Gluing Theorems for Subharmonic Functions

TL;DR

This paper advances the technique of gluing subharmonic functions, enhancing its theoretical framework and potential applications in analyzing the distribution of roots and masses of holomorphic and subharmonic functions.

Contribution

It develops and improves the gluing technique for subharmonic functions, providing a stronger foundation for future applications.

Findings

Enhanced method for gluing subharmonic functions

Improved understanding of root and mass distribution

Foundation for future research applications

Abstract

In our articles of recent years, the technique of gluing two subharmonic functions turned out to be very useful in studying the distribution of the roots or masses of holomorphic or subharmonic functions, respectively. Here we develop and improve this technique. Its applications will be given in our further works.

Equations

$$\underset{y\to x}{\mathrm{limsup}}{u}_0(y)=u(x)\text{for each x∈O∩∂O0},$$

$$U:=\{\begin{array}{ll}\max \{u,u_0\}& \text{ on O0},\\ u& \text{ on O∖O0}\end{array}$$

$${\mathbb{R}}_{-\infty }:=\{-\infty \}\cup \mathbb{R},{\mathbb{R}}_{+\infty }$$

$${\mathbb{R}}^{+}:=\{x\in \mathbb{R}:x\ge 0\},x^{+}$$

$$S^{+}:=\{x\ge 0:x\in S\},S_{\ast }$$

$$x\cdot (\pm \infty ):=\pm \infty =:$$

$$\frac{x}{\pm \infty }:=0$$

$$B(x,r):=\{x^{\prime }\in {\mathbb{R}}^d:|x^{\prime }-x|<r\},$$

$$B(\infty ,r):=\{x\in {\mathbb{R}}_{\infty }^d:|x|>1/r\},$$

$$B(r):=B(0,r),\mathbb{B}:=B(0,1),$$

$$B_{\circ }(x,r):=B(x,r)\setminus \{x\},$$

$${\mathbb{R}}_{-\infty }^X\stackrel{\text{eqref}df:Rr}{:=}({\mathbb{R}}_{-\infty }{)}^X,{\mathbb{R}}_{+\infty }^X\stackrel{\text{eqref}df:Rr}{:=}({\mathbb{R}}_{+\infty }{)}^X,{\mathbb{R}}_{\pm \infty }^X\stackrel{\text{eqref}df:Rr}{:=}({\mathbb{R}}_{\pm \infty }{)}^X.$$

$$\begin{array}{rl}{\mathrm{Dom}}_{-\infty }:=f^{-1}({\mathbb{R}}_{-\infty })\subset X,& {\mathrm{Dom}}_{+\infty }f:=f^{-1}({\mathbb{R}}_{+\infty })\subset X,\\ \mathrm{Dom}f:=f^{-1}({\mathbb{R}}_{\pm \infty })& ={\mathrm{Dom}}_{-\infty }f\bigcup {\mathrm{Dom}}_{+\infty }f\subset X,\\ \mathrm{dom}f:=f^{-1}(\mathbb{R})=& {\mathrm{Dom}}_{-\infty }f\bigcap {\mathrm{Dom}}_{+\infty }f\subset X,\end{array}$$

$${\mathrm{sbh}}_{\ast }(O):=\mathrm{sbh}(O)\setminus \{-\infty \},{\mathrm{sbh}}^{+}(O):=(\mathrm{sbh}(O){)}^{+}.$$

$${\star }_o:x⟼x^{{\star }_o}$$

$$u^{{\star }_o}(x^{{\star }_o})$$

$$\displaystyle\Bigl{(}u\in\operatorname{sbh}(O)\Bigr{)}\Longleftrightarrow\Bigl{(}u^{\star_{o}}\in\operatorname{sbh}(O^{\star_{o}})\Bigr{)}.$$

$$\underset{\stackrel{y\to x}{y\in O_0\cap O}}{\mathrm{limsup}}v(y)$$

$$\underset{\stackrel{y\to x}{y\in O_0\cap O}}{\mathrm{limsup}}{v}_0(y)$$

$$V:=\{\begin{array}{ll}{v}_0& \text{ on O0∖O},\\ \sup \{v_0,v\}& \text{ on O0∩O},\\ v& \text{ on O∖O0,}\end{array}$$

$$u$$

$$u_0$$

$$u$$

$$u_0$$

$$-\infty <m_v\le$$

$$\underset{x\in O_0\cap \partial O}{\sup }\underset{\stackrel{y\to x}{y\in O_0\cap O}}{\mathrm{limsup}}v(y)$$

