| Sbornik: Mathematics |
|
|
|
|
|
This article is cited in 1 scientific paper (total in 1 paper) Integrals of a difference of subharmonic functions against measures and the Nevanlinna characteristic B. N. Khabibullinab a Faculty of Mathematics and Information Technologies, Bashkir State University, Ufa, Russia b Institute of Mathematics with Computing Centre, Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa, Russia
Bibliographic databases:
     § 1. Introduction1.1. The aim and the originsOur main task is to find upper estimates for the integrals of a difference of two subharmonic functions against Borel measures on subsets of the complex plane $\mathbb C$ or finite-dimensional Euclidean space in terms of the product of the Nevanlinna characteristic and certain numerical parameters of the measure and/or its support. In the case of $\mathbb C$ they produce new inequalities for integrals of the logarithm of the modulus of a meromorphic function against certain measures. Our starting point is the Edrei-Fuchs small arcs lemma (see [1]), which has important applications in the theory of meromorphic functions. These were reflected, for instance, in [2], Ch. I, Theorem 7.4. We reproduce the statement of this lemma from [2], but we use the notation $\lambda_{\mathbb R}$, rather than the widely used notation $\operatorname{mes}$ (see [2]–[6] and other papers), for the linear Lebesgue measure on $\mathbb R$. For a meromorphic function $f\neq \infty$ in the unit disc
$$
\begin{equation}
D(R):=\bigl\{z\in \mathbb C\mid |z|<R\bigr\}\subset \mathbb C
\end{equation}
\tag{1.1}
$$
of radius $R\in \overline{\mathbb R}^+:= \mathbb R^+\,{\cup}\, \{+\infty\}$, where $\mathbb R^+:=\{x\in \mathbb R\mid x\geqslant 0\}$ is the positive ray of $\mathbb R$, its Nevanlinna characteristic is the function on $[0,R):=\{r\in \mathbb R^+\mid r<R\}$ taking values on the extended real line $\overline{\mathbb R}:=\overline{\mathbb R}^+\cup (-\overline{\mathbb R}^+)$ which is denoted and defined (following [2]) by
$$
\begin{equation}
T(r, f):=m(r,f)+N(r,f) \quad\text{for } 0\leqslant r<R,
\end{equation}
\tag{1.2}
$$
where
$$
\begin{equation}
m(r,f)=\frac{1}{2\pi}\int_0^{2\pi} \ln^+|f(re^{i\varphi})|\,\mathrm{d} \varphi, \qquad \ln^+x:=\max\{0, \ln x\},\quad x\in \mathbb R^+,
\end{equation}
\tag{1.3}
$$
and
$$
\begin{equation}
N(r,f):=\int_{0}^{r}\frac{n(t,f)-n(0,f)}{t}\,\mathrm{d} t+n(0,f)\ln r, \qquad \ln 0:=-\infty;
\end{equation}
\tag{1.4}
$$
$n(r,f)$ is the number of poles of $f$ in the closed disc $\overline D(r):=\{z\in \mathbb C\mid |z|\leqslant r\}\subset \mathbb C$, counted with multiplicities, where we agree that $0\cdot \ln 0:=0$. The following result, going back to the early 1960s, is often called the Edrei-Fuchs lemma (or theorem) on small arcs. Theorem 1 (see [1], Lemma III, and [2], Theorem 7.3). Let $f$ be a meromorphic functions, $k$ and $\delta$ be numbers such that $k>1$ and $0< \delta \leqslant 2\pi$, and let $r>1$. Then there exists a constant $c_1(k,\delta)$ such that for any $\lambda_{\mathbb R}$-measurable set $E_r\subset [-\pi,\pi]$ such that $\lambda_{\mathbb R}(E_r)=\delta$, where $c_1(k,\delta)=\dfrac{6k}{k-1}\delta \ln \dfrac{2\pi e}{\delta}\to 0$ as $\delta \to 0$ for fixed $k$. The key point in Theorem 1 is the order of smallness of $c_1(k,\delta)$ as $\delta \to 0$. The questions of extending this lemma to differences of subharmonic functions subject to the same integration and of dropping the condition $r>1$ by adding a term of order $O(\ln r)$ to the right-hand side of (1.5) was discussed by Girnyk (see [7], Theorem E). The Edrei-Fuchs small arcs lemma is just the starting point of our investigations. We discuss the contents of the paper and prospects for applications of our results in greater detail in § 1.5. 1.2. A synopsis on meromorphic functionsHere are some illustrations to results in this paper as applied to meromorphic functions in a neighbourhood of the closed disc $\overline D(R)\subset \mathbb C$ of radius $R>0$ (that is, on $\overline D(R)$), which all the more hold for meromorphic functions on $\mathbb C$. Let $\overline D_z(r):=z+\overline D(r)\subset \mathbb C$ denote the closed disc with centre $z\in \mathbb C$ and radius $r\in \mathbb R^+$. For a Borel measure $\mu$ on $\mathbb C$ set
$$
\begin{equation}
\mu^{\operatorname{rad}}_z(r):=\mu(\overline D_z(r))\quad\text{and} \quad {\mathrm N}^{\mu}_z(r):=\int_0^r\frac{\mu^{\operatorname{rad}}_z(t)}{t}\,\mathrm{d} t, \qquad r\in \mathbb R^+.
\end{equation}
\tag{1.6}
$$
Main Corollary. Let $0<r\in \mathbb R^+$, let $\mu$ be a Borel measure on $\overline D(r)\subset \mathbb C$. Then the following two assertions are equivalent. I. There exist $r_0>0$ and $R>r$ such that $\sup_{z\in \overline D(R)}{\mathrm N}_z^{\mu}(r_0)<+\infty$. II. For each $R>r$ and each meromorphic function $f\neq \infty$ on $\overline D(R)$ the function $\ln^+|f|$ is $\mu$-integrable and It is obvious from the definition (1.4) that $N(r, f)$ is an increasing function, so in the first inequality in assertion II we can replace $r$ in $N( r, f)$ by any $r'\in [0,r]$ and for $r'\neq 0$ the right-hand side will still be finite. Furthermore, for $r\geqslant 1$ or $n(0,f)=0$, that is, for $f(0)\in \mathbb C$ we have $N( r, f)\stackrel{(1.4)}{\geqslant}0$, and thus we can remove the subtrahend $N( r, f)$ in parentheses from the inequality. As assertion II states that the middle part of the inequality is finite, assertion I follows from II in an obvious way by taking $r_0:=R$. That assertion II follows from I is a consequence of the Main Theorem in § 3, which states that five assertions on integrals against measures in the disc $\overline D(r)$ or ball (in the multidimensional case) of a difference of two subharmonic functions are equivalent. For a function $h\colon \mathbb R^+\to \mathbb R^+$ we define the Hausdorff $h$-content of a set in $\mathbb C$ to be the infimum of the sums of the values of the function $h$ at the radii of various systems of discs covering this set. Corollary 1. Let $\mu$ be a Borel measure on the disc $\overline D(r)\subset \mathbb C$ with total mass $M:=\mu(\overline D(r))$, and let $f\neq\infty$ be a meromorphic function on the disc $\overline D(R)$ of radius $R>r$. I. If the modulus of continuity ${\mathrm h}_{\mu}\colon t\mapsto \sup_{z\in D(R)} \mu_z^{\operatorname{rad}}(t)$ of $\mu$ satisfies the Dini condition at zero, namely,
$$
\begin{equation}
\int_0 \frac{{\mathrm h}_{\mu}(t)}{t}\,\mathrm{d} t< +\infty,
\end{equation}
\tag{1.7}
$$
then the following Lebesgue integral exists:
$$
\begin{equation}
\int_{\overline D(r)} \ln^+|f|\,\mathrm{d} \mu \leqslant 5\frac{R+r}{R-r} \bigl(T(R, f)-N(r, f)\bigr) \biggl(M+ \int_0^{r}\frac{{\mathrm h}_{\mu}(t)}{t}\,\mathrm{d} t\biggr);
\end{equation}
\tag{1.8}
$$
here the right-hand side is finite, and the total mass $M$ of $\mu$ on the right-hand side can be replaced by the Hausdorff ${\mathrm h}_{\mu}$-content of the support $\operatorname{supp} \mu$ of $\mu$. II. If $h\colon [0,r]\to \mathbb R^+$ is a continuous function satisfying $h(0)=0$ and differentiable on the open interval $(0,r)$,
$$
\begin{equation}
{{\mathrm s}_h}:=\sup_{t\in (0,r)}\frac{h(t)}{th'(t)}<+\infty,
\end{equation}
\tag{1.9}
$$
and ${\mathrm h}_{\mu}(t)\leqslant h(t)$ for all $t\in [0,r]$, then the following Lebesgue integral exists and satisfies
$$
\begin{equation}
\int_{\overline D(r)} \ln^+|f|\,\mathrm{d} \mu \leqslant 5\frac{R+r}{R-r}\bigl(T(R, f)-N(r, f)\bigr)M \ln\frac{e^{1+{\mathrm s}_h}r}{h^{-1}(M)};
\end{equation}
\tag{1.10}
$$
here on the right-hand side $M$ can be replaced by the Hausdorff $ h$-content of the support $\operatorname{supp} \mu$. Moreover, $r$ in $N(r,f)$ can be replaced by any $r'\in [0,r]$ in (1.8) and (1.10), while if $f(0)\in \mathbb C$ or $r\geqslant 1$, then the term $-N(r,f)$ can be removed from (1.8) and (1.10) altogether. We verify Corollary 1 in several steps in the form of comments to our theorems. Here from part II of the corollary we deduce the proof of the Edrei-Fuchs small arcs lemma. Proof of Theorem 1. Consider the meromorphic function on the unit circle $\partial \overline D(1)$ and let $\sigma$ be the arc length measure on $\partial \overline D(1)$. The integral on the left-hand side of (1.5) is precisely the integral of $\ln^+|f_r|$ with respect to the restriction $\sigma_\llcorner$ of $\sigma$ to the set $e^{iE_r}:=\{e^{i\theta}\mid \theta \in E_r\}\subset \partial \overline D(1)$. The condition $\lambda_{\mathbb R}(E_r)=\delta$ means that the total mass of this restriction $\sigma_\llcorner$ is $\delta=:M$. It is elementary geometry that for each disc $\overline D_z(t)$ we have $\sigma(\overline D_z(t))\leqslant \pi t$ . Hence for the modulus of continuity ${\mathrm h}_{\sigma_\llcorner}$ of the restriction $\sigma_\llcorner$ we all the more have ${\mathrm h}_{\sigma_\llcorner}(t)\leqslant \pi t$ for $t\in \mathbb R^+$. Thus conditions (1.9) with ${\mathrm s}_h=1$ and $h^{-1}(x)=\frac{1}{\pi}x$ hold for $h(t)\equiv \pi t$. Taking $r=1$ and $k>1$ in place of $R>1$ in Corollary 1, from (1.10) in part II we obtain
$$
\begin{equation*}
\begin{aligned} \, \int_{E_{r}} \ln^+|f(re^{i\varphi})|\,\mathrm{d} \varphi &\stackrel{(1.11)}{=}\int_{\overline D(1)} \ln^+|f_r|\,\mathrm{d} \sigma_\llcorner \\ &\stackrel{(1.10)}{\leqslant} 5\frac{k+1}{k-1}\, T(k, f_r) \delta\ln\frac{e^2}{\frac{1}{\pi}\delta}= 5\frac{k+1}{k-1} \delta\ln\frac{\pi e^2}{\delta} T(kr, f). \end{aligned}
\end{equation*}
\notag
$$
The proof is complete. In the above inequality we see the same order of smallness $\delta\ln \frac{1}{\delta}$ as $\delta\to 0$ for fixed $k>1$ as in the Edrei-Fuchs small arcs lemma. The absolute constants are slightly greater due to the significantly more general nature of Corollary 1. 1.3. On integrals involving the maximum modulus of a meromorphic functionIn the case when we only integrate over sets lying on the positive half-axis $\mathbb R^+$, one can find upper estimates for integrals of $\ln^+ M(r,f)$ for meromorphic functions $f$ in terms of $T(r,f)$, where
$$
\begin{equation}
M(r,f):=\sup\bigl\{|f(z)| \mid |z|=r\bigr\},\qquad r\in \mathbb R^+.
\end{equation}
\tag{1.12}
$$
Taking results of this kind into consideration, we can say that a classical result of R. Nevanlinna preceded the Edrei-Fuchs lemma. Theorem 2 (see [8] and [2], Ch. I, Theorem 7.2; also see the discussion in [6], the introduction, § 1.1). Let $1<k\in \mathbb R^+$ and $0<r_0\in \mathbb R^+$. Then there exists $c_0(k)\in \mathbb R^+$ such that for each meromorphic function $f\neq \infty$ on $\mathbb C$ A ‘straightened’ version of the Edrei-Fuchs small arcs lemma is given by the following small-intervals lemma due to Grishin and Sodin. Theorem 3 (see [3], Lemma 3.1). There exists $C\in \mathbb R^+$ such that for any meromorphic function $f\neq \infty$, any $r>1$ and $k>1$ and any $\lambda_{\mathbb R}$-measurable subset of $E\subset [1,r)$ Remark 1. Estimates, as in Nevanlinna’s theorem and Theorem 3, for integrals of the logarithm of the maximum modulus (1.12) of a meromorphic function $f$ over subsets of the positive half-axis are a separate special problem, which relies heavily on the fact that the sets $E$ of integration lie in $\mathbb R^+$ and involves a composition of two operations: taking the supremum over circles as in (1.12) and integrating over a part of $\mathbb R^+$. In view of the first operation, such results cannot be derived directly from inequalities for integrals of the functions $\ln^+ |f|$ themselves, so it must be investigated separately. We carried out these studies in [9], but we do not discuss them here and cannot use them. 1.4. Preceding results on subharmonic functionsIn a joint paper Grishin and Malyutina proved (see [4], Theorem 8) and used repeatedly (see [4], Theorems 2 and 4) a version of the Grishin-Sodin lemma for subharmonic functions of formal proximate order $\boldsymbol \rho$ in the sense of Valiron (see [10], Ch. III, § 6, [11], Ch. I, § 12, [2], Ch. II, § 2, [12], § 7.4), which can be defined in an equivalent way by a single condition, as a differentiable function $\boldsymbol \rho\geqslant 0$ on $\mathbb R^+\setminus \{0\}$ such that the finite limit
$$
\begin{equation*}
\rho :=\lim_{r\to +\infty}r(\boldsymbol \rho(r)\ln r)'\in \mathbb R^+
\end{equation*}
\notag
$$
exists. (This was also mentioned in the corollary in [13], and, for much more general proximate, or model, growth functions, in the theorem in [13].) We call the following result the Grishin-Malyutina small intervals theorem. Theorem 4 (see [4], Theorem 8). Let $v\not\equiv -\infty$ be a subharmonic function on $\mathbb C$ such that $\sup_{z\in \mathbb C}v(z)|z|^{-\boldsymbol \rho (|z|)}<+\infty$ for some proximate order $\boldsymbol \rho $. Then there exists $C\in \mathbb R^+$ such that for any $r>1$ and any $\lambda_{\mathbb R}$-measurable set $E\subset [1,r)$ The Grishin-Malyutina small intervals theorem is an easy consequence of results obtained by Gabdrakhmanova and this author in a recent paper (see [5], § 1.3), where this theorem was extended to arbitrary subharmonic functions on $\mathbb C$. For an extended number-valued function $v$ on the circle $\partial \overline D(r)$ taking values in $\overline{\mathbb R}$ let
$$
\begin{equation}
{\mathrm M}_v(r):=\sup_{0\leqslant \theta < 2\pi } v(re^{i\theta})
\end{equation}
\tag{1.16}
$$
be the maximum radial characteristic of $v$ and
$$
\begin{equation}
{\mathrm C}_v(r):=\frac{1}{2\pi}\int_0^{2\pi} v(re^{i\theta})\,\mathrm{d} \theta
\end{equation}
\tag{1.17}
$$
be the mean value of $v$ on this circle, provided that the integral converges. Then we have the following theorem on small intervals. Theorem 5 (see [5], Theorem 1). There exists $a\geqslant 1$ such that for an arbitrary subharmonic function $u\not\equiv -\infty$ on $\mathbb C$, and numbers $b\in (0,1]$ and ${0\,{\leqslant}\,r_0\,{\leqslant}\,r\,{<}\,R\,{<}\,+\infty}$, any $\lambda_{\mathbb R}$-measurable set $E\subset [r,R]$, and any function $g\colon E\to \overline{\mathbb R}$ with essential supremum Another general result in [6], the main theorem, can be called a theorem on small intervals with $L^p$-weight. To state it we need some preliminary work. For a Borel measure $\mu$ on $\overline D(R)$ set
$$
\begin{equation}
\mu^{\operatorname{rad}}(r):=\mu(\overline D(r))\stackrel{(1.6)}{=} \mu_0^{\operatorname{rad}}(r)\in \overline{\mathbb R}^+, \qquad r\in \mathbb R^+,
\end{equation}
\tag{1.20}
$$
and
$$
\begin{equation}
{\mathrm N}_{\mu}(r,R):=\int_{r}^{R}\frac{\mu^{\operatorname{rad}}(t)}{t}\,\mathrm{d} t \stackrel{(1.6)}{=}{\mathrm N}_0^{\mu}(R)-{\mathrm N}_0^{\mu}(r) \in \overline{\mathbb R}^+, \qquad 0\leqslant r<R\leqslant +\infty,
\end{equation}
\tag{1.21}
$$
where the last equality holds only when ${\mathrm N}_0^{\mu}(R)$ is finite. A ‘plus’ subscript on a number or a function means its positive part, while the ‘minus’ subscript means the negative part of the quantity. Let $U=u-v$ be the difference of subharmonic functions $u\not\equiv -\infty$ and $ v\not\equiv -\infty$ in a neighbourhood of the closed disc $\overline D(R)$, which have Riesz measures $\varDelta_u\geqslant 0$ and $\varDelta_v\geqslant 0$, respectively, so that $U$ is a nontrivial ($\not\equiv\pm\infty$) $\delta$-subharmonic function (see [14], [15], [16] and [17], § 3.1) with Riesz charge $\varDelta_{U}=\varDelta_u-\varDelta_v$. In [6] and [18] we used the following difference Nevanlinna characteristic of $U$:
$$
\begin{equation}
{\mathrm T}_U(r,R)={\mathrm C}_{U^+}(R)-{\mathrm C}_{U^+}(r)+ {\mathrm N}_{\varDelta_U^-}(r,R) , \qquad 0<r< R\in \mathbb R^+,
\end{equation}
\tag{1.22}
$$
where the Borel measure $\varDelta_U^-\geqslant 0$ is the lower variation of the Riesz charge $\varDelta_{U}$. For a $\lambda_{\mathbb R}$-measurable set $E\subset \mathbb R$ and a $\lambda_{\mathbb R}$-measurable function $g\colon E\to \overline{\mathbb R}$, along with the essential supremum $\|g\|_{\infty}$ of $g$ on $E$ (see (1.18)) we also use the $L^p$-seminorm of $g$ on $E$:
$$
\begin{equation*}
\|g\|_p:=\biggl(\int_E |g|^p\,\mathrm{d}\lambda_{\mathbb R}\biggr)^{1/p} \quad\text{for } 1\leqslant p\in \mathbb R^+ \quad\text{on } E.
