Some Notions of Subharmonicity over the Quaternions

Abstract

This work introduces several notions of subharmonicity for real-valued functions of one quaternionic variable. These notions are related to the theory of slice regular quaternionic functions introduced by Gentili and Struppa in 2006. The interesting properties of these new classes of functions are studied and applied to construct the analogs of Green’s functions.

Dedicated to Wolfgang Sprößig on the occasion of his 70th birthday

Appendix

For the reader’s convenience, we include in this appendix some classical results and definitions, which are used in the present work. We begin with some theorems concerning subharmonic functions of m real variables, along with some instrumental definitions. Let λ denote the Lebesgue measure on ${\mathbb {R}}^m$ and σ denote the surface area measure.

Theorem 4.51 ([9, Theorem 2.4.1 (iii)])

Let D be an open subset of ${\mathbb {R}}^m$ and let υ : D → [−∞, +∞) be an upper semicontinuous function which is not identically −∞ on any connected component of D. The function υ is subharmonic in D if, and only if, for any Euclidean ball B(a, R) such that $\overline {B(a,R)}\subset D$ , it holds

$$\displaystyle \begin{aligned}\upsilon(a)\leq \mathbf{L}(\upsilon;a,R),\end{aligned}$$

where

$$\displaystyle \begin{aligned}\mathbf{L}(\upsilon;a,R):=\frac{1}{s_mR^{m-1}}\int_{\partial B(a,R)}u(x)d\sigma(x),\quad s_m:=\sigma(\partial B(0,1)).\end{aligned}$$

Theorem 4.52 ([9, Theorem 2.4.2])

Let D be a bounded connected open subset of ${\mathbb {R}}^m$ and let υ : D → [−∞, +∞) be subharmonic in D. Then either υ is constant or, for each x ∈ D,

$$\displaystyle \begin{aligned}\upsilon(x)<\sup_{z\in\partial D}\left\{\limsup_{D\ni y\to z}\upsilon(y)\right\}.\end{aligned}$$

Corollary 4.53 ([9, Corollary 2.4.3])

u is harmonic if, and only if, u and − u are both subharmonic.

Definition 4.54 ([9, §2.5])

Define $h:{\mathbb {R}}\to {\mathbb {R}}$ by the formula

$$\displaystyle \begin{aligned}h(t):=\left\{ \begin{array}{ll} \exp(-1/t)&\ \text{if}\ t>0\\ 0&\ \text{if}\ t\leq0 \end{array} \right.\end{aligned}$$

and define $\chi :{\mathbb {R}}^m\to {\mathbb {R}}$ by the formula

$$\displaystyle \begin{aligned}\chi(x):=\frac{1}{c}\,h(1-\Vert x\Vert^2),\quad c:=\int_{B(0,1)}h(1-\Vert x\Vert^2)d\lambda(x).\end{aligned}$$

The standard smoothing kernels $\chi _\varepsilon :{\mathbb {R}}^m\to {\mathbb {R}}$ are defined, for all ε > 0, by the formula

$$\displaystyle \begin{aligned}\chi_\varepsilon(x):=\frac{1}{\varepsilon^m}\,\chi\left(\frac{x}{\varepsilon}\right).\end{aligned}$$

Given a function υ on a open subset D of ${\mathbb {R}}^m$, the convolution

$$\displaystyle \begin{aligned}\upsilon*\chi_\varepsilon(x)=\chi_\varepsilon*\upsilon(x):=\int_{{\mathbb{R}}^m}\chi_\varepsilon(x-y)\upsilon(y)d\lambda(y)\end{aligned}$$

is well-defined on

$$\displaystyle \begin{aligned}D_{\varepsilon}:=\left\{ \begin{array}{ll} \{x\in D : \mathrm{dist}(x, \partial D)>\varepsilon\}&\ \text{if}\ D\neq{\mathbb{R}}^m\\ \,{\mathbb{C}}^n&\ \text{if}\ D={\mathbb{R}}^m \end{array} \right.\end{aligned}$$

Theorem 4.55 ([9, Theorem 2.5.5])

Let D be an open subset of ${\mathbb {R}}^m$ and let υ : D → [−∞, +∞) be subharmonic. For all ε > 0 such that D _ε is not empty, υ ∗ χ _ε is C ^∞ and subharmonic in D _ε . Moreover, υ ∗ χ _ε monotonically decreases with decreasing ε and

$$\displaystyle \begin{aligned} \lim_{\varepsilon \to 0^+} \upsilon*\chi_\varepsilon (x) = \upsilon(x) \end{aligned} $$

(4.16)

for each x ∈ D.

We now recall some properties of plurisubharmonic functions of n complex variables.

Theorem 4.56 ([9, Theorem 2.9.1])

Let D be an open subset of ${\mathbb {C}}^n$ and let υ : D → [−∞, +∞) be an upper semicontinuous function which is not identically −∞ on any connected component of D. υ is plurisubharmonic in D if, and only if, for any $a\in D,b\in {\mathbb {C}}^n$ such that $\{a+\lambda b : \lambda \in {\mathbb {C}}, |\lambda |\leq 1\}\subset D$ , it holds

$$\displaystyle \begin{aligned}\upsilon(a)\leq l(\upsilon;a,b)\,,\end{aligned}$$

where

$$\displaystyle \begin{aligned}l(\upsilon;a,b):=\frac{1}{2\pi}\int_0^{2\pi}\upsilon(a+e^{it}b)dt.\end{aligned}$$

Moreover, plurisubharmonicity is a local property.

Theorem 4.57 ([9, Theorem 2.9.2])

Let D be an open subset of ${\mathbb {C}}^n$ and let υ : D → [−∞, +∞) be plurisubharmonic. For all ε > 0 such that D _ε is not empty, υ ∗ χ _ε is C ^∞ and plurisubharmonic in D _ε . Moreover, υ ∗ χ _ε monotonically decreases with decreasing ε and

$$\displaystyle \begin{aligned} \lim_{\varepsilon \to 0^+} \upsilon*\chi_\varepsilon (z) = \upsilon(z) \end{aligned} $$

(4.17)

for each z ∈ D.

Theorem 4.58 ([9, Corollary 2.9.9])

Let D be a bounded connected open subset of ${\mathbb {C}}^n$ and let υ be a plurisubharmonic function on D. Then either υ is constant or, for each z ∈ D,

$$\displaystyle \begin{aligned}\upsilon(z)<\sup_{w\in\partial D}\left\{\limsup_{D\ni y\to w}\upsilon(y)\right\}.\end{aligned}$$

Remark 4.59 ([9, §3.1])

Let D be an open subset of ${\mathbb {C}}$ and let υ be a plurisubharmonic function on D. The function υ is harmonic if, and only if, it is maximal among plurisubharmonic functions on D.

Proposition 4.60 ([9, Proposition 6.1.1 (iv)])

Let D be a bounded domain in ${\mathbb {C}}^n$ , let w ∈ D and let $\gamma ^D_w$ denote the pluricomplex Green function of D with pole at w. Then $\gamma ^D_w$ is a negative plurisubharmonic function with a logarithmic pole at w.