$$-\infty <\underset{x\in O\cap \partial O_0}{\sup }\underset{\stackrel{y\to x}{y\in O\cap O_0}}{\mathrm{limsup}}g(y)\le m_g$$

$$v_0:=\frac{M_v^{+}+m_v^{-}}{M_g-m_g}(2g-M_g-m_g)\in \mathrm{sbh}(O_0),$$

$$\limsup_{\stackrel{{\scriptstyle y\to x}}{{y\in O_{0}\cap O}}}v(y)\overset{\eqref{g01vOm}-\eqref{g01vOM}}{\leq}M_{v}^{+}+m_{v}^{-}=\frac{M_{v}^{+}+m_{v}^{-}}{M_{g}-m_{g}}(2M_{g}-M_{g}-m_{g})\\ \overset{\eqref{g01g}}{\leq}\frac{M_{v}^{+}+m_{v}^{-}}{M_{g}-m_{g}}\Bigl{(}2\inf_{x\in O_{0}\cap\partial O}g(x)-M_{g}-m_{g}\Bigr{)}\\ =\inf_{x\in O_{0}\cap\partial O}\frac{M_{v}^{+}+m_{v}^{-}}{M_{g}-m_{g}}\Bigl{(}2g(x)-M_{g}-m_{g}\Bigr{)}\\ \overset{\eqref{v0g}}{=}\inf_{O_{0}\cap\partial O}v_{0}\leq v_{0}(x),\quad\forall x\in O_{0}\cap\partial O.$$

$$\limsup_{\stackrel{{\scriptstyle y\to x}}{{y\in O_{0}\cap O}}}v_{0}(y)\overset{\eqref{v0g}}{\leq}\frac{M_{v}^{+}+m_{v}^{-}}{M_{g}-m_{g}}\biggl{(}2\limsup_{\stackrel{{\scriptstyle y\to x}}{{y\in O_{0}\cap O}}}g-M_{g}-m_{g}\biggr{)}\overset{\eqref{g01g}}{\leq}\frac{M_{v}^{+}+m_{v}^{-}}{M_{g}-m_{g}}(2m_{g}-M_{g}-m_{g})\\ =-(M_{v}^{+}+m_{v}^{-})\leq-m_{v}^{-}\leq m_{v}\overset{\eqref{g01vOm}}{\leq}\inf_{x\in O\cap\partial O_{0}}v(x)\leq v(x),\quad\forall x\in O\cap\partial O_{0}.$$

$$k_q(t)$$

$$K_{d-2}(x,y)$$

$$\Bigl{(}E\subset(-\infty)_{u}:=\{x\in\mathbb{R}^{d}\colon u(x)=-\infty\}\Bigr{)}\Longleftrightarrow\Bigl{(}\text{Cap}^{*}E=0\Bigr{)},$$

$${\text{Cap}}^{\ast }(S):=\underset{S\subset O=\mathrm{int}O}{\inf }\underset{\stackrel{C=\mathrm{clos}C⋐O}{\mu \in {\mathrm{Meas}}^{1+}(C)}}{\sup }{k}_{d-2}^{-1}\left(\iint K_{d-2}(x,y)\mathrm{d}\mu (x)\mathrm{d}\mu (y)\right)$$

$${\mathbb{R}}^d\ni o\in \mathrm{int}{S}_0\subset S_0⋐S\subset \mathrm{int}\mathcal{O}=\mathcal{O}\subset {\mathbb{R}}_{\infty }^d\ne \mathcal{O}.$$

$$o\stackrel{\text{eqref}x0S}{\in }\mathrm{int}{S}_0\subset S_0⋐D⋐S\subset \mathcal{O}.$$

$$g_D(\cdot ,o)$$

Full text

MSC 2010: 31B5, 3A05, 3C05, 32A60

Keywords: subharmonic function, Green’s function, potential, Riesz measure, harmonic continuation, plurisubharmonic function

1 Introduction

As usual, $\mathbb{N}:=\{1,2,\dots\}$, $\mathbb{R}$ and $\mathbb{C}$ are the sets of all *natural, real * and complex numbers, respectively; $\mathbb{N}_{0}:=\{0\}\cup\mathbb{N}$ is French natural series, and $\mathbb{Z}:=\mathbb{N}_{0}\cup\mathbb{N}_{0}$.