\end{equation*}
\notag
$$
Theorem 6 (see [6], the main theorem). For ${0< r_0< r\in\mathbb R^+}$, ${1<k\in\mathbb R^+}$, ${1<p\leq+\infty}$, ${1/p+1/q=1}$, ${1\leqslant q<+\infty}$, an $\lambda_{\mathbb R}$-measurable set $E\subset [0,r]$ and a $\lambda_{\mathbb R}$-measurable function $g\colon E\to \overline{\mathbb R}$, let $U\not\equiv \pm\infty$ be a $\delta$-subharmonic function on $\mathbb C$ and $u\not\equiv -\infty$ be a subharmonic function on $\mathbb C$. Then and In is easy to see that for $p = + \infty$ this theorem on small intervals with $L_p$-weight contains and refines Theorems 2 and 3 and Theorems 4–7 alike. However, the Edrei-Fuchs small arcs lemma cannot be deduced from it, neither can be deduced the qualitatively new theorem on small planar sets, which also covers sets of small Lebesgue measure $\lambda_{\mathbb C}$ in $\mathbb C$. Theorem 7 (see [18], Theorem 2). Let $0< r_0< r\in \mathbb R^+$ and $1<k\in \mathbb R^+$. Then for any $\delta$-subharmonic function $U\not\equiv \pm\infty$, any subharmonic function $u\not\equiv {-}\infty$ on $\mathbb C$, and any $\lambda_{\mathbb C}$-measurable set $E\subset \overline D(r)$, and 1.5. The contents and structure of the paper; prospects for applicationsAll final upper estimates for integrals and their proofs in this paper are performed for $\delta$-subharmonic functions and stated in terms of the modified difference Nevanlinna characteristic
$$
\begin{equation*}
\boldsymbol{T}_U(r,R)\stackrel{(1.22)}{:=}{\mathrm T}_U(r,R)+{\mathrm C}_{U^+}(r),
\end{equation*}
\notag
$$
which we discuss below in § 2. The equalities in § 2 which connect ${\boldsymbol T}_U$ with the classical Nevanlinna characteristic of meromorphic functions, allow us to adapt these upper estimate easily to the case of meromorphic functions and traditional notation (1.2)–(1.4) and (1.12), just as in the Main Corollary and Corollary 1. The Main Theorem, which is stated and proved in § 3, has a more extended form of a criterion than the Main Corollary to it which is stated in § 1.2; it contains five equivalent statements. Theorems 8 and 9 in § 4, which are based on the concept of modulus of continuity of a Borel measure $\mu$ (see Definition 2) and on a priori bounds for this modulus in terms of a sufficiently general function $h$, enable us to put the right-hand sides of the integral estimates in the Main Theorem into a clear explicit form. In § 5 we estimate integrals of $U^+$ against measures which are majorized by the Hausdorff $h$-contents or $h$-measures of the possibly fractal supports: in particular, by the $p$-dimensional Hausdorff contents or measures. Very special and extreme examples of such measures are given by the linear Lebesgue measure $\lambda_{\mathbb R}$ on $\mathbb R$, which coincides with the one-dimensional Hausdorff measure, and by the plane Lebesgue measure $\lambda_{\mathbb C}$ on $\mathbb C$, which coincides with the two-dimensional Hausdorff measure. Only these measures have been considered previously. The results in § 5 rely significantly on Frostman’s classical theorem in potential theory. Theorem 11 shows that in integral estimates qualitative parameters of the measure of integration $\mu$ can always be replaced by the Hausdorff $h$-contents and $h$-measures of the support $\operatorname{supp} \mu$ on the right-hand sides of integral bounds. Furthermore, it follows from Theorem 12 that for each choice of the support $S$ of the measure we can find measures $\mu$ such that estimates in terms of $\mu$ itself (as in the Main Theorem and Theorems 8 and 9) and estimates in terms of the Hausdorff $h$-contents and $h$-measures of the support $\operatorname{supp} \mu=S$ (as in Theorem 11) are equivalent to within constants depending only on the dimension of the space. In § 6 we collect particular cases of our main results and their consequences of independent interest. In § 6.1 we consider separately cases when we can give estimates just in terms of the $p$-dimensional Hausdorff contents or measures of the support of $\operatorname{supp} \mu$; these are quite special, but the most common cases of Hausdorff $h$-contents and $h$-measures. In § 6.2 we present estimates for functions on the whole complex plane or the whole space. In § 6.3 we present similar estimates for functions on the unit disc or ball; to our knowledge, this has never been done in the context of estimates for integrals of meromorphic or $\delta$-subharmonic functions in terms of the Nevanlinna characteristic. In § 6.4 we limit ourselves to the restrictions of the plane Lebesgue measure $\lambda_{\mathbb C}$ or the space Lebesgue measure in the multidimensional case to certain subsets taken as the measure $\mu$; as explained in Remark 6, this improves the theorem on small planar sets in § 1.4. In § 6.5, as a significant immediate development of the Edrei-Fuchs small arcs lemma, we present Corollaries 7 and 8 on integration over subsets of bi-Lipschitz curves or curves with bounded slope, and, as a direct generalization of these results, we also state Corollary 9 on integration over subsets of bi-Lipschitz hypersurfaces, which concludes the paper. The importance of new integral estimates for meromorphic functions and differences of subharmonic functions that we obtain here does not reduce to covering internal needs of Nevanlinna theory. They can also be used in other questions relating to the growth theory of meromorphic and (pluri)subharmonic functions, and in applications of this theory. For instance, our integrals bounds are closely connected with estimating holomorphic or subharmonic functions from below outside small exceptional subsets of a disc, the plane $\mathbb C$, a ball or a space, which is a subject of whole chapters in some classical monographs (for instance, Chs. 6 and 7 in [19]). Such lower estimates are used in various ways in function theory and its applications. Moreover, as justly noted by a referee, the integral estimates in this paper are even equivalent to lower estimates. The transition from lower estimates to integral inequalities was reflected, for instance, by the methods used in the proofs in [4] and [5], while the idea of a reverse transition was touched upon in a certain form in [20], Theorem 2. Another kind of weak-type estimate for meromorphic functions $f$ on a disc or the plane $\mathbb C$ can be extracted from combining Theorems 11 and 12. In the zeroth approximation these estimates show that, on a fractal set $E\subset D(R)$ of dimension $p>0$ whose $p$-dimensional Hausdorff measure is distinct from zero, we can always find a point $z\in E$ at which $\ln |f(z)|$ does not exceed the product of the Nevanlinna characteristic $T(r,f)$ and the logarithm of the reciprocal value of the Hausdorff $p$-measure of $E$. For holomorphic functions on a disc or $\mathbb C$, as well as for subharmonic functions on a ball or the space, this enables one to have similar lower estimates at a point in $E$ in terms of the maximum values of the function on circles or spheres. We have discarded our initial intention to include such results on lower bounds and weak-type estimates on fractal sets in this paper because a full-scale analysis of such questions must be carried out separately in a natural combination with results on integral estimates involving the integrals of $M(r,f)$ discussed in § 1.3. Neither do we consider integral inequalities for plurisubharmonic functions or differences of such functions, although we have in fact laid a considerable basis for such an expansion, because the constants in estimates depend in fact only on the dimension. Thus from meromorphic functions of several variables and plurisubharmonic functions in a ball or the space we can go over to their restrictions to complex lines or subspaces through the origin, and then we can give uniform integral estimates on each line or subspace and average over spheres so as to preserve the form of integral estimates. Such integrals estimates can be used directly to extend Liouville-type theorems, considered in [21] and [22], on the global boundedness or constancy of entire, holomorphic or (pluri)subharmonic functions on a disc, the plane, a ball or space, provided that they have bounded growth outside small sets considered in [21] and [22]. AcknowledgementI am deeply grateful to the referees for their valuable and useful observations, which contributed to a number of refinements of the presentation of results and to a broader view of our subject. § 2. Nevanlinna difference characteristicGiven a meromorphic function $f\neq 0,\infty$ on $\mathbb C$, the logarithm of its modulus $\ln |f|\not\equiv \pm \infty$ is a $\delta$-subharmonic function on ${\mathbb{C}}$. The following relationships between standard characteristics of $f$ and the characteristics of $\delta$-subharmonic functions which we introduced in § 1.4 are obvious:
$$
\begin{equation}
\ln M(r, f) \stackrel{(1.12),(1.16)}{=}{{\mathrm M}}_{\ln|f|}(r), \qquad r\in {\mathbb{R}}^+,
\end{equation}
\tag{2.1}
$$
$$
\begin{equation}
m(r, f) \stackrel{(1.3),(1.17)}{=}{\mathrm C}_{\ln^+|f|}(r), \qquad r\in {\mathbb{R}}^+,
\end{equation}
\tag{2.2}
$$
$$
\begin{equation}
N(R, f)-N(r, f) \stackrel{(1.4),(1.21)}{=} {\mathrm N}_{\varDelta_{\ln|f|}^-}(r,R), \qquad 0<r< R\in \mathbb R^+,
\end{equation}
\tag{2.3}
$$
$$
\begin{equation}
T(R, f)-T(r, f) \stackrel{(1.2),(1.22)}{=}{\mathrm T}_{\ln|f|}(r,R), \qquad 0<r<R\in {\mathbb{R}}^+.
\end{equation}
\tag{2.4}
$$
In view of the form of the right-hand sides of (1.23) and (1.25) in what follows it will be more convenient to use a modified difference Nevanlinna characteristic, which can be defined in terms of the previous version of difference Nevanlinna characteristic ${\mathrm T}_{U}$ introduced in (1.22) by
$$
\begin{equation}
\begin{aligned} \, \notag {\boldsymbol T}_U(r,R) &:={\mathrm T}_{U}(r,R)+{\mathrm C}_{U^+}(r) \\ &\!\!\stackrel{(1.22)}{=}{\mathrm C}_{U^+}(R)+{\mathrm N}_{\varDelta_U^-}(r,R), \qquad 0<r<R\in \mathbb R^+. \end{aligned}
\end{equation}
\tag{2.5}
$$
Now we can define the Nevanlinna characteristic by
$$
\begin{equation}
{\boldsymbol T}_U(R):={\boldsymbol T}_U(0,R)\stackrel{(1.21)}{:=}{\mathrm C}_{U^+}(R)+{\mathrm N}_{\varDelta_U^-}(0,R)\in \overline{\mathbb R}^+.
\end{equation}
\tag{2.6}
$$
Then in view of (2.2), (2.5) and (2.3) we can replace (2.4) by
$$
\begin{equation}
T(R, f)-N(r, f)\stackrel{(2.6)}{=}{\boldsymbol T}_{\ln|f|}(r,R), \qquad 0< r<R\in {\mathbb{R}}^+.
\end{equation}
\tag{2.7}
$$
We go over to definitions for finite-dimensional Euclidean space. In the notation for one-point sets we still often remove curly brackets if this cannot lead to ambiguity. The extended number line $\overline{\mathbb R}:=\mathbb R\cup \{\pm \infty\}$ is the two-point compactification of $\mathbb R$ obtained by the addition of the two endpoints
$$
\begin{equation*}
\inf \mathbb R=:-\infty=:\sup \varnothing\quad\text{and}\quad \sup \mathbb R=:+\infty=:\inf \varnothing,
\end{equation*}
\notag
$$
where $\varnothing$ is an empty set; we supplement it by the order relations $-\infty \leqslant x\leqslant +\infty$ for all $x\in \overline{\mathbb R}$ and the operations $-(\pm\infty)=\mp\infty$, $|\pm\infty|:=+\infty$, $x\pm(\pm \infty)=+ \infty$ for $x\in \overline{\mathbb R}\setminus-\infty$, $x\pm (\mp \infty)=-\infty$ for $x\in \overline{\mathbb R}\setminus +\infty$, $x\cdot (\pm\infty):=\pm\infty=:(-x)\cdot (\mp\infty)$ for $x\in \overline{\mathbb R}^+\setminus 0$, ${\pm x}/{0}:=\pm\infty$ for $x\in \overline{\mathbb R}^+\setminus0$, ${x}/{\pm\infty}:=0$ for $x\in \mathbb R$; on the other hand
$$
\begin{equation}
0\cdot\pm \infty:=0=:\pm \infty \cdot 0, \quad \text{unless otherwise stated}.
\end{equation}
\tag{2.8}
$$
Only the following pairs of sums and differences and five operations of division are not well defined:
$$
\begin{equation*}
\nexists!\bigl( (\pm\infty)+(\mp\infty)\bigr), \quad \nexists!\,\bigl((\pm\infty)-(\pm \infty)\bigr), \quad \nexists!\,\frac{0}{0}, \quad \nexists!\,\frac{\pm\infty}{\pm\infty}, \quad \nexists!\,\frac{\pm\infty}{\mp\infty}.
