delta (\eta /2)\), where \(\mathrm{dist} (\xi ,\zeta )=\Vert \xi ^{-1}\zeta \Vert \).
We hope to apply Proposition 5.3 to \(G(\eta ,\delta )\) to get a contradict, but its boundary may not be smooth. We need to regularize them. Recall that \(u_\varepsilon \rightarrow u\) and \(v_\varepsilon \rightarrow v\) uniformly as \(\varepsilon \rightarrow 0+\) on any compact subset of \(\Omega \). Define
which satisfies \( G(\eta ,\delta ,\varepsilon )\subset G(3\eta /4,\delta )\subset G( \eta /2,\delta )\) if \(0<\varepsilon <\alpha (\eta ,\delta )\) is sufficiently small, since \(|v(x,t)-v_\varepsilon (x,t)|\le \eta /4\) for \((x,t)\in G(\eta /2, \delta )\). Now choose \(\tau \) so small that
is a non-empt, open, relatively compact subset of \(\Omega \). At last we can choose positive numbers \( \eta _1<\eta _2, \delta _0 , \varepsilon _0, \tau _0\) such that for any \(\eta \in [\eta _1,\eta _2] \), \(0<\varepsilon <\varepsilon _0\), \(0<\tau <\tau _0\), \( G(\eta ,\delta _0 ,\varepsilon ,\tau )\) is a non-empt, open, relatively compact subset of \(\Omega \).
For fixed \(\varepsilon ,\tau \), by Sard’s theorem, almost all values of the \(C^\infty \) function \(v_\varepsilon ( x ,t )+\delta _0 |x-x_0|^2 -u_\tau (x ,t ) \) are regular, i.e. \( G(\eta ,\delta _0 ,\varepsilon ,\tau )\) has smooth boundary for almost all \(\eta \). Consequently, we can take sequence of numbers \(\tau _k\rightarrow 0\) and \(\varepsilon _k\rightarrow 0\) such that \( G(\eta ,\delta _0 ,\varepsilon _k,\tau _k)\) has a smooth boundary for each k and almost all \(\eta \in [\eta _1,\eta _2]\). Now apply Proposition 5.3 to the domain \( G(\eta ,\delta _0 ,\varepsilon _k,\tau _k)\) to get
by using Proposition 5.2 (2), where integrals are taken over \(G(\eta ,\delta _0 ,\varepsilon _k,\tau _k)\), and
by the expression of \(\triangle _{l(n+l)}\) in (3.7). Since \((\triangle u)^n\le (\triangle v)^n\) and \(\eta \rightarrow (\triangle v)^n ( G(\eta ,\delta _0 ))\) is decreasing in \(\eta \), we can choose a continuous point \(\eta \) such that \( G(\eta ,\delta _0 ,\varepsilon _k,\tau _k)\) has a smooth boundary. For any \(\eta _1<\eta '<\eta<\eta ''<\eta _2\), \(G(\eta ',\delta _0 )\supset G(\eta ,\delta _0 ,\varepsilon _k,\tau _k)\supset G(\eta '',\delta _0 )\) for large k. So we have
by (5.7). Thus,
by convergence of quaternionic Monge–Ampère measures by Theorem 1.1. At the continuous point \(\eta \), we have
This is a contradict since \(G(\eta '',\delta _0 )\) is a nonempt open subset of \(\Omega \) for \(\eta ''\) close to \(\eta \). \(\square \)
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Wang, W. The quaternionic Monge–Ampère operator and plurisubharmonic functions on the Heisenberg group. Math. Z. 298, 521–549 (2021). https://doi.org/10.1007/s00209-020-02608-3
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DOI: https://doi.org/10.1007/s00209-020-02608-3