For $d\in\mathbb{N}$ we denote by $\mathbb{R}^{d}$ the $d$-dimensional real Euclidean space with the standard Euclidean norm $|x|:=\sqrt{x_{1}^{2}+\dots+x_{d}^{2}}$ for $x=(x_{1},\dots,x_{d})\in\mathbb{R}^{d}$ and the distance function $\operatorname{dist}(\cdot,\cdot)$.

For a subset $S\subset\mathbb{R}^{d}$, we denote by $\operatorname{har}(S)$ and $\operatorname{sbh}(S)$ the classes of all harmonic (affine for m = 1) and subharmonic (locally convex for $m=1$) functions on an open set $O\supset S$, respectively. The basis of our note is

Gluing Theorem A** ([14, Theorem 2.4.5], [9, Corollary 2.4.4]).**

Let $\mathcal{O}$ be an open set in $\mathbb{R}^{d}$, and let $\mathcal{O}_{0}$ be a subset of ${\mathcal{O}}$. If $u\in\operatorname{sbh}({\mathcal{O}})$, $u_{0}\in\operatorname{sbh}({\mathcal{O}}_{0})$, and

[TABLE]

then the formula

[TABLE]

defines a subharmonic function on ${\mathcal{O}}$.

Important applications of Theorem A can be found in our articles [10], [11], [13].

2 Basic definitions, notations and conventions

The reader can skip this Section 2 and return to it only if necessary.

2.1 Sets, order, topology.

For the real line $\mathbb{R}$ with Euclidean norm-module $|\cdot|$,

[TABLE]

unless otherwise specified. An open connected (sub-)set of $\mathbb{R}_{\pm\infty}$ is a (sub-)interval of $\mathbb{R}_{\pm\infty}$. The Alexandroff one-point compactification of $\mathbb{R}^{d}$ is denoted by $\mathbb{R}^{d}_{\infty}:=\mathbb{R}^{d}\cup\{\infty\}$.

The same symbol [math] is used, depending on the context, to denote the number zero, the origin, zero vector, zero function, zero measure, etc.

Given $x\in\mathbb{R}^{d}$ and $r\overset{\eqref{df:R+}}{\in}\mathbb{R}_{+\infty}$, we set

[TABLE]

Thus, the basis of open (respectively closed) neighborhood of the point $x\in\mathbb{R}_{\infty}^{d}$ is open (respectively closed) balls $B(x,r)$ (respectively $\overline{B}(x,r)$) centered at $x$ with radius $r>0$.

Given a subset $S$ of $\mathbb{R}^{d}_{\infty}$, the closure $\operatorname{clos}S$, the* interior* $\operatorname{int}S$ and the boundary $\partial S$ will always be taken relative $\mathbb{R}^{d}_{\infty}$. For $S^{\prime}\subset S\subset\mathbb{R}^{d}_{\infty}$ we write $S^{\prime}\Subset S$ if $\operatorname{clos}S^{\prime}\subset\operatorname{int}S$. An open connected (sub-)set of $\mathbb{R}^{d}_{\infty}$ is a (sub-)domain of $\mathbb{R}^{d}_{\infty}$.

2.2 Functions.

Let $X,Y$ are sets. We denote by $Y^{X}$ the set of all functions $f\colon X\to Y$. The value $f(x)\in Y$ of an arbitrary function $f\in X^{Y}$ is not necessarily defined for all $x\in X$. The restriction of a function f to $S\subset X$ is denoted by f\bigm{|}_{S}. We set

[TABLE]

A function $f\in\mathbb{R}_{\pm\infty}^{X}$ is said to be extended numerical. For extended numerical functions $f$, we set

[TABLE]

For $f,g\in\mathbb{R}_{\pm\infty}^{X}$ we write $f=g$ if $\operatorname{Dom}f=\operatorname{Dom}g=:D$ and $f(x)=g(x)$ for all $x\in D$, and we write $f\leq g$ if $f(x)\leq g(x)$ for all $x\in D$. For $f\in\mathbb{R}_{\pm\infty}^{X}$, $g\in\mathbb{R}_{\pm\infty}^{Y}$ and a set $S$, we write ‘‘$f=g$ on $S$ ’’ or ‘‘$f\leq g$ on $S$ ’’ if f\bigm{|}_{S\cap D}=g\bigm{|}_{S\cap D} or f\bigm{|}_{S\cap D}\leq g\bigm{|}_{S\cap D} respectively.