\end{equation*}
\notag
$$
An interval $I$ of $\overline{\mathbb R}$ is a closed subset of $\overline{\mathbb R}$ with left-hand endpoint $\inf I$ and right-hand endpoint $\sup I$. As usual, $[a,b]:=\{x\in \overline{\mathbb R}\mid a\leqslant x\leqslant b\}$ is a closed interval, $(a,b]:=[a,b]\setminus a$ and $[a,b):=[a,b]\setminus b$ are half-open intervals, and $(a,b):=(a,b]\cap [a,b)$ is an open interval. We let $x^+:=\sup \{0,x\}$ denote the positive part of $x\in \overline{\mathbb R}$ and $x^-:=(-x)^+$ denote its negative part. In general, in what follows positivity corresponds to $\geqslant 0$ and negativity to $\leqslant 0$. If $0<x\in \overline{\mathbb R}$, then $x$ is strictly positive, and if $0>x\in \overline{\mathbb R}$, then $x$ is strictly negative. Given an extended number function $f\colon X\to \overline{\mathbb R}$, its value $f(x)$ at some point $x$ may not be defined in general, and its positive part $f^+\colon x\mapsto(f(x))^+$, $x\in X$, is defined at the same points as $f$ itself is. The same holds for its negative part $f^{-}:=(-f)^+$. The function $f$ is said to be positive on $X$ (and we write $f\geqslant 0$) if ${f=f^+}$. A function $f$ is increasing on $X\subset \overline{\mathbb R}$ if the relations $x'\in X$, $x\in X$ and $x'< x$ imply that $f(x')\leqslant f(x)$; it is strictly increasing if they imply the strict inequality $f(x')< f(x)$. The situation with decreasing functions is similar. Throughout, $n\in \mathbb N:=\{1,2,\dots\}$, but Euclidean space $\mathbb R^n$ has dimension $n\geqslant 2$. The Euclidean norm of a vector $x:=(x_1,\dots ,x_n)$ in $\mathbb R^n$ is $|x|:=\sqrt{x_1^2+\dots +x_n^2}$. As concerns the theory of measure and integration, we use the terminology in the monographs by Federer [23], Evans and Gariepy [24] and Landkof [25] (the introduction and § 1), but not their notation. For instance, an extended positive number function $\mu$ on the set of subsets of a sets $X$ is called an (outer) measure on $X$ if it is countably additive and $\mu(\varnothing)=0$. The concepts of a Borel measure and a regular measure on subsets of $\mathbb R^n$ are standard. A Radon measure is a regular Borel measure that take finite values on compact sets. We denote the restriction of a measure $\mu$ to $S\subset \mathbb R^n$ by $\mu{\lfloor}_S$. A measure $\mu$ is concentrated on a set $S\subset \mathbb R^n$ if $\mu(\mathbb R^n\setminus S)=0$. As usual, $\operatorname{supp} \mu$ is the support of the Borel measure $\mu$. We call differences of Radon measure charges, and denote their upper, lower and total variation by $\nu^+:=\sup\{\nu,0\}$, $\nu^-:=(-\nu)^+$ and $|\nu|:=\nu^++\nu^-$, respectively. As in [24], if an integral of a function against a measure $\mu$ exists and takes a value in $\overline{\mathbb R}$, then the function is said to be $\mu$-integrable; if this integrals is also finite, that is, takes a value in $\mathbb R$, then the function is said to be $\mu$-summable. The Stieltjes (Riemann-Stieltjes or Lebesgue-Stieltjes) integral over an interval with endpoints $a<b$ of a function $g$ with bounded variation on this interval is meant to be the integral over $(a,b]\subset \overline{\mathbb R}$ unless otherwise stated:
$$
\begin{equation*}
\int_a^b \dots \,\mathrm{d} g:=\int_{(a,b]} \dots \,\mathrm{d} g.
\end{equation*}
\notag
$$
We let $B_x(r):=\bigl\{y\in \mathbb R^n\mid |y-x|<r \bigr\}$, ${\overline B_x(r):=\bigl\{y\in \mathbb R^n\mid |y-x|\leqslant r \bigr\}}$ and $\partial \overline B_x(r):=\overline B_x(r)\setminus B_x(r)$ denote the open and closed balls and sphere of radius $r\in \mathbb R^+$ with centre $x\in \mathbb R^n$, respectively. Thus, $B_x(0)=\varnothing$ is an empty set, but $\overline B_x(0)=\partial \overline B_x(0)=\{x\}$. Furthermore, a ball can have radius $r=+\infty$; by definition $\overline B_x(+\infty):= B_x(+\infty):=\mathbb R^n$. For balls and spheres with centre at the origin we do not usually write the subscript $0$:
$$
\begin{equation}
B(r):=B_0(r), \qquad \overline B(r):= \overline B_0(r), \qquad \partial \overline B(r)=\partial \overline B_0(r).
\end{equation}
\tag{2.9}
$$
Throughout, we identify $\mathbb R^{2}$ and $\mathbb C\ni z=x+iy \longleftrightarrow (x,y)\in \mathbb R^{2}$, $x,y\in \mathbb R$; in the plane $B_z(r)=D_z(r)$, $\overline B_z(r)=\overline D_z(r)$ and $\partial \overline B_z(r)=\partial \overline D_z(r)$ are discs and circles with centre $z$, while $B(r)\stackrel{(1.1)}{=}D(r)$, $\overline B(r)=\overline D(r)$, $\partial \overline B(r)=\partial \overline D(r)$ are ones with centre at zero. We denote the area of the unit sphere $\partial \overline B(1)$ in $\mathbb R^n$ by
$$
\begin{equation}
s_{n-1}=\frac{2\pi^{n/2}}{\Gamma (n/2)}, \qquad s_{1}=2\pi, \quad s_{2}=4\pi, \quad s_{3}=\pi^2, \quad \dots\,.
\end{equation}
\tag{2.10}
$$
For the surface measure $\sigma_{n-1}^r$ on $ \partial \overline B(r)\subset \mathbb R^n$ and an integrable function $U$: ${\partial \overline B(r)\to \overline{\mathbb R}}$ with respect to $\sigma_{n-1}^r$ the mean value of $U$$\sigma_{n-1}^r$ on $\partial \overline B(r)$ is denoted similarly to (1.17) by
$$
\begin{equation}
{\mathrm C}_U(r):=\frac{1}{s_{n-1}r^{n-1}}\int_{\partial \overline B(r)}U\,\mathrm{d} \sigma_{n-1}^r.
\end{equation}
\tag{2.11}
$$
We often treat Borel measures $\mu$ on a Borel subset of $\mathbb R^n$ as measures extended to the whole of $\mathbb R^n$; as in (1.6),
$$
\begin{equation}
\mu_y^{\operatorname{rad}}(t):=\mu(\overline B_y(t))\in \overline{\mathbb R}^+,\qquad t\in \mathbb R^+,
\end{equation}
\tag{2.12}
$$
is the radial counting function of $\mu$ with centre $y\in \mathbb R^n$. For $y=0$ the subscript $0$ is normally suppressed, as in formula (1.20) for $\mathbb C$. The following integer related to the dimension $n\in \mathbb N$:
$$
\begin{equation}
\widehat{n}:=\max\{{1,n-2}\}=1+({n-3})^+\in \mathbb N
\end{equation}
\tag{2.13}
$$
will be used repeatedly. As in formula (1.21) for $\mathbb C$,
$$
\begin{equation}
{\mathrm N}_{\mu}(r,R):={\widehat{n}}\int_r^R \frac{\mu^{\operatorname{rad}}(t)}{t^{n-1}}\,\mathrm{d} t\in \overline{\mathbb R}^+ \quad\text{for } 0\leqslant r<R<R_0\in \overline{\mathbb R}^+
\end{equation}
\tag{2.14}
$$
is the radial difference integrated counting function of the measure, and
$$
\begin{equation}
{\mathrm N}_y^{\mu}(r)\stackrel{(1.6)}{:=}\widehat{n}\int_0^r \frac{\mu_y^{\operatorname{rad}}(t)}{t^{n-1}}\,\mathrm{d} t\in \overline{\mathbb R}^n, \qquad r\in \overline{\mathbb R}^+,
\end{equation}
\tag{2.15}
$$
is its radial integrated counting function with centre $y\in \mathbb R^n$. For references on subharmonic functions we use [26]–[28] and [25]. A domain in $\mathbb R^n$ is a connected open set. For a connected subset $S$ of $ \mathbb R^n$ we let $\operatorname{sbh}(S)$ denote the set of subharmonic functions defined on some domains $D_u$ containing $S$, or in other words, on $S$. By taking the restriction of a measure to $S$, given a subharmonic function $u\not\equiv -\infty$ on $D_u\supset S$, the Laplace operator ${\bigtriangleup}$ defined in the sense of the theory of distributions, produces the Riesz measure of $u$ on $S$, which we denote and define by
$$
\begin{equation}
\varDelta_u\stackrel{(2.13)}{:=} \frac{1}{s_{n-1}{\widehat{n}}} {\bigtriangleup} u.
\end{equation}
\tag{2.16}
$$
The subset $\operatorname{sbh}_*(S)$ of $ \operatorname{sbh}(S)$ consists of the functions distinct from identical $-\infty$ in domains $D_u\supset S$; then we write $u\not\equiv -\infty$ on $S$. The class $\operatorname{dsh}(S)$ of $\delta$-subharmonic functions on $S$ consists of the functions representable as the difference $U=u-v$ of two functions $u, v\in \operatorname{sbh}(S)$, except when both these functions are identically equal to $-\infty$, and $\operatorname{dsh}_*(S)$ is formed by the differences ${U=u-v}$ of functions $u, v\in \operatorname{sbh}_*(S)$. For these we write $U\not\equiv \pm\infty$ on $S$. Various equivalent forms of the definition of $\delta$-subharmonic functions, their consistency and main properties were considered in [14], [15], [29], § 2.8.2, [16], [17], § 3.1, and [30]. For a $\delta$-subharmonic function $U\not\equiv \pm\infty$ on $S$, its Riesz charge $\varDelta_U:=\varDelta_u-\varDelta_v$ is well defined as the difference of the Riesz measures of the functions $u,v\in \operatorname{sbh}_*(S)$. Definition 1. The difference Nevanlinna characteristic ${\boldsymbol T}_U(r,R)$ of a $\delta$-subharmonic function $U\not\equiv \pm\infty$ on a ball $B(R_0) \stackrel{(2.9)}{\subset} \mathbb R^n$ is the sum in (2.5), where we use the notation and definitions from (2.11) and (2.14), namely, We can give another definition of the difference Nevanlinna characteristic ${\boldsymbol T}_U$. A $\delta$-subharmonic function $U\not\equiv \pm\infty$ with Riesz signed measure $\varDelta_U$ on a ball $\overline B(R)\stackrel{(2.9)}{\subset} \mathbb R^n$ has canonical representations where $u_*\not\equiv -\infty$ and $v_*\not\equiv -\infty$ are subharmonic functions on $\overline B(R)$ with Riesz measures $\varDelta_{u_*}=\varDelta_U^+:=\sup\{0,\varDelta_U\}$ (the upper variation of the signed measure $\varDelta_U$) and $\varDelta_{v_*}=\varDelta_U^-$ (the lower variation of $\varDelta_U$). Such canonical representations are well defined up to the addition of the same harmonic function to both terms. From the obvious equalities and the definition (2.11) we obtain
$$
\begin{equation}
{\mathrm C}_{U^+}(R)={\mathrm C}_{\sup\{u_*,v_*\}}(R)-{\mathrm C}_{v_*}(R) \quad\text{for all } 0<R<+\infty,
\end{equation}
\tag{2.18}
$$
where $\sup\{u_*,v_*\}$ is a subharmonic function on $\overline B(R)$, and by the Poisson-Jensen-Privalov formula (see [31], [32], [33], Ch. II, § 2, and [27], § 3.7)
$$
\begin{equation}
{\mathrm N}_{\varDelta_{v_*}}(r,R)={\mathrm C}_{v_*}(R)-{\mathrm C}_{v_*}(r) \quad\text{for all } 0<r<R<+\infty.
\end{equation}
\tag{2.19}
$$
Adding (2.18) and (2.19) we obtain
$$
\begin{equation}
{\boldsymbol T}_U(r,R)\stackrel{(2.17)}{=} {\mathrm C}_{\sup\{u_*,v_*\}}(R)-{\mathrm C}_{v_*}(r) \in \mathbb R^+, \qquad 0< r<R\in \mathbb R^+.
\end{equation}
\tag{2.20}
$$
Now, for each $\delta$-subharmonic function $U\not\equiv \pm\infty$ on $\overline B(R)$ we have
$$
\begin{equation}
\begin{aligned} \, \notag {\boldsymbol T}_U(r,R) &\stackrel{(2.20)}{=}{\mathrm C}_{\sup\{v_*, u_*\}}(R)-{\mathrm C}_{u_*}(r) +{\mathrm C}_{u_*}(r) -{\mathrm C}_{v_*}(r) \\ &\stackrel{(2.20)}{=} {\boldsymbol T}_{-U}(r,R)+{\mathrm C}_{u_*-v_*}(r) ={\boldsymbol T}_{-U}(r,R)+{\mathrm C}_U(r). \end{aligned}
\end{equation}
\tag{2.21}
$$
In particular, if $u$ is a subharmonic functions on $\overline B(R)$ with Riesz measure $\varDelta_u$, then
$$
\begin{equation}
{\boldsymbol T}_{-u}(r,R)\stackrel{(2.21)}{=} {\boldsymbol T}_u(r,R)-{\mathrm C}_u(r)\stackrel{(2.17)}{=}{\mathrm C}_{u^+}(R)-{\mathrm C}_u(r)\in \mathbb R^+.
\end{equation}
\tag{2.22}
$$
The function $f\colon I\to \mathbb R$ is said to be convex (concave) on an open interval $I\subset \mathbb R$ with respect to a strictly increasing continuous function $k\colon I\to \mathbb R$ if the composition $f \circ k^{-1}$ is convex (concave, respectively) on the set $k(I)\subset \mathbb R$. The extended number function
$$
\begin{equation}
\Bbbk_{n-2} \colon t\mapsto \begin{cases} \ln t &\text{for } n=2, \\ -\dfrac{1}{t^{n-2}} &\text{for } n>2, \end{cases} \quad 0<t\in \mathbb R^+,\quad \Bbbk (0):=-\infty \in \overline{\mathbb R},
\end{equation}
\tag{2.23}
$$
is clearly strictly increasing and continuous on $ \mathbb R^+$. The spherical means (2.11) of subharmonic functions are increasing functions which are convex with respect to $\Bbbk_{n-2}$ (see [26], Theorem 2.6.8, and [27], § 3.9), and directly from the representation (2.20) for the Nevanlinna difference characteristic we obtain the following. Proposition 1. The Nevanlinna difference characteristic ${\boldsymbol T}_U$ of a $\delta$-subharmonic function $U\not\equiv \pm \infty$ on a ball with centre at zero is positive, increasing and convex with respect to $\Bbbk_{n-2}$ in the second (larger) variable, and it is decreasing and convex with respect to $\Bbbk_{n-2}$ in the first variable. § 3. Main resultMain Theorem. For $0<r\in \mathbb R^+$ let $\mu$ be a Borel measure on $\overline B(r)\subset \mathbb R^n$. Then the following five statements are equivalent. I. In the notation from (2.15) the following relation holds:
$$
\begin{equation}
\sup_{y\in \overline B(R)}{\mathrm N}_y^{\mu}(r_0)<+\infty \quad\text{for some } r_0>0\quad\text{and} \quad R>r.
\end{equation}
\tag{3.1}
$$
II. For $R>r$ any function $U\in {\operatorname{dsh}}_*(\overline B(R))$ is $\mu$-summable and
$$
\begin{equation}
\int_{\overline B(r)} U^+\,\mathrm{d} \mu \leqslant A_n(r,R){\boldsymbol T}_U( r, R) \Bigl(\mu^{\operatorname{rad}}(r)\max\{1, r^{2-n}\} +\sup_{y\in \overline B(r)}{\mathrm N}_y^{\mu}(r)\Bigr),
\end{equation}
\tag{3.2}
$$
where the right-hand side is finite,
$$
\begin{equation}
A_n(r,R):=5 \max\{1, n-2\} \biggl(\frac{R+r}{R-r}\biggr)^{n-1}\max\{1, (R-r)^{n-2}\},
\end{equation}
\tag{3.3}
$$
and the first argument $r$ of ${\boldsymbol T}_U( r, R)$ can be replaced by any $r'\in [0,r]$. III. There exist $R>r$ and $T>0$ such that all $\delta$-subharmonic functions $U\not\equiv \pm\infty$ on $\overline B(R)$ are $\mu$-integrable and
$$
\begin{equation}
\sup\biggl\{\int_{\overline B(r)} U^+\,\mathrm{d} \mu \biggm| U\in \operatorname{dsh}_*(\overline B(R)), \, {\boldsymbol T}_U( r, R)\leqslant T \biggr\}<+\infty.
\end{equation}
\tag{3.4}
$$
IV. The measure $\mu$ is finite and the $\mu$-potential, which is denoted and defined by
$$
\begin{equation}
{\operatorname{pt}}_{\mu}\colon x\mapsto \int_{\mathbb R^n} \Bbbk_{n-2}(|y-x|)\,\mathrm{d} \mu(y),\qquad x\in \mathbb R^n,
\end{equation}
\tag{3.5}
$$
is bounded below on the support $\operatorname{supp} \mu\subset \overline B(r)$. V. The measure $\mu$ is finite and Proof. For $\mu= 0$ everything is obvious, so we assume that $\mu\neq 0$.
I $\Rightarrow$ II. In the proof of this implication we use repeatedly the following elementary lemma. Lemma 1 (see [34], Proposition 2.2). For $0<r\in \mathbb R^+$ let $h\colon (0,r]\to \mathbb R^+$ be an increasing function. If the following Riemann integral is convergent: Conversely, if (3.9) holds, then the following limits exist:
$$
\begin{equation}
h(0):=\lim_{0<t\to 0} h(t)\in \mathbb R^+, \qquad \lim_{0<t\to 0}\bigl(h(t)-h(0)\bigr)\Bbbk_{n-2}( t)=0
\end{equation}
\tag{3.10}
$$
and the following integral is convergent:
$$
\begin{equation}
\int_{0}^{r}\frac{h(t)-h(0)}{t^{n-1}}\,\mathrm{d} t<+\infty.
\end{equation}
\tag{3.11}
$$
For $\widehat{n}\stackrel{(2.13)}{=}\max \{1, {n-2}\}$, if either of (3.7) and (3.9) holds, then
$$
\begin{equation}
{\widehat{n}}\int_{0}^{r}\frac{h(t)-h(0)}{t^{n-1}}\,\mathrm{d} t= \int_{0}^{r}\bigl(\Bbbk_{n-2}(r)-\Bbbk_{n-2}(t)\bigr)\,\mathrm{d} h(t),
\end{equation}
\tag{3.12}
$$
and if (3.7) holds and $r_0\in (0,r]$, then We also use the following result several times. Lemma 2. If (3.1) holds, then $\mu$ is a finite measure, Proof. By relation (3.1) and the definition (2.15) the $\mu$-measures of the spherical shells ${B_y(r_0)\setminus B_y(\min\{R-r,r_0\}/2)}$ are uniformly bounded for $y$ in $\overline B(R)$. Such shells cover the compact set $\overline B(r)$, so $\mu$ is finite. From (3.13), for $h:=\mu_y^{\operatorname{rad}}$ we obtain
$$
\begin{equation*}
{\mathrm N}_y^{\mu}(t)\leqslant \mu(\mathbb R^n) \bigl(\Bbbk_{n-2}(t)-\Bbbk_{n-2}(r_0)\bigr)+{\mathrm N}_y^{\mu}(r_0) \quad\text{for any } y\in \mathbb R^n, \quad t\geqslant r_0.
\end{equation*}
\notag
$$
Taking the suprema of both sides over $y\in \overline B(r)$ yields the inequality $<\!+\infty$ in (3.14). For each $y$ outside $\overline B(r)$, by the triangle inequality $\overline B_y(t)\cap \overline B(r)$ lies in the ball $\overline B_{y'}(t)$ with centre $y':=ry / |y| \in \overline B(r)$, so we see from the definition (2.15) we see that ${\mathrm N}_y^{\mu}(t)\leqslant {\mathrm N}_{y'}^{\mu}(t)$ for $|y|>r$, and we obtain the first equality in (3.14).
From (3.1) and Lemma 1 for $h:=\mu_y^{\operatorname{rad}}$, as the limits
$$
\begin{equation}
\lim_{0<t\to 0} \mu_y^{\operatorname{rad}}(t)=0\quad\text{and} \quad \lim_{0<t\to 0} \mu_y^{\operatorname{rad}}(t)\Bbbk_{n-2}( t)=0
\end{equation}
\tag{3.15}
$$
exist (see (3.8)), and from equality (3.12), by the definition (2.15), for each $R\in \mathbb R^+\setminus 0$ we obtain
$$
\begin{equation*}
\begin{aligned} \, {\mathrm N}_y^{\mu}(R) &\stackrel{(2.15)}{=}\widehat{n}\int_0^R \frac{\mu_y^{\operatorname{rad}}(t)}{t^{n-1}}\,\mathrm{d} t \stackrel{(3.12)}{=} \int_{0}^{R}\bigl(\Bbbk_{n-2}(R)-\Bbbk_{n-2}(t)\bigr)\,\mathrm{d} \mu_y^{\operatorname{rad}}(t) \\ &\ \ =\mu^{\operatorname{rad}}(R)\Bbbk_{n-2}(R)- \int_{\overline B_y(R)} \Bbbk_{n-2}(|x-y|)\,\mathrm{d} \mu(x) \quad\text{for all } y\in \mathbb R^n. \end{aligned}
\end{equation*}
\notag
$$
However, if we take $R\geqslant 2r$, then the ball $\overline B_y(R)$ contains $\overline B(r)$, so that
$$
\begin{equation*}
{\mathrm N}_y^{\mu}(R)\stackrel{(3.5)}{=} \mu^{\operatorname{rad}}(R)\Bbbk_{n-2}(R) -{\operatorname{pt}}_{\mu}(y) \quad\text{for all } y\in \mathbb R^n, \quad R\geqslant 2r.
\end{equation*}
\notag
$$
Hence, for a finite measure $\mu$ satisfying (3.5) we have
$$
\begin{equation*}
\inf_{y\in \mathbb R^n} {\operatorname{pt}}_{\mu}(y)>-\infty,
\end{equation*}
\notag
$$
that is, the potential ${\operatorname{pt}}_{\mu}$ is bounded below on the whole of $\mathbb R^n$. Then the energy integral
$$
\begin{equation*}
I[\mu]:=\int_{\mathbb R^n}{\operatorname{pt}}_{\mu}\,\mathrm{d} \mu
\end{equation*}
\notag
$$
is finite and $\mu$ is a measure with finite energy. Any Borel polar set is a nullset for such a measure (see [26], Theorem 3.2.3, and [28], Theorem II.2).
Lemma 2 is proved. Let $u\not\equiv -\infty$ and $v\not\equiv -\infty$ be two subharmonic functions on $\overline B(R)$, and let $U:=u-v$. Then this difference takes well-defined finite values at each point $x\in \overline B(R)\setminus E$ outside the Borel polar set
$$
\begin{equation}
E=\bigl\{x\in B(R)\mid u(x)=-\infty\bigr\}\cup \bigl\{x\in B(R)\mid v(x)=-\infty\bigr\}.
\end{equation}
\tag{3.16}
$$
The function $U^+$ is well defined everywhere in $\overline B(r)\setminus E$ as the positive part of the difference $u-v$ of upper semicontinuous functions $u$ and $v$ taking values in $\mathbb R$. It is measurable with respect to the restriction $\mu{\lfloor}_{\overline B(r)\setminus E}$ of the Borel measure $\mu$ to $\overline B(r)\setminus E$, so it is also integrable with respect to this restriction. On the other hand it follows from the final part of Lemma 2 that $\mu(E)=0$, so $U^+$ is $\mu$-integrable on the whole of $\overline B(r)$, and it is also $\mu$-summable because of (3.2), as the right-hand of this inequality is finite by (3.14) in Lemma 2. We go over to inequality (3.2) for $U=u-v$. Using the Poisson-Jensen formula (see [27], formula (3.7.3)) in $B(R)$ for $u(x)$ and $v(x)$ at points $x\in \overline B (r)$, and subtracting one of the equalities from the other at each point $x\in \overline B(R)$ outside the set $E$ in (3.16) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag U(x)&=\frac{1}{{\mathrm s}_{n-1}}\int_{\partial \overline B(R)} \frac{R^2-|x|^2}{R|y-x|^n}U(y)\,\mathrm{d} \sigma_{n-1}^R (y) \\ &-\int_{B(R)}\biggl( \Bbbk_{n-2}\biggl(\biggl|\frac{R}{|y|}y-\frac{|y|}{R}x\biggr| \biggr)-\Bbbk_{n-2}(|y-x|)\biggr)\,\mathrm{d} \varDelta_U(y), \qquad x\in \overline B(r)\setminus E. \end{aligned}
\end{equation}
\tag{3.17}
$$
Here for the positive Poisson kernel (see [27], § 1.5.4) we have
$$
\begin{equation*}
\frac{1}{{\mathrm s}_{n-1}} \frac{R^2-|x|^2}{R|y-x|^n}\leqslant \frac{1}{{\mathrm s}_{n-1}} \frac{R+r}{R(R-r)^{n-1}} \quad\text{for all } y\in \partial \overline B(R)\quad\text{and} \quad x\in \overline B(r),
\end{equation*}
\notag
$$
and for the positive Green’s function we have (see [27], Theorem 1.10)
$$
\begin{equation*}
\begin{aligned} \, &\Bbbk_{n-2}\biggl(\biggl|\frac{R}{|y|}y-\frac{|y|}{R}x\biggr| \biggr)-\Bbbk_{n-2}(|y-x|) \\ &\qquad \leqslant \Bbbk_{n-2}(R+r)-\Bbbk_{n-2}(|y-x|) \quad \text{for all } y\in B(R)\quad\text{and} \quad x\in \overline B(r). \end{aligned}
\end{equation*}
\notag
$$
Thus it follows from (3.17) that for all $x\in\overline B(r)\setminus E$, in view of (3.16),
$$
\begin{equation*}
U^+(x)\leqslant \frac{R^{n-2}(R+r)}{(R-r)^{n-1}}{\mathrm C}_{U^+}(R) +\int_{B(R)}\bigl(\Bbbk_{n-2}(R+r)-\Bbbk_{n-2}(|y-x|)\bigr) \,\mathrm{d}\varDelta_U^-(y).
\end{equation*}
\notag
$$
Now, as $U^+$ is $\mu$-integrable, we can integrate this inequality against $\mu$ and use Fubini’s theorem on iterated integrals:
$$
\begin{equation}
\begin{aligned} \, \notag \int_{\overline B(r)}U^+\,\mathrm{d} \mu &\leqslant \int_{\overline B(r)}\frac{R^{n-2}(R+r)}{(R-r)^{n-1}}{\mathrm C}_{U^+}(R)\,\mathrm{d} \mu(x) \\ \notag &\qquad+\int_{\overline B(r)}\int_{B(R)} \bigl(\Bbbk_{n-2}(R+r)-\Bbbk(|y-x|)\bigr) \,\mathrm{d}\varDelta_U^-(y)\,\mathrm{d} \mu (x) \\ \notag &\!\!\!\!\stackrel{(2.17)}{\leqslant} \frac{R^{n-2}(R+r)}{(R-r)^{n-1}}{\boldsymbol{T}_U}(r, R)\mu^{\operatorname{rad}}(r) \\ &\qquad+\int_{B(R)}\int_{\overline B(r)} \bigl(\Bbbk_{n-2}(R+r)-\Bbbk(|y-x|)\bigr) \,\mathrm{d} \mu (x)\,\mathrm{d}\varDelta_U^-(y), \end{aligned}
\end{equation}
\tag{3.18}
$$
where in the last inequality we used that ${\mathrm C}_{U^+}(R)\stackrel{(2.17)}{\leqslant} {\boldsymbol{T}_U}(r, R)$, which follows from the definition of the Nevanlinna difference characteristic (2.17). Furthermore, for $y\in B(R)$ and $x\in \overline B(r)$ we have $|y-x|<R+r$, and we can write the last inner integral against $\mu$ as a Riemann-Stieltjes integral over $(0,R+r)$ with respect to the increasing function $\mu_y^{\operatorname{rad}}$, so that we can extend these inequalities as follows:
$$
\begin{equation*}
\begin{aligned} \, \int_{\overline B(r)}U^+\,\mathrm{d} \mu &\stackrel{(2.12)}{\leqslant} \frac{R^{n-2}(R+r)}{(R-r)^{n-1}}{\boldsymbol{T}_U}(r, R)\mu^{\operatorname{rad}}(r) \\ &\qquad +\int_{B(R)}\int_0^{R+r}\bigl(\Bbbk_{n-2}(R+r)-\Bbbk_{n-2}(t)\bigr) \,\mathrm{d} \mu_y^{\operatorname{rad}} (t)\,\mathrm{d}\varDelta_U^-(y), \end{aligned}
\end{equation*}
\notag
$$
where the last integral is one of the expressions for the mutual energy of the measures $\mu$ and $\varDelta_U^-$ (see [25], Ch. I, § 4, and [28], Ch. 11). As we consider arbitrary $\delta$-subharmonic functions $U$, the Riesz charge $\varDelta_U$ and its lower variation $\varDelta_U^-$ can be arbitrary: for instance, they can have infinite energy. So the standard estimates for the mutual energy in terms of the product of energies can turn out useless, and we use a rougher estimate for the last integral, in terms of the measure $(\varDelta_U^-)(\overline B(R))=(\varDelta_U^-)^{\operatorname{rad}}(R)$:
$$
\begin{equation}
\begin{aligned} \, \notag &\int_{\overline B(r)}U^+\,\mathrm{d} \mu +\int_{B(R)}\leqslant \frac{R^{n-2}(R+r)}{(R-r)^{n-1}}{\boldsymbol T}_{U}(r,R)\mu^{\operatorname{rad}}(r) \\ &\qquad\qquad +(\varDelta_U^-)^{\operatorname{rad}}(R) \sup_{y\in B(R)} \int_0^{R+r}\bigl(\Bbbk_{n-2}(R+r)-\Bbbk_{n-2}(t)\bigr) \,\mathrm{d} \mu_y^{\operatorname{rad}} (t). \end{aligned}
\end{equation}
\tag{3.19}
$$
For the last Riemann-Stieltjes integral, using equalities (3.12) and inequality (3.13) from Lemma 1 for $h:=\mu_y^{\operatorname{rad}}$ in succession, and using also the definition (2.15) in combination with the obvious inequality $\mu_y^{\operatorname{rad}} (r)\leqslant \mu^{\operatorname{rad}} (r)$, we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\int_0^{R+r}\bigl(\Bbbk_{n-2}(R+r)-\Bbbk_{n-2}(t)\bigr) \,\mathrm{d} \mu_y^{\operatorname{rad}} (t) \stackrel{(3.12)}{=} \widehat{n}\int_0^{R+r}\frac{\mu_y^{\operatorname{rad}} (t)}{t^{n-1}}\,\mathrm{d} t \\ &\qquad \stackrel{(3.13)}{\leqslant} \bigl(\Bbbk_{n-2}(R+r)-\Bbbk_{n-2}(r)\bigr) \mu^{\operatorname{rad}} (r)+{\mathrm N}_y^{\mu}(r). \end{aligned}
\end{equation}
\tag{3.20}
$$
All the relations established above also hold for any $R_*>r$ not exceeding $R$:
$$
\begin{equation}
r<R_*<R , \qquad \overline B(r)\subset B(R_*)\subset \overline B(R).