For $f\in F\subset\mathbb{R}_{\pm\infty}^{X}$, we set $f^{+}\colon x\mapsto\max\{0,f(x)\}$, $x\in\operatorname{Dom}f$, $F^{+}:=\{f\geq 0\colon f\in F\}$. So, $f$ is positive on $X$ if $f=f^{+}$, and we write ‘‘$f\geq 0$ on $X$’’.

The class $\operatorname{sbh}(O)$ contains the minus-infinity function $\boldsymbol{-\infty}\colon x\mapsto-\infty$ identically equal to $-\infty$;

[TABLE]

If $o\notin O\ni\infty$, then we can to use the inversion in the sphere $\partial B(o,1)$ centered at the point $o\in\mathbb{R}^{d}$:

[TABLE]

For a subset $S\subset\mathbb{R}_{\infty}^{d}$, the classes $\operatorname{har}(S)$ and $\operatorname{sbh}(S)$ consist of the restrictions to $S$ of harmonic and subharmonic functions in some (in general, its own for each function) open set $O\subset\mathbb{R}_{\infty}^{d}$ containing $S$. The class $\operatorname{sbh}_{*}(S)$ are defined like previous class (2.5).

By ${\rm const}_{a_{1},a_{2},\dots}\in\mathbb{R}$ we denote constants, and constant functions, in general, depend on $a_{1},a_{2},\dots$ and, unless otherwise specified, only on them, where the dependence on dimension $d$ of $\mathbb{R}_{\infty}^{d}$ will be not specified and not discussed; ${\rm const}^{+}_{\dots}\geq 0$.

3 General Gluing Theorems

Gluing Theorem 1**.**

Let $O$ and $O_{0}$ be a pair of open subsets in $\mathbb{R}^{d}$, $v\in\operatorname{sbh}(O)$ and $v_{0}\in\operatorname{sbh}(O_{0})$ be a pair of functions such that

[TABLE]

Then the function

[TABLE]

is subharmonic on $O_{0}\cup O$.

Proof.

It is enough to apply Gluing Theorem A twice:

1. [O0]

to one pair of functions

[TABLE]

under condition (3.10) realizing condition (1.1); 2. [O]

to another pair of functions

[TABLE]

under condition (3.11) realizing condition (1.1).

These two glued subharmonic functions coincide at the open intersection $O\cap O_{0}$ and give subharmonic function $V$ defined in (3.2).

∎

Gluing Theorem 2** (quantitative version).**

Let $O$ and $O_{0}$ be a pair of open subsets in $\mathbb{R}^{d}$, and $v\in\operatorname{sbh}(O)$ and $g\in\operatorname{sbh}(O_{0})$ be a pair of functions such that

[TABLE]

If we choose the function

[TABLE]

then the function $V$ from (3.2) is subharmonic on $O_{0}\cup O$.

Proof.

The function $v_{0}$ from definition (3.6) is subharmonic on $O_{0}$ since this function $v_{0}$ has a form ${\rm const}^{+}g+{\rm const}$ with ${\rm const}^{+}\in\mathbb{R}^{+}$, ${\rm const}\in\mathbb{R}$. In addition, by construction (3.6), for each $x\in O_{0}\cap\partial O$, we obtain

[TABLE]

Thus, we have (3.10). Besides, by construction (3.6), for each $x\in O\cap\partial O_{0}$, we obtain

[TABLE]

Thus, we have (3.11), and our Gluing Theorem 2 follows from Gluing Theorem 1. ∎

Remark 1**.**

Theorems of this section can be easily transferred to the cone of plurisubharmonic functions [9, Corollary 2.9.15]. We sought to formulate our theorems and their proofs with the possibility of their fast transport to the plurisubharmonic functions and to abstract potential theories with more general constructions based on the theories of harmonic spaces and sheaves in the spirit of books [4], [3], [5], [2], [1], etc.

4 Gluing with the Green Function

Definition 1** ([14], [7], [12]).**

For $q\in\mathbb{R}$, we set

[TABLE]

Reminder, that a set $E\subset\mathbb{R}^{d}$ is called polar if there is a function $u\in\operatorname{sbh}_{*}(\mathbb{R}^{d})$ such that

[TABLE]

where the set $(-\infty)_{u}$ is minus-infinity $G_{\delta}$-set for the function $u$,

[TABLE]

is the outer capacity of $S\subset\mathbb{R}^{d}$.

Let $\mathcal{O}$ be an open proper subset in $\mathbb{R}_{\infty}^{d}$.