\end{equation}
\tag{3.21}
$$
Thus, in view of (3.20) we can write (3.19) as
$$
\begin{equation}
\begin{aligned} \, \notag &\int_{\overline B(r)}U^+\,\mathrm{d} \mu \leqslant \frac{R_*^{n-2}(R_*+r)}{(R_*-r)^{n-1}}{\boldsymbol{T}_U}(r,R)\mu^{\operatorname{rad}}(r) \\ &\qquad\qquad +(\varDelta_U^-)^{\operatorname{rad}}( R_*)\Bigl( \mu^{\operatorname{rad}}(r)\bigl(\Bbbk_{n-2}(R_*+r)-\Bbbk_{n-2}(r)\bigr)+ \sup_{y\in \overline B(r)}{\mathrm N}_y^{\mu}(r)\Bigr), \end{aligned}
\end{equation}
\tag{3.22}
$$
where we have used the equality
$$
\begin{equation*}
\sup_{y\in B(R)}{\mathrm N}_y^{\mu}(r)=\sup _{y\in \overline B(r)}{\mathrm N}_y^{\mu}(r),
\end{equation*}
\notag
$$
which is a consequence of (3.14) in Lemma 2. Lemma 3. Let $\varDelta$ be Borel measure on $\overline B(R)\subset \mathbb R^n$ and let $0<R_*<R$. Then Proof. Since the counting function $\varDelta^{\operatorname{rad}}$ is increasing, we have
$$
\begin{equation*}
\varDelta^{\operatorname{rad}}(R_*)\leqslant \int_{R_*}^{R}\frac{\varDelta^{\operatorname{rad}}(t)}{t^{n-1}}\,\mathrm{d} t\biggm/ \int_{R_*}^{R} \frac{\mathrm{d} t}{t^{n-1}} \stackrel{(2.14)}{=} \frac{{\mathrm N}_{\varDelta} (R_*,R)}{\Bbbk_{n-2}(R)-\Bbbk_{n-2}(R_*)},
\end{equation*}
\notag
$$
which completes the proof of the lemma. We return to the proof of the main theorem. By Lemma 3 applied to $(\varDelta_U^-)^{\operatorname{rad}}(R_*)$, in view of the inequalities
$$
\begin{equation*}
{\mathrm N}_{\varDelta_U^-}(R_*,R)\stackrel{(2.14)}{\leqslant} {\mathrm N}_{\varDelta_U^-}(r,R)\stackrel{(2.17)}{\leqslant} {\boldsymbol{T}}_U(r,R)
\end{equation*}
\notag
$$
we can estimate the right-hand side of (3.22) from above by
$$
\begin{equation}
\begin{aligned} \, \notag &{\boldsymbol{T}}_U(r,R) \frac{1}{\Bbbk_{n-2}(R)-\Bbbk_{n-2}(R_*)} \\ &\qquad\qquad \times\Bigl(\mu^{\operatorname{rad}}(r)\bigl(\Bbbk_{n-2}(R_*+r)-\Bbbk_{n-2}(r)\bigr) +\sup_{y\in \overline B(r)}{\mathrm N}_y^{\mu}(r)\Bigr). \end{aligned}
\end{equation}
\tag{3.24}
$$
We consider the cases $n=2$ and $n>2$ separately. The case $n=2$. In (3.21) we take $R_*:=\sqrt{rR} $ equal to the geometric mean of $r$ and $R$. Then by (3.24) the right-hand side of (3.24) is estimated above by
$$
\begin{equation*}
\begin{aligned} \, &{\boldsymbol{T}}_U(r,R) \frac{1}{\ln \sqrt{R/r}}\biggl(\mu^{\operatorname{rad}}(r)\ln \frac{\sqrt{Rr}+r}{r}+ \sup_{z\in \overline D(r)}{\mathrm N}_z^{\mu}(r)\biggr) \\ &\qquad={\boldsymbol{T}}_U(r,R) \biggl(\mu^{\operatorname{rad}}(r)+\mu^{\operatorname{rad}}(r)\frac{\ln(1+\sqrt{r/R})}{\ln \sqrt{R/r}}+\frac{2}{\ln (R/r)}\sup_{z\in \overline D(r)}{\mathrm N}_z^{\mu}(r)\biggr) \\ &\qquad\leqslant {\boldsymbol{T}}_U(r,R) \biggl(\mu^{\operatorname{rad}}(r)+\mu^{\operatorname{rad}}(r)\frac{2\ln 2}{\ln (R/r)}+ \frac{2}{\ln (R/r)}\sup_{z\in \overline D(r)}{\mathrm N}_z^{\mu}(r) \biggr). \end{aligned}
\end{equation*}
\notag
$$
Now, using the fact that
$$
\begin{equation*}
\frac{1}{\ln (R/r)}=\biggl(\int_r^R\frac{\mathrm{d} t}{t}\biggr)^{-1} \leqslant \frac{R}{R-r}\geqslant 1 \quad\text{for } 0<r<R,
\end{equation*}
\notag
$$
we can estimate the right-hand side of (3.22) above by
$$
\begin{equation*}
\begin{aligned} \, &{\boldsymbol{T}}_U(r,R) \biggl(\frac{R}{R-r}\mu^{\operatorname{rad}}(r) +\frac{R}{R-r}\mu^{\operatorname{rad}}(r)\, 2\ln 2+2\frac{R}{R-r}\sup_{z\in \overline D(r)}{\mathrm N}_z^{\mu}(r) \biggr) \\ &\qquad \leqslant \frac{R}{R-r}{\boldsymbol{T}}_U(r,R) \Bigl((1+2\ln 2)\mu^{\operatorname{rad}}(r) +2\sup_{z\in \overline D(r)}{\mathrm N}_z^{\mu}(r)\Bigr), \end{aligned}
\end{equation*}
\notag
$$
so that by (3.22) we have
$$
\begin{equation*}
\begin{aligned} \, &\int_{\overline B(r)}U^+\,\mathrm{d} \mu \leqslant {\boldsymbol{T}_U}(r,R) \biggl(\frac{\sqrt R+\sqrt r}{\sqrt R-\sqrt r}\mu^{\operatorname{rad}}(r) \\ &\qquad\qquad +\frac{R}{R-r}\Bigl((1+2\ln 2)\mu^{\operatorname{rad}}(r)+2\sup_{z\in D(R)}{\mathrm N}_z^{\mu}(r) \Bigr)\biggr) \\ &\qquad\leqslant {\boldsymbol{T}_U}(r,R) \biggl(\biggl(2\frac{R+r}{ R- r}+\frac{(1+2\ln 2)R}{R-r}\biggr)\mu^{\operatorname{rad}}(r)+\frac{2R}{R-r}\sup_{z\in D(R)}{\mathrm N}_z^{\mu}(r) \biggr) \\ &\qquad\leqslant 5\frac{R+r}{R-r}\Bigl(\mu^{\operatorname{rad}}(r)+\sup_{z\in D(R)}{\mathrm N}_z^{\mu}(r) \Bigr). \end{aligned}
\end{equation*}
\notag
$$
This yields the required inequality (3.2) with as in (3.3). The case $n>2$. From the obvious inequality
$$
\begin{equation*}
\Bbbk_{n-2}(R_*+r)-\Bbbk_{n-2}(r)\leqslant \frac{1}{r^{n-2}}
\end{equation*}
\notag
$$
for $n>2$ and the elementary inequality
$$
\begin{equation*}
{\Bbbk_{n-2}(R)-\Bbbk_{n-2}(R_*)}=\frac{1}{n-2}\int_{R_*}^R\frac{\mathrm{d} t}{t^{n-1}}\geqslant \frac{1}{n-2}\frac{R-R_*}{R^{n-1}},
\end{equation*}
\notag
$$
taking $R_*:=\frac12(R+r) $ equal to the arithmetic mean of $r$ and $R$ on the right-hand side of (3.22), we obtain an upper estimate by
$$
\begin{equation*}
{\boldsymbol{T}}_U(r,R)\frac{2(n-2)R^{n-1}}{R-r}\biggl( \mu^{\operatorname{rad}}(r)\frac{1}{r^{n-2}}+ \sup_{z\in \overline B(r)}{\mathrm N}_z^{\mu}(r)\biggr) \quad \text{for } n>2,
\end{equation*}
\notag
$$
so that
$$
\begin{equation*}
\begin{aligned} \, &\int_{\overline B(r)}U^+\,\mathrm{d} \mu \leqslant 2\biggl(\frac{R+r}{R-r}\biggr)^{n-1} {\boldsymbol{T}_U}(r,R)\mu^{\operatorname{rad}}(r) \\ &\qquad\qquad+2(n-2){\boldsymbol{T}}_U(r,R) \frac{R^{n-1}}{R-r}\biggl(\mu^{\operatorname{rad}}(r)\frac{1}{r^{n-2}}+ \sup_{z\in \overline B(r)}{\mathrm N}_z^{\mu}(r) \biggr) \\ &\qquad\leqslant 2\biggl(\frac{R+r}{R-r}\biggr)^{n-1}{\boldsymbol{T}}_U(r,R) \Bigl(\bigl(1+(n-2)(R-r)^{n-2}\bigr)\mu^{\operatorname{rad}}(r)\max\{1, r^{2-n}\} \\ &\qquad\qquad+(n-2)(R-r)^{n-2} \sup_{z\in \overline B(r)}{\mathrm N}_z^{\mu}(r) \Bigr) \\ &\qquad\leqslant 2\biggl(\frac{R+r}{R-r}\biggr)^{n-1} {\boldsymbol{T}}_U(r,R) \cdot 2(n-2)\max\{1, (R-r)^{n-2}\} \\ &\qquad\qquad\times\Bigl(\mu^{\operatorname{rad}}(r)\max\{1, r^{2-n}\}+ \sup_{z\in \overline B(r)}{\mathrm N}_z^{\mu}(r) \Bigr), \end{aligned}
\end{equation*}
\notag
$$
which yields the required inequality (3.2) with $A_n(r,R)$ as in (3.3) for $n>2$. The right-hand side of (3.2) is finite in view of (3.14). II $\Rightarrow$ III. Fix some $R>r$ and look at inequality (3.2) for the harmonic function $U\equiv 1$; it is obvious that ${\boldsymbol{T}}_1(r,R)=1$. As the right-hand side of (3.2) is finite for ${U\equiv 1}$, there exists $M>0$ independent of $U\in \operatorname{dsh}_*(\overline B(R))$ such that
$$
\begin{equation*}
A_n(r,R)\Bigl(\mu^{\operatorname{rad}}(r)\max\{1, r^{2-n}\}+\sup_{y\in \overline B(R)} {\mathrm N}_y^{\mu}(r)\Bigr)\leqslant M<+\infty.
\end{equation*}
\notag
$$
Hence for each function $U\in \operatorname{dsh}_*(\overline B(R))$ such that $ {\boldsymbol T}_U( r, R)\leqslant T$, using (3.2) again we obtain
$$
\begin{equation*}
\int_{\overline B(r)} U^+\,\mathrm{d} \mu \leqslant MT<+\infty,
\end{equation*}
\notag
$$
where the product $MT$ is independent of $U$. This proves (3.4). III $\Rightarrow$ IV. Relation (3.4) holds for each $T>0$ because when we multiply $U\in \operatorname{dsh}_*(\overline B(R))$ by a positive scalar, we also multiply the Nevanlinna difference characteristic by this scalar. Moreover, it follows from (3.4) that the harmonic function identically equal to one is $\mu$-summable, and thus $\mu$ is a finite measure. For $y\in \mathbb R^n$ we look at the subharmonic functions
$$
\begin{equation}
k_y\colon x\overset{(2.10)}{\longmapsto} \Bbbk_{n-2} (|x-y|),\quad x \in \mathbb R^n,\quad \text{with Riesz measure } \varDelta_{k_y}=\boldsymbol{\delta}_y,
\end{equation}
\tag{3.25}
$$
where $\boldsymbol{\delta}_y$ is the Dirac probability measure at $y$, so that $\operatorname{supp} \boldsymbol{\delta}_y= \{y\}$. Applying Poisson-Jensen formula (3.17) to the values of the functions $U:= k_y$ at zero we immediately obtain
$$
\begin{equation}
{\mathrm C}_{k_y}(R)= \Bbbk_{n-2}(R) \quad\text{for all } y\in B(R), \quad 0<R\in \mathbb R^+.
\end{equation}
\tag{3.26}
$$
For $R>r$ under consideration in III, consider the family of subharmonic functions $\{K_y\}_{y\in \mathbb R^n}$ on the whole of $\mathbb R^n$ that are defined by
$$
\begin{equation}
K_y\colon x\mapsto \Bbbk_{n-2}(R+r)-\Bbbk_{n-2}(|x-y|),\qquad x\in \mathbb R^n.
\end{equation}
\tag{3.27}
$$
If $y$ lies on $\overline B(r)$, then the functions $K_y$ are positive on $\overline B(R)$ and
$$
\begin{equation}
{\mathrm C}_{K_y^+}(R)={{\mathrm C}_{K_y}}(R)\stackrel{(3.26)}{=}\Bbbk_{n-2}(R+r)-\Bbbk_{n-2}(R), \qquad y\in \overline B(r).
\end{equation}
\tag{3.28}
$$
In addition, the Riesz charge of $K_y$ is the opposite of the Dirac measure at $y$, and we have
$$
\begin{equation*}
\varDelta_{K_y}=-\boldsymbol{\delta}_y=-\varDelta_{K_y}^- \quad\text{and} \quad \varDelta_{K_y}^-(t)= \begin{cases} 0&\text{for } t\in [0,|y|), \\ 1&\text{for } t\in [|y|, +\infty). \end{cases}
\end{equation*}
\notag
$$
Hence for $|y|\leqslant r$ we obtain
$$
\begin{equation*}
{\mathrm N}_{\varDelta_{K_y}^-}(r, R)=\widehat{n}\int_{r}^R\frac{1}{t^{n-1}}\,\mathrm{d} t \stackrel{}{=}\Bbbk_{n-2}(R)-\Bbbk_{n-2}(r), \qquad y\in \overline B(r).
\end{equation*}
\notag
$$
In combination with (3.28), by the definition of the Nevanlinna difference characteristic this yields
$$
\begin{equation}
{\boldsymbol{T}_{K_y}}(r,R)={\mathrm C}_{K_y^+}(R)+ {\mathrm N}_{\varDelta_{K_y}^-}(r, R)= \Bbbk_{n-2}(R+r)-\Bbbk_{n-2}(r),
\end{equation}
\tag{3.29}
$$
where the right-hand side is strictly positive and independent of $y\in \overline B(r)$. We take $T$ equal to the right-hand side of (3.29). Then by relation (3.4) for $U\stackrel{(3.27)}{:=}K_y$ there exists $C$ such that
$$
\begin{equation*}
\sup_{y\in \overline B(r)} \int_{\overline B(r)}K_y(x)\,\mathrm{d} \mu(x)\leqslant C<+\infty,
\end{equation*}
\notag
$$
and from the explicit expression for $K_y$ in (3.27) we obtain
$$
\begin{equation*}
\inf_{y\in \overline B(r)} \int_{\overline B(r)}\Bbbk_{n-2}(|x-y|)\,\mathrm{d} \mu(x)\geqslant \Bbbk_{n-2}(R+r)\mu^{\operatorname{rad}}(r)-C > -\infty.
\end{equation*}
\notag
$$
IV $\Rightarrow$ I. We take some $R>r$. As the potential ${\operatorname{pt}}_{\mu}$ is bounded below on ${\operatorname{supp} \mu\subset \overline B(r)}$, it is bounded below on $\mathbb R^n$ (see [26], Theorem 3.1.4, or [25], Theorem 1.10). Using the representation of ${\operatorname{pt}}_{\mu}$ by a Riemann-Stieltjes integral we obtain
$$
\begin{equation}
\inf_{y\in \overline B(R)}\int_0^{+\infty}\Bbbk_{n-2}(t) \,\mathrm{d} \mu_y^{\operatorname{rad}}(t)>-\infty.
\end{equation}
\tag{3.30}
$$
However, for $y\in \overline B(R)$ the ball $B_y(R+r)$ contains $\overline B(r)$, so that $\mu_y^{\operatorname{rad}}(t)\equiv \mu^{\operatorname{rad}}(r)$ for $t\geqslant R+r$, and the upper limit of integration in (3.30) can be replaced by $R+r$. This yields
$$
\begin{equation*}
\sup_{y\in \overline B(R)}\int_0^{R+r}\bigl( \Bbbk_{n-2}(R+r)- \Bbbk_{n-2}(t) \bigr)\,\mathrm{d} \mu_y^{\operatorname{rad}}(t)<+\infty,
\end{equation*}
\notag
$$
and by Lemma 1 for $h:=\mu_y^{\operatorname{rad}}$, under the assumption (3.9), from (3.10) and (3.12) we obtain
$$
\begin{equation*}
\sup_{y\in \overline B(R)} \widehat{n}\int_0^{R+r}\frac{\mu_y^{\operatorname{rad}}(t)}{t^{n-1}} \,\mathrm{d} t <+\infty.
\end{equation*}
\notag
$$
Taking $r_0:=R+r$ this yields relation (3.1) required in statement I. I $\Rightarrow$ V. That $\mu$ is a finite measure follows from Lemma 2. On the other hand (3.6) is a special case of (3.1). V $\Rightarrow$ IV. Set $R:=r_0+2r>2r>r$. Since $\mu$ is finite, we have
$$
\begin{equation}
\sup_{y\in \operatorname{supp} \mu}{\mathrm N}_y^{\mu}(R) \leqslant \sup_{y\in \operatorname{supp} \mu}{\mathrm N}_y^{\mu}(r_0)+ \mu^{\operatorname{rad}}(r)\int_{r_0}^R\frac{\mathrm{d} t}{t^{n-1}} <+\infty.
\end{equation}
\tag{3.31}
$$
Hence we obtain, in particular, condition (3.7) from Lemma 1 for $h:=\mu_y^{\operatorname{rad}}$ where $y\in \operatorname{supp} \mu$ is arbitrary. Therefore, from (3.8) and (3.12) we obtain
$$
\begin{equation*}
{\mathrm N}_y^{\mu}(R)={\widehat{n}}\int_{0}^{R}\frac{\mu_y^{\operatorname{rad}}(t)}{t^{n-1}}\,\mathrm{d} t\stackrel{(3.12)}{=} \int_{0}^{R}\bigl(\Bbbk_{n-2}(R)-\Bbbk_{n-2}(t)\bigr)\,\mathrm{d} \mu_y^{\operatorname{rad}}(t)
\end{equation*}
\notag
$$
and, as a consequence,
$$
\begin{equation}
\inf_{y\in \operatorname{supp} \mu}\int_{0}^{R}\Bbbk_{n-2}(t)\,\mathrm{d} \mu_y^{\operatorname{rad}}(t)\geqslant -|\Bbbk_{n-2}(R)| \mu^{\operatorname{rad}}(r) -\sup_{y\in \operatorname{supp} \mu}{\mathrm N}_y^{\mu}(R).
\end{equation}
\tag{3.32}
$$
However, for all $y\in \operatorname{supp} \mu\subset \overline B(r)$ we have $\overline B_y(R)\supset \overline B(r)$, so that $\mu_y(t)\equiv \mu_y(R)$ for $t\geqslant R$. For any $y\in \operatorname{supp} \mu$ this yields
$$
\begin{equation*}
\int_{0}^{R}\Bbbk_{n-2}(t)\,\mathrm{d} \mu_y^{\operatorname{rad}}(t)=\int_{0}^{+\infty}\Bbbk_{n-2}(t)\,\mathrm{d} \mu_y^{\operatorname{rad}}(t) =\int_{\mathbb R^n}\Bbbk_{n-2}(|x-y|)\,\mathrm{d} \mu(x)={\operatorname{pt}}_{\mu}(y).
\end{equation*}
\notag
$$
In combination with (3.32) and (3.31) this ensures that the potential ${\operatorname{pt}}_{\mu}$ is bounded below on $\operatorname{supp} \mu$ and statement IV holds. The proof of the Main Theorem is complete. Remark 2. The implication I $\Rightarrow$ II in the Main Corollary is a consequence of the implication I $\Rightarrow$ II in the Main Theorem. Remark 3. Using (2.20), here and in what follows, from estimates (3.2) for integrals of $U^+$ we can deduce directly estimates for integrals of $|U|$ in terms of twice the right-hand side. In a similar way, for subharmonic functions $u$ equalities (2.21) enable us to estimate the corresponding integrals of $|u|$ from above in terms of a mean value of ${\mathsf C}_{u^+}(R)$ or, by implication, of the positive part of the radial maximum ${\mathsf M}_u(R)\stackrel{(1.16)}{:=}\sup\{u(x)\mid |x|=R\}$ of $u$ in place of the Nevanlinna characteristic ${\boldsymbol T}_u(r,R)$, similarly to Theorems 4–7. In the results that follow we do not explicitly mention this possibility; however, this observation is related directly to the lower bounds discussed in § 1.5. § 4. Modulus of continuity of a measure and integral inequalitiesProposition 2. Let $\mu$ be a Borel measure on $\mathbb R^n$ and Then ${\mathrm h}_{\mu}$ is an increasing function satisfying and if the support $\operatorname{supp} \mu$ of $\mu$ lies in the ball $\overline B(r)$, then Proof. The increase of ${\mathrm h}_{\mu}$ and inequality (4.3) are obvious consequences of (4.2) and the definition (4.1). Since $\overline B(r)\subset \overline B(t)$ for $t\geqslant r$, we have
$$
\begin{equation*}
{\mathrm h}_{\mu}(t)\geqslant {\mathrm h}_{\mu}(r)\stackrel{(4.1)}{=} \sup_{y\in \mathbb R^n}\mu (\overline B_y(t))\geqslant \mu(\overline B(r))\stackrel{(4.2)}{=}M \quad\text{for all } t\geqslant r.
\end{equation*}
\notag
$$
In combination with (4.3) this yields (4.4) and proves Proposition 2. The following result is an easy consequence of the implication I $\Rightarrow$ II. Theorem 8. For $0<r\in \mathbb R^+$ let $\mu$ be a Borel measure on $\overline B(r)\subset \mathbb R^n$ with total mass (4.2) and modulus of continuity ${\mathrm h}_{\mu}$ (see (4.1)) and assume that Proof. It is obvious from the definition (2.15) that
$$
\begin{equation*}
{\mathrm N}_y^{\mu}(x)\stackrel{(2.15)}{:=}\widehat{n}\int_0^x \frac{\mu_y^{\operatorname{rad}}(t)}{t^{n-1}}\,\mathrm{d} t \stackrel{(4.1)}{\leqslant} \widehat{n}\int_0^x \frac{{\mathrm h}_{\mu}(t)}{t^{n-1}}\,\mathrm{d} t
\end{equation*}
\notag
$$
for each $x\in \mathbb R^+$. Since here the right-hand side is independent of $y\in \overline B(r)$ and by (4.5) it is finite for some $x>0$, condition (3.1) in the Main Theorem is satisfied. Moreover, $\mu$ is a finite measure by Lemma 2, so that
$$
\begin{equation*}
\sup_{y\in B(r)}{\mathrm N}_y^{\mu}(r)\leqslant \widehat{n}\int_0^r \frac{{\mathrm h}_{\mu}(t)}{t^{n-1}}\,\mathrm{d} t<+\infty.
\end{equation*}
\notag
$$
In view of this, the implication I $\Rightarrow$ II in the Main Theorem and inequality (3.2) ensure that $U\in \operatorname{dsh}_*(\overline B(R))$ is $\mu$-summable on $\overline B(R)$, and the right-hand side of (4.6) is finite.
Theorem 8 is proved. The next theorem yields upper estimates for integrals with more explicit expressions on the right-hand sides. Theorem 9. For $0<r\in \mathbb R^+$ let $h\colon [0,r]\to \mathbb R^+$ be a continuous function which is differentiable on $(0,r)$, satisfies $h(0)=0$, and such that Proof. If the infimum in (4.7) is $+\infty$, then $h=0$ on $(0,r)$, the measure $\mu$ is infinite by condition (4.8), and inequalities (4.9) and (4.10) hold trivially by (2.8). So ${\mathrm s}_h>0$ in what follows.
We note some properties of $h$. First of all, it follows from (4.8) that $h'$ is strictly positive on $(0,r)$, so that $h$ is strictly increasing on $(0,r)$, and by continuity it is also strictly increasing on the closed interval $[0,r]$. In particular, $h(t)>0$ for $t\in (0,r]$. Based on the definition of ${\mathrm s}_h>0$ in (4.8), we verify by direct calculation that
$$
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d} t}\biggl(\frac{h(t)}{t^{n-2}}\biggr)= \biggl(\frac{th'(t)}{h(t)}\,{+}\,{2\,{-}\,d}\biggr)\frac{h(t)}{t^{n-1}}\stackrel{(4.8)}{\geqslant} \frac{1}{{\mathrm s}_h}\frac{h(t)}{t^{n-1}}>0 \quad\text{for all } t\in (0,r),
\end{equation}
\tag{4.11}
$$
and $t\mapsto h(t)/t^{n-2}$ is a strictly increasing function on $(0,r]$. The fact that this function extends to $0$ by continuity as $\lim_{0<t\to 0}h(t)/t^{n-2}\geqslant 0$ while keeping its strict increasing property is obvious. In addition,
$$
\begin{equation}
\begin{aligned} \, \notag \int_0^{x}\frac{h(t)}{t^{n-1}}\,\mathrm{d} t &\leqslant {\mathrm s}_h\int_0^x \frac{\mathrm{d}}{\mathrm{d} t}\biggl(\frac{h(t)}{t^{n-2}}\biggr)\,\mathrm{d} t ={\mathrm s}_h \frac{h(x)}{x^{n-2}}-{\mathrm s}_h \lim_{0<t\to 0}\frac{h(t)}{t^{n-2}} \\ &\leqslant {\mathrm s}_h \frac{h(x)}{x^{n-2}} <+\infty \quad \text{for } x\in [0,r]. \end{aligned}
\end{equation}
\tag{4.12}
$$
Hence it follows, in particular, that when (4.8) is satisfied, the modulus of continuity ${\mathrm h}_{\mu}$ satisfies (4.5), and furthermore, $h$ attains value $M$ no farther to the right than ${\mathrm h}_{\mu}$ does. In view of (4.4) this means that the quantity $h^{-1}(M)\leqslant r$ is well defined, By Theorem 8, for each $\delta$-subharmonic function $U\not\equiv \pm\infty$ we have (4.6), where the first two terms on the right-hand side are as in (4.9) and (4.10), and only the last factor in parentheses needs additional treatment by means of upper bounds. It is defined as the sum
$$
\begin{equation*}
\begin{aligned} \, &M \max\{1, r^{2-n}\}+\widehat{n}\int_0^{r}\frac{{\mathrm h}_{\mu}(t)}{t^{n-1}}\,\mathrm{d} t \\ &\qquad =M\max\{1, r^{2-n}\}+\widehat{n}\int_0^{h^{-1}(M)}\frac{{\mathrm h}_{\mu}(t)}{t^{n-1}}\,\mathrm{d} t +\widehat{n}\int_{h^{-1}(M)}^{r}\frac{{\mathrm h}_{\mu}(t)}{t^{n-1}}\,\mathrm{d} t \\ &\!\!\!\!\!\!\!\qquad\stackrel{(4.8),(4.3)}{\leqslant} M\max\{1, r^{2-n}\}+\widehat{n}\int_0^{h^{-1}(M)}\frac{h(t)}{t^{n-1}}\,\mathrm{d} t +\widehat{n}\int_{h^{-1}(M)}^{r}\frac{M}{t^{n-1}}\,\mathrm{d} t \\ &\!\!\qquad \stackrel{(4.12)}{\leqslant} M\max\{1, r^{2-n}\}+\widehat{n}\,{\mathrm s}_h\frac{h(h^{-1}(M))}{(h^{-1}(M))^{n-2}} +\widehat{n}\int_{h^{-1}(M)}^{r}\frac{M}{t^{n-1}}\,\mathrm{d} t \\ &\!\!\qquad \stackrel{(2.23)}{=}M\biggl(\max\{1, r^{2-n}\}+\frac{{\mathrm s}_h\widehat{n}}{(h^{-1}(M))^{n-2}} +\bigl(\Bbbk_{n-2}(r)-\Bbbk_{n-2}(h^{-1}(M))\bigr)\biggr). \end{aligned}
\end{equation*}
\notag
$$
For $n=2$ the right-hand side is
$$
\begin{equation*}
M\biggl(1+{\mathrm s}_h+\ln\frac{r}{h^{-1}(M)}\biggr) =M\ln\frac{e^{1+{\mathrm s}_h}r}{h^{-1}(M)}
\end{equation*}
\notag
$$
and coincides with the part of the right-hand side of (4.9) that contains $M$. For ${n\!>\!2}$ it is
$$
\begin{equation*}
\begin{aligned} \, &M\biggl(\max\{1, r^{2-n}\}+\frac{{\mathrm s}_h(n-2)}{(h^{-1}(M))^{n-2}} +\biggl(\frac{1}{(h^{-1}(M))^{n-2}} -\frac{1}{r^{n-2}}\biggr)\biggr) \\ &\qquad =M\biggl((1- r^{2-n})^++\frac{{1+\mathrm s}_h(n-2)}{(h^{-1}(M))^{n-2}} \biggr) \leqslant M\biggl(1+\frac{1+(n-2){\mathrm s}_h}{(h^{-1}(M))^{n-2}}\biggr) \end{aligned}
\end{equation*}
\notag
$$
and coincides with the part of the right-hand side of (4.10) that contains $M$.