Consider a point $o\in\mathbb{R}^{d}$ and subsets $S_{0},S\subset\mathbb{R}_{\infty}^{d}$ such that

[TABLE]

Let $D$ be a domain in $\mathbb{R}_{\infty}^{d}$ such that

[TABLE]

Such domain $D$ possesses the generalized Green’s function $g_{D}(\cdot,o)$ (see [7, 5.7.2], [8, Ch. 5, 2]) with pole at the point $o\overset{\eqref{Dg}}{\in}D$ described by the following properties:

[TABLE]

Properties (4.5) for the generalized Green’s function $g_{D}(\cdot,o)$ from (4.3)–(4.4) are well known [14, 4.4], [7, 5.7], and property (4.3) follows from 0<g(\cdot,o)\in C\bigl{(}D\setminus\{o\}\bigr{)} on $D\setminus\{o\}$.

Gluing Theorem 3**.**

Under conditions (4.3) suppose that a function $v\in\operatorname{sbh}(\mathcal{O}\setminus S_{0})$ satisfy constraints above and below in the form

[TABLE]

Every domain $D$ with inclusions (4.4) possesses the generalized Green’s function $g_{D}(\cdot,o)$ with pole $o\in\operatorname{int}S_{0}$, properties (4.5) and constant $M_{g}$ of (4.5M) such that the choice of function

[TABLE]

Proof.

It is enough to apply Gluing Theorem 2 with

[TABLE]

according to the references written above the relations in (4.6)–(4.7). ∎

Given $S\subset\mathbb{R}^{d}$ and $r\in\mathbb{R}^{+}$, a set

[TABLE]

is called a outer $r$-parallel set [15, Ch. I,§ 4]. Easy to install the following

Proposition 1**.**

Let a subset $S_{0}\Subset\mathbb{R}^{d}$ be connected, and $r\in\mathbb{R}_{*}^{+}$. Then $S^{\cup r}$ is connected, $S_{0}\Subset S_{0}^{\cup(r/3)}$, and there is a regular for the Dirichlet problem domain $D\subset S_{0}^{\cup r}$ such that

[TABLE]

For v\in L^{1}\bigl{(}\partial B(x,r)\bigr{)}, we define the averaging value of $v$ at the point $x$ as

[TABLE]

where $\sigma_{d-1}$ the normalized by $1$ surface measure on the sphere $\partial B(x,r)$.

Gluing Theorem 4**.**

Let $\mathcal{O}\subset\mathbb{R}^{d}$ be an open subset, and $S_{0}\subset\mathbb{R}^{d}$ be a connected set such that there is a point

[TABLE]

Let $r\in\mathbb{R}^{+}$ be a number such that

[TABLE]

and $D$ be a domain from Proposition 1 satisfying (4.9). Let $v\in\operatorname{sbh}_{*}(\mathcal{O}\setminus S_{0})$ be a function satisfying constraints above and below in the form

[TABLE]

Then $m_{v}>-\infty$, and there is a function $V\in\operatorname{sbh}_{*}(\mathcal{O}\setminus\{o\})$ satisfying (4.7h)–(4.7+), i. e.,

[TABLE]

Proof.

We have $m_{v}>-\infty$ since the function $v^{\circ(r/3)}$ is continuous in $S_{0}^{\cup(2r/3)}\setminus S_{0}^{\cup(r/3)}$ [8, Theorem 1.14]. The function $v$ can be transformed using the Perron – Wiener – Brelot method (into the open ‘‘layer’’ $S_{0}^{\circ r}\setminus\operatorname{clos}S_{0}$ from boundary of this layer) to a new subharmonic function $\widetilde{v}\geq v$ on $\mathcal{O}\setminus S_{0}$ such that $\widetilde{v}\in\operatorname{har}(S_{0}^{\circ r}\setminus\operatorname{clos}S_{0})$ and $\widetilde{v}=v$ on $\mathcal{O}\setminus\operatorname{clos}S$. This follow from the principle of subordination (domination) for harmonic continuations and the maximum principle that

[TABLE]

If we choose in Gluing Theorem 3 for the role a set $S_{0}$ the set $S_{0}^{\cup(r/3)}$, and instead of $S$ the set $S_{0}^{\cup(2r/3)}$, then, by construction (4.7v)–(4.7V) and conditions (4.7h)–(4.7o) , we get series of conclusions (4.14) of Theorem 4. ∎