Theorem 9 is proved. Remark 4. Condition (4.7) in Theorem 9 can also be written as § 5. Hausdorff measure and content in integral inequalities Definition 3 (see [35], [23] and [36]–[39]). Given a function $h\colon \mathbb R^+\to \mathbb R^+$ and a quantity $t\in \overline{\mathbb R}^+\setminus 0$, we call the set function Here and in what follows we often use without explicit references the well-known classical properties of Hausdorff contents and Hausdorff measures, which can be found in the basic references at the beginning of Definition 3. Example. The linear Lebesgue measure $\lambda_{\mathbb R}$ on $\mathbb R$ and the plane Lebesgue measure $\lambda_{\mathbb C}$ on $\mathbb C$, which we used in §§ 1 and 2, coincide with the one-dimensional Hausdorff measure $1\text{-}{\mathfrak m}^{0}$ on $\mathbb R$ and the two-dimensional Hausdorff measure $2\text{-}{\mathfrak m}^{0}$ on $\mathbb C$, while the cardinality of a set $S$ is equal to its zero-dimensional Hausdorff measure $0\text{-}{\mathfrak m}^{0}(S)$; however, the reader is advised to take account of [40] for better accuracy. Furthermore, the $n$-dimensional space Lebesgue measure $\lambda_{\mathbb R^n}$ on $\mathbb R^n$ coincides with the $n$-dimensional Hausdorff measure $n\text{-}{\mathfrak m}^{0}(S)$. If $p>n$, then the $p$-dimensional Hausdorff measure $p\text{-}{\mathfrak m}^{0}$ in $\mathbb R^n$ is the null measure. We will repeatedly use the following theorem of Frostman, which is fundamental for potential theory. Theorem 10 (see [35], Theorem II.1, and [37], Theorem 5.1.12). I. If $h\colon \mathbb R^+\to \mathbb R^+$ is a function and $\mu$ is a Borel measure on $\mathbb R^n$ with modulus of continuity then II. In any dimension $n$ there exists $A>0$ such that for any increasing function $h\colon \mathbb R^+\to \mathbb R^+$, $h(0)=0$, and any compact set $E\subset \mathbb R^n$ there exists a Radon measure $\mu$ on $E$ such that (5.5) and, as a consequence, both (5.6) and the reverse inequality hold.Remark 5. In the formulations of Theorem 10 that we know parts I and II have common assumptions, and, as a result, in part I the function $h$ is overburdened with conditions. Part I is proved by taking $N$, $x_j$ and $r_j$ as in curly brackets in (5.1) and then applying the operation $\inf$ over all such $N$, $x_j$ and $r_j$ to both sides of the inequalities Part II of Theorem 10 is much deeper. It is of interest here, for instance, because in the case of Theorems 8 and 9 it ensures that measures $\mu$ satisfying the assumptions of these theorems exist. Theorem 11. Let $\mu$ be a Borel measure with total mass $M$ and modulus of continuity ${\mathrm h}_{\mu}$ which is concentrated on a $\mu$-measurable set $S\subset \overline B(r)$. Then and for each function $h\colon [0,r]\to \mathbb R^+$ such that $h\geqslant {\mathrm h}_{\mu}$ on $[0,r]$, provided that $h$ is extended to $(r,+\infty)$ by $h(r)$, the inequalities In particular, (i) on the right-hand side of inequality (4.6) in Theorem 8 the total mass $M$ can be replaced by the ${\mathrm h}_{\mu}$-content ${\mathfrak m}_{{\mathrm h}_{\mu}}^{t}(S)$ with any radius $t\in \overline{\mathbb R}^+$; (ii) in Theorem 9 the pairs of occurrences of the total mass $M$ on the right-hand sides of (4.9) and (4.10) can simultaneously be replaced by the $h$-content ${\mathfrak m}_h^{t}(S)$ of $S$ with any radius $t\in \overline{\mathbb R}^+$. Proof. As concerns part I of Theorem 10, for $h:={\mathrm h}_{\mu}$ we have
$$
\begin{equation}
M=\mu(\mathbb R^n)=\mu(S)\leqslant {\mathfrak m}_{{\mathrm h}_{\mu}}^{{\infty}}(S) \stackrel{(5.2)}{\leqslant} {\mathfrak m}_{{\mathrm h}_{\mu}}^{t}(S),
\end{equation}
\tag{5.10}
$$
which yields (5.9) immediately. Given a content radius $t\geqslant r$, the ball $\overline B(r)$ contains $S$ but on the other hand ${\mathfrak m}_{{\mathrm h}_{\mu}}^{t}(S)\leqslant {\mathrm h}_{\mu}(r)$ by the definition (5.1), where the right-hand side is at most $M$ by inequality (4.3) in Proposition 2. In combination with (5.10) this yields (5.8). Inequalities (5.9) imply (i) in an obvious way.
To prove (ii) we need the following lemma. Lemma 4. Let $H$ be a function as in Theorem 9, and let ${\mathrm s}_h$ in (4.7) be positive, so that $h$ is strictly increasing on $[0,r]$, where $r\leqslant B\in \mathbb R^+$. Then and are increasing functions on $[0, h(r)]$. Proof. We make the change of variables $y:=h^{-1}(x)\in [0,r]$ and, from the functions in (5.11) and (5.12), go over to the following functions:
$$
\begin{equation}
y \stackrel{(5.11)}{\longmapsto} \frac{h(y)}{y^{n-2}},\qquad y\in [0,r], \quad \text{for } n>2,
\end{equation}
\tag{5.13}
$$
$$
\begin{equation}
y \stackrel{(5.12)}{\longmapsto} h(y)\ln \frac{Be^{{\mathrm s}_h}}{y},\qquad y\in [0,r], \quad \text{for } n=2.
\end{equation}
\tag{5.14}
$$
As the continuous function $h$ is strictly increasing on $[0,r]$, it is sufficient to show that the functions in (5.13) and (5.14) are increasing. For the function in (5.13) this was verified in (4.11) and the part of the proof of Theorem 9 following (4.11). As for the function in (5.14), differentiation on $(0,r)$ yields
$$
\begin{equation*}
\begin{aligned} \, \frac{\mathrm{d}}{\mathrm{d} y}h(y)\ln \frac{Be^{{\mathrm s}_h}}{y} &=h'(y)\ln \frac{Be^{{\mathrm s}_h}}{y}-\frac{h(y)}{y} \\ &\!\!\stackrel{(4.7)}{\geqslant} h'(y)\ln \frac{Be^{{\mathrm s}_h}}{y}-{\mathrm s}_hh'(y)=h'(y)\ln \frac{B}{y}\geqslant 0 \end{aligned}
\end{equation*}
\notag
$$
on $(0,r)$ for $B\geqslant r$, which means that the function in (5.14) is increasing on $(0,r)$ and, since it is continuous, also on $[0,r]$.
The proof is complete. Now Lemma 4 shows that, as the function (5.11) is increasing, we can replace $M$ by the $h$-content ${\mathfrak m}_h^{t}(S) \stackrel{(5.9)}{\geqslant} M$ of $S$ with any content radius $t\in \overline{\mathbb R}^+$ in inequality (4.10) in Theorem 9. As concerns (4.9) in Theorem 9, using Lemma 4 again, since the function in (5.12) with $B:=er\geqslant r$ is increasing, we can replace $M$ by the $h$-content ${\mathfrak m}_h^{t}(S)\stackrel{(5.9)}{\geqslant} M$ of $S$ with any content radius $t\in \overline{\mathbb R}^+$. Theorem 11 is proved. The replacement of $M$ in inequality (4.6) in Theorem 8 by Hausdorff $h$-content ${\mathfrak m}_{h}^{{\infty}}(S)$ with radius $t\geqslant r$, which is allowed by Theorem 11 (i), does not make this inequality weaker in view of (5.8). Moreover, in some cases, after the replacement of the total mass $M$ by the Hausdorff $h$-content ${\mathfrak m}_{h}^{{\infty}}(S)$ with radius $+\infty$, inequalities (4.10) and (4.9) with arbitrary $h$ in Theorem 9 only become weaker by a multiplicative constant. This is reflected in the following theorem. Theorem 12. There exists $A\geqslant 1$ which only depends on the dimension $n$ such that for each $r\in \mathbb R^+\setminus 0$, each compact set $S\subset \overline B(r)$ and any function $h\colon {[0,r]\to \mathbb R^+}$ satisfying the assumptions of Theorem 9 and such that the constant ${\mathrm s}_h $ defined in (4.7) is positive, there exists a Borel measure $\mu$ on $\overline B(r)$ with total mass $M>0$, support $\operatorname{supp} \mu \subset S$ and modulus of continuity satisfying (4.8) such that, apart from inequalities (4.9) and (4.10) involving $M$ and ${\mathfrak m}_{h}^{{\infty}}(S)$ in place of ${M}$ alike, for an arbitrary $\delta$-subharmonic function $U\not\equiv\pm\infty$ on the ball $\overline B(R)$ of radius $R>r$ the following inequalities hold, with $M$ multiplied by $A$: and Proof. Let $E:=S\subset \overline{B}(r)$ and let $h$ be the function from the statement of Theorem 9 extended by the constant $h(r)$ to $(r,+\infty)$. By part II of Theorem 10 we can find a Radon measure $\mu\neq 0$ with all the properties specified there. Then the assumptions of Theorems 9 and 11 are satisfied, so their conclusions (4.9), (4.10) and (ii), respectively, hold. At the same time it follows form (5.9) and (5.7) that
$$
\begin{equation*}
M=\mu(S)\stackrel{(5.9)}{\leqslant} {\mathfrak m}_h^{{\infty}}(S) \stackrel{(5.7)}{\leqslant} A\mu(S)=AM.
\end{equation*}
\notag
$$
Hence, bearing in mind that $h^{-1}$ in the denominators of the left-hand sides of (5.16) and (5.15) is an increasing function, we obtain the inequalities in (5.15) and (5.16).
The theorem is proved. § 6. Particular cases of inequalities for integrals of differences of subharmonic functions6.1. The case of Hausdorff $p$-dimensional contents and measures Theorem 13. Let $0<r\leqslant t\in \overline{\mathbb R}^+$, $p\in (n-2,n]$ and $b\in \mathbb R^+$. Then for each Borel measure $\mu$ on $\overline B(r)$ with support $\operatorname{supp} \mu\subset S \subset \overline B(r)$ and modulus of continuity Proof. Set $h(x):=bx^p$ for $x\in [0,r]$. Then
$$
\begin{equation}
\frac1{{\mathrm s}_h}\stackrel{(4.7)}{=}p-({n-2})>0, \qquad h^{-1}(y)= \biggl(\frac{y}{b}\biggr)^{1/p}\quad\text{and} \quad h\stackrel{(5.3)}{=}\frac{b}{c_p}h_p.
\end{equation}
\tag{6.4}
$$
By (6.1) condition (4.8) in Theorem 9 is satisfied. From that theorem and Theorem 11, (ii) we obtain
$$
\begin{equation*}
\int U^+\,\mathrm{d} \mu \leqslant 5\frac{R+r}{R-r} {\boldsymbol T}_U(r,R) \, {\mathfrak m}_h^{t}(S) \ln\frac{e^{1+1/p}r}{h^{-1}({\mathfrak m}_h^{t}(S))} \quad\text{for } n=2
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\int U^+\,\mathrm{d} \mu \leqslant A_n(r,R) {\boldsymbol T}_U(r,R)\, {\mathfrak m}_h^{t}(S) \biggl(1+\frac{1+p/(p-(n-2))}{(h^{-1}({{\mathfrak m}_h^{t}(S)}))^{n-2}}\biggr) \quad\text{for } n>2.
\end{equation*}
\notag
$$
By the equality
$$
\begin{equation*}
h^{-1}({{\mathfrak m}_h^{t}(S)})\stackrel{(6.4)}{=} \biggl(\frac{{\mathfrak m}_h^{t}(S)}{b}\biggr)^{1/p}
\end{equation*}
\notag
$$
this can also be written as
$$
\begin{equation}
\int U^+\,\mathrm{d} \mu \leqslant 5\frac{R+r}{R-r} {\boldsymbol T}_U(r,R) \, {\mathfrak m}_h^{t}(S)\frac{1}{p} \ln\frac{be^{p+1}r^p}{{\mathfrak m}_h^{t}(S)} \quad\text{for } n=2\quad (\text{that is, for}\ \mathbb C)
\end{equation}
\tag{6.5}
$$
and
$$
\begin{equation}
\begin{aligned} \, \notag \int U^+\,\mathrm{d} \mu &\leqslant A_n(r,R) {\boldsymbol T}_U(r,R)\, {\mathfrak m}_h^{t}(S) \\ &\qquad\times \biggl(1+\frac{b^{(n-2)/p}(2p-({n-2}))}{(p-({n-2}))({{\mathfrak m}_h^{t}(S)})^{(n-2)/p}}\biggr) \quad\text{for } n>2. \end{aligned}
\end{equation}
\tag{6.6}
$$
In addition, by the definitions of Hausdorff $h$-content (5.1) and $p$-dimensional Hausdorff content (5.4) (see Definition 3), from the last equality in (6.4) we obtain
$$
\begin{equation*}
{\mathfrak m}_h^{t}\stackrel{(6.4)}{=}\frac{b}{c_p} {\mathfrak m}_{h_p}^{t}\stackrel{(5.4)}{=} \frac{b}{c_p}p\text{-}{\mathfrak m}^{t},
\end{equation*}
\notag
$$
and substituting this into the right-hand sides of (6.5) and (6.6) we have
$$
\begin{equation}
\int U^+\,\mathrm{d} \mu \leqslant \frac{5b}{pc_p}\frac{R+r}{R-r} {\boldsymbol T}_U(r,R)\, p\text{-}{\mathfrak m}^{t}(S) \ln\frac{c_pe^{1+p}r^p}{p\text{-}{\mathfrak m}^{t}(S)} \quad\text{for } n=2
\end{equation}
\tag{6.7}
$$
and
$$
\begin{equation}
\begin{aligned} \, \notag \int U^+\,\mathrm{d} \mu &\leqslant bA_n(r,R) {\boldsymbol T}_U(r,R)p\text{-}{\mathfrak m}^{t}(S) \\ &\qquad\times\biggl(\frac{1}{c_p}+\frac{c_p^{(n-2)/p-1}(n+2)}{(p-({n-2})) (p\text{-}{\mathfrak m}^{t}(S))^{(n-2)/p}}\biggr) \quad\text{for } n>2, \end{aligned}
\end{equation}
\tag{6.8}
$$
respectively, because $p\leqslant n$. For $n=2$, from the definition (5.3) we have the upper bound
$$
\begin{equation*}
\pi \geqslant c_p\stackrel{(5.3)}{:=}\dfrac{\pi^{p/2}}{\Gamma(p/2+1)}\geqslant 1 \quad\text{for } p\in (0,2].
\end{equation*}
\notag
$$
By (6.7) this yields (6.2).
For $n\geqslant 3$ and $p\in (n-2,n]$, as $(n-2)/p-1<0$, we obtain
$$
\begin{equation*}
\begin{aligned} \, c_p^{(n-2)/p-1} &= \biggl(\frac{\Gamma(p/2+1)}{\pi^{p/2}}\biggr)^{1-(n-2)/p} \leqslant \biggl(\Gamma\biggl(\frac p2+1\biggr)\biggr)^{1-(n-2)/p} \\ &\leqslant \biggl(\Gamma\biggl(\frac n2+1\biggr)\biggr)^{2/n}\leqslant \frac{n}{2}, \end{aligned}
\end{equation*}
\notag
$$
so for the expression in parentheses in (6.8) we have
$$
\begin{equation*}
\begin{aligned} \, \biggl(\frac{1}{c_p}+\frac{c_p^{(n-2)/p-1}({n+2})}{(p-({n-2})) (p\text{-}{\mathfrak m}^{t}(S))^{(n-2)/p}}\biggr) &\leqslant \biggl(\frac n2\biggr)^{n/2}\!+\frac{n({n+2})/2}{(p-({n-2})) (p\text{-}{\mathfrak m}^{t}(S))^{(n-2)/p}} \\ &\leqslant n^{n}\biggl(\!1+\frac{1} {(p-({n-2})) (p\text{-}{\mathfrak m}^{t}(S))^{(n-2)/p}} \biggr). \end{aligned}
\end{equation*}
\notag
$$
By (6.8) this yields (6.3). Theorem 13 is proved. 6.2. Functions on the complex plane or $ {\mathbb R}^n$In upper estimates the factor $(R+r)/(R-r)$ near the beginning of the right-hand side in the planar case, as well as the factor $A_n$ explicitly given in (3.3) in the case of $\mathbb R^n$ with $n>2$, allow one to take the closeness of $R>r$ to $r$ explicitly into account. The case when $R$ is significantly distant from $r$, when functions and measures are considered on the whole of $\mathbb R^n$ and the Nevanlinna characteristic grows sufficiently slowly, can be no less important. Corollary 2. Let $\mu$ be a Radon measure on $\mathbb R^n$, let $U\not\equiv \pm \infty$ be a $\delta$-subharmonic function on the whole of $\mathbb R^n$, and let $s\colon \mathbb R^+\to \mathbb R^+\setminus 0$ be a function. Then the following hold. I. If for each $R\in \mathbb R^+$ (cf. (3.1))
$$
\begin{equation*}
\sup_{y\in \overline B(R)}{\mathrm N}_y^{\mu}(r_0)<+\infty \quad\textit{for some } r_0>0,
\end{equation*}
\notag
$$
then $U$ is a locally integrable function with respect to $\mu$ and
$$
\begin{equation*}
\begin{aligned} \, \int_{\overline B(r)} U^+\,\mathrm{d} \mu &\leqslant 5n \biggl(1+\frac{2r}{s(r)}\biggr)^{n-1} (1+s(r))^{n-2}{\boldsymbol T}_U(r,r+s(r)) \\ &\qquad \times \Bigl(\mu^{\operatorname{rad}}(r)\max\{1, r^{2-n}\} +\sup_{y\in \mathbb R^n}{\mathrm N}_y^{\mu}(r)\Bigr) \quad\textit{for each } r\in \mathbb R^+. \end{aligned}
\end{equation*}
\notag
$$
II. If $h\colon \mathbb R^+\to \mathbb R^+$ is a continuous function which is differentiable on $\mathbb R^+\setminus 0$ and satisfies $h(0)=0$, and if conditions (4.7) and (4.8) hold for each $r\in \mathbb R^+$ (that is, for $r:=+\infty$), then $U$ is locally $\mu$-summable and
$$
\begin{equation*}
\begin{aligned} \, \int_{\overline D(r)} U^+\,\mathrm{d} \mu &\leqslant 5\biggl(1+\frac{2r}{s(r)}\biggr) {\boldsymbol T}_U(r,r+s(r)) \\ &\qquad\times \mu^{\operatorname{rad}}(r)\ln\frac{e^{1+{\mathrm s}_h}r}{h^{-1}( \mu^{\operatorname{rad}}(r))}\quad\textit{for } n=2 \quad(\textit{that is, on}\ \mathbb C) \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \int_{\overline B(r)} U^+\,\mathrm{d} \mu &\leqslant 5n \biggl(1+\frac{2r}{s(r)}\biggr)^{n-1} (1+s(r))^{n-2}{\boldsymbol T}_U(r,r+s(r)) \\ &\qquad \times \mu^{\operatorname{rad}}(r)\biggl(1+\frac{1+(n-2){\mathrm s}_h}{(h^{-1}(\mu^{\operatorname{rad}}(r)))^{n-2}}\biggr) \quad\textit{for } n>2, \end{aligned}
\end{equation*}
\notag
$$
where the pairs of occurrences of $\mu^{\operatorname{rad}}(r)$ on the right-hand side can simultaneously be replaced by the Hausdorff $h$-content ${\mathfrak m}_h^{{\infty}}(\overline B(r)\cap \operatorname{supp} \mu)$ with infinite content radius or by the Hausdorff $h$-measure ${\mathfrak m}_h^{0}(\overline B(r)\cap \operatorname{supp} \mu)$ of the part of the support of $\mu$ lying in the disc $\overline D(r)\subset \mathbb C$ or ball $\overline B(r)\subset \mathbb R^n$ for $n>2$, respectively. On the right-hand sides of the above inequalities the first argument $r$ of ${{\boldsymbol T}_U(r,r+s(r))}$ can be replaced by any $r'\in [0,r]$. Proof. Part I is inequality (3.2) in the Main Theorem for the restriction $\mu{\lfloor}_{\overline B(r)}$ of $\mu$ to $\overline B(r)$ which is written for and slightly simplified (coarsened) for
$$
\begin{equation}
\begin{aligned} \, \notag A_n(r,r+s(r)) &\stackrel{(3.3)}{:=} 5\max\{1, n-2\}\biggl(\frac{(r+s(r))+r}{s(r)}\biggr)^{n-1}\max\bigl\{1,(s(r))^{n-2}\bigr\} \\ &\stackrel{(6.9)}{\leqslant} 5(n-1) \biggl(1+\frac{2r}{s(r)}\biggr)^{n-1} \max\bigl\{1,(s(r))^{n-2}\bigr\}, \end{aligned}
\end{equation}
\tag{6.10}
$$
where the last maximum can be replaced by the larger sum $(1+s(r))^{n-2}$.
Part II is obtained by applying Theorem 9 for $r\in \mathbb R^+$ and inequalities (4.9) and (4.10) in it to the restrictions of the Radon measure $\mu$ to the discs $\overline D(r)$ or balls $\overline B(r)$, taking (6.10) into account and using Theorem 11 (ii) in the final part of the argument. This completes the proof of Corollary 2. 6.3. Functions on the disc $ {\mathbb D}:=D(1)$ or ball $ {\mathbb B}:=B(1)\subset {\mathbb R}^n$Theorems 1–7 are poorly adapted to meromorphic functions and differences of subharmonic functions on the unit disc. Throughout this subsection $\mu$ is the Borel measure in $\mathbb B\subset \mathbb R^n$ that is finite in $r\overline{\mathbb B}=\overline B(r)$ for each $r\in [0,1)$, $U\not\equiv \pm \infty$ is a $\delta$-subharmonic function on $\mathbb B$, and $s\colon [0,1)\to \mathbb R^+$ is a function such that Corollary 3. Let $n\geqslant 2$ and let $\mu_r:=\mu{\lfloor}_{r\mathbb B}$ be the restriction of $\mu$ to $r\mathbb B$. If Proof. By (6.12) we have statement V of the Main Theorem, with inequality (3.6) for $\mu_r$ in place of $\mu$. By the implication V $\Rightarrow$ II in the Main Theorem, taking inequalities
$$
\begin{equation}
\begin{aligned} \, \notag &A_n(r,r+s(r)) \\ &\qquad \stackrel{(6.10)}{\leqslant} 5{(n}-1) \biggl(1+\frac{2r}{s(r)}\biggr)^{n-1} \max\bigl\{1,(s(r))^{n-2}\bigr\} \stackrel{(6.11)}{\leqslant} \frac{5(n-1)3^{n-1}}{(s(r))^{n-1}} \notag \\ &\qquad\ \, \leqslant \frac{1}{(s(r))^{n-1}} \begin{cases} 15 &\text{for } n=2, \\ 3^{2n} &\text{for } n\geqslant 2, \end{cases} \qquad \text{for all } r\in [0,1) \end{aligned}
\end{equation}
\tag{6.14}
$$
into account, from (3.2) with $\mu_r$ in place of $\mu$ and $r+s(r)$ as $R$ we obtain (6.13), which completes the proof. Corollary 4. If $h\colon [0,1]\to \mathbb R^+$ is a continuous function which is differentiable on $(0,1)$ and satisfies $h(0)=0$, and (4.7) and (4.8) hold for $r:=1$, then $U$ is $\mu$-summable on each disc $r\overline{\mathbb D}$ or ball $r\overline{\mathbb B}$ of radius $r<1$, respectively, and the following inequalities hold: The proof consists of two steps: at the first step we apply Theorems 9 and 11, (ii) to the restriction $\mu{\lfloor}_{r\mathbb B}$ of the measure $\mu$ in place of $\mu$ for $r+s(r)$ as $R$; at the second we simplify the constant in estimates by taking avail of (6.14). 6.4. The case of the $n$-dimensional space Lebesgue measure on $ {\mathbb R}^n$In Theorem 13 we take as $\mu$ the restriction of $\lambda_{\mathbb C}$ to a $\lambda_{\mathbb C}$-measurable set $E\subset \overline D(r)$ or the restriction of $\lambda_{\mathbb R^n}$ to a $\lambda_{\mathbb R^n}$-measurable set $E\subset \overline B(r)$. Then by Theorem 13, taking $p:=n$, $t:=0$ and choosing $b$ in (6.1) to be the area $\pi$ of the unit disc $D(1)\subset \mathbb C$ for $n=2$ or the volume of the unit ball $B(1)\subset \mathbb R^n$, which is
$$
\begin{equation}
\frac{\pi^{n/2}}{\Gamma(n/2+1)}\leqslant \frac{\pi^{{5}/2}}{\Gamma({5}/2+1)}=\frac{8}{15}\pi^2< 6 \quad\text{for } n>2,
\end{equation}
\tag{6.15}
$$
since $\lambda_{\mathbb R^n}$ coincides with the $n$-dimensional Hausdorff measure $n\text{-}{\mathfrak m}^{0}$ in $\mathbb R^n$, as mentioned in the example, we obtain the following corollary. Corollary 5. Given $r$, $0<r\in \mathbb R^+$, and a $\lambda_{\mathbb R^n}$-measurable set $E\subset \overline B(r)$, for any $\delta$-subharmonic function $U\not\equiv \pm \infty$ on the ball $\overline B(R)$ of radius $R>r$, and where the argument $r$ of ${\boldsymbol T}_U(r,R)$ can be replaced by any $r'\in [0,r]$. Remark 6. Inequality (6.16) even improves slightly inequality (1.25) in Theorem 7 on small plane sets, and not just on the level of absolute constants. For example, if $R:=kr$, where $k>1$, then on the right-hand sides of (1.25) we can remove $k$ under the logarithm sign, which can be essential for large $k$ when we consider $\delta$-subharmonic or meromorphic functions with very slowly growing Nevanlinna characteristic on $\mathbb C$. The next result is a combination of Corollaries 5 and 2 taking (6.10) into account. Corollary 6. Let $U\not\equiv \pm \infty$ be a $\delta$-subharmonic function on the whole of $\mathbb R^n$, and let $s\colon \mathbb R^+\to \mathbb R^+\setminus 0$ be an arbitrary function. Then for each $\lambda_{\mathbb R^n}$-measurable set $E\subset \mathbb R^n$ and each $r\in \mathbb R^+$, 6.5. The case of the $(n-1)$-dimensional surface measure in $ {\mathbb R}^n$Here we just touch upon the integration of differences of subharmonic functions in $\mathbb C$ against the arc length measure in $\mathbb C$ or against the surface measure on a hypersurface in $\mathbb R^n$, because more general corollaries concerning integration of $\delta$-subharmonic functions over Riemann surfaces (see [30]) or manifolds of fractal dimension can also be stated. In using known facts on Lipschitz functions and their relationships with rectifiability and Hausdorff measures we mostly follow [23], § 3.2, [41], [24], § 3.3, and [42], § 3.7, but leave out precise references. 6.5.1. The case of a curve in $\mathbb C$Let $O\neq \varnothing $ be an open set in $\mathbb R$ and $l\colon O\to \mathbb C$ be an injective Lipschitz function with Lipschitz constant
$$
\begin{equation}
\operatorname{Lip}(l):=\sup_{\stackrel{x_1\neq x_2}{x_1,x_2\in O}}\frac{|l(x_2)-l(x_1)|}{|x_2-x_1|}\in \mathbb R^+;
\end{equation}
\tag{6.18}
$$
in this subsection we occasionally call it a Lipschitz curve (without self-intersections and endpoints). By Rademacher’s theorem this function is differentiable almost everywhere with respect to the linear Lebesgue measure $\lambda_{\mathbb R}$, and the modulus of its derivative has a well-defined essential supremum (cf. (1.18)):
$$
\begin{equation}
\|l'\|_{\infty}:=\inf \bigl\{a\in \mathbb R\mid \lambda_{\mathbb R}(\{x\in E\mid |l'(x)|>a\})=0 \bigr\}\leqslant \,\operatorname{Lip}(l) \in \mathbb R^+.
\end{equation}
\tag{6.19}
$$
For a Borel set $E\subset l(O)$, its length
$$
\begin{equation}
\sigma(E):=\int_{l^{-1}(E)}|l'|\,\mathrm{d} \lambda_{\mathbb R}
\end{equation}
\tag{6.20}
$$
coincides with its one-dimensional Hausdorff measure $1\text{-}{\mathfrak m}^{0}(E)$. In particular, the length measure $\sigma$ on $l(O)$ is a regular Borel measure in $\mathbb C$. We say that a Lipschitz curve $l\colon O\to \mathbb C$ is bi-Lipschitz if
$$
\begin{equation}
\operatorname{Lip}(l^{-1}):= \sup_{\stackrel{z_1\neq z_2}{z_1,z_2\in l(O)}}\frac{|l^{-1}(z_2)-l^{-1}(z_1)|}{|z_2-z_1|}= \sup_{\stackrel{x_1\neq x_2}{x_1,x_2\in O}}\frac{|x_2-x_1|}{|l(x_2)-l(x_1)|}\in \mathbb R^+.
\end{equation}
\tag{6.21}
$$
Corollary 7. Let $l\colon O\to \mathbb C$ be a bi-Lipschitz curve and let $l(O)\subset \overline D(r)$ for some $r\in \mathbb R^+$. Then each $\delta$-subharmonic function $U\not\equiv \pm\infty$ on the disc $\overline D(R)$ of radius $R>r$ is summable with respect to the arc length measure $\sigma$ on $l(O)$, and for each Borel subset $E$ of $ l(O)$ Proof. By (6.20) the intersection of the disc $\overline D_z(t)$ with a Borel set $E\subset l(O)$, where $z\in \mathbb C$ is arbitrary, has length
$$
\begin{equation*}
\int_{l^{-1}(E\cap \overline D_z(t) )}|l'|\,\mathrm{d} \lambda_{\mathbb R} \stackrel{(6.19)}{\leqslant} \operatorname{Lip}(l) \lambda_{\mathbb R}\bigl(l^{-1}(\overline D_z(t))\bigr),
\end{equation*}
\notag
$$
where, bearing in mind that $\operatorname{Lip}(l^{-1})\stackrel{(6.21)}{<}+\infty$, the diameter of $l^{-1}(\overline D_z(t))=l^{-1}(l(O)\cap \overline D_z(t))$ is at most $\operatorname{Lip}(l^{-1}) \cdot 2t$. Hence, for any $z\in \mathbb C$ and $t\in \mathbb R^+$ the length $\sigma(l(O)\cap \overline D_z(t))$ (as defined in (6.20)) of the part of $l(O)$ occurring in $\overline D_z(t)$ is at most $\operatorname{Lip}(l) \,\operatorname{Lip}(l^{-1}) \cdot 2t$. Thus, for the modulus of continuity ${\mathrm h}_{\sigma_E}$ of the restriction $\sigma_E:=\sigma{\lfloor}_E$ of $\sigma$ to $E$ we have
$$
\begin{equation*}
{\mathrm h}_{\sigma_E}(t)\leqslant {\mathrm h}_{\sigma}(t)\leqslant 2\,\operatorname{Lip}(l) \,\operatorname{Lip}(l^{-1})\, t,
\end{equation*}
\notag
$$
which means that condition (6.1) in Theorem 13 holds for
$$
\begin{equation*}
p:=1, \qquad b:=2\,\operatorname{Lip}(l) \,\operatorname{Lip}(l^{-1})
\end{equation*}
\notag
$$
and the Borel measure $\mu:=\sigma_E$. By Theorem 13 the required inequality (6.22) is just explicitly written relation (6.2), where the equality $1\text{-}{\mathfrak m}^{0}(E)=\sigma (E)=\sigma_E(E)$ for $E\subset l(O)$ is taken into account.
The proof is complete. Remark 7. When $l$ is differentiable on $O$, then by Lagrange’s mean value theorem we have the rather rough estimate Remark 8. We can also consider curves $l$ with self-intersections, dropping the condition that $l$ is injective and using the so-called multiplicity function for the curve and the area formula (see [23], § 3.2, [24], § 3.3, and [42], § 3.7). Now we consider also Lipschitz curves with special parametrization. As before, let $O\subset \mathbb R$ be an open subset of $\mathbb R$, and let $y\colon O\to \mathbb R$ be a Lipschitz function with Lipschitz constant $\operatorname{Lip}(y)\in \mathbb R^+$. The corresponding Lipschitz curve in $\mathbb C$ (without self-intersections and endpoints)
$$
\begin{equation}
l_y\colon x\mapsto x+iy(x)\in \mathbb C,\qquad x\in O,
\end{equation}
\tag{6.24}
$$
with Lipschitz constant obviously equal to
$$
\begin{equation}
\operatorname{Lip}(l_y)=\sqrt{1+(\operatorname{Lip}(y))^2}\in \mathbb R^+
\end{equation}
\tag{6.25}
$$
is called a curve with bounded slope $q:=\operatorname{Lip}(y)$ in $\mathbb C$; it is often (not quite accurately) treated as the image $l_y(O)\subset \mathbb C$ of the function (6.24) or its graph $\{x+iy(x)\mid x\in O\}\subset \mathbb C$. The Lipschitz curve (6.24) is a fortiori bi-Lipschitz because
$$
\begin{equation}
\operatorname{Lip}(l_y^{-1})\stackrel{(6.21)}{=} \sup_{\stackrel{x_1\neq x_2}{x_1,x_2\in O}}\frac{|x_2-x_1|}{\sqrt{(x_2-x_1)^2+(y(x_2)-y(x_1))^2}} \leqslant 1.
\end{equation}
\tag{6.26}
$$
Corollary 8. Let $l_y$ in (6.24) be a curve with bounded slope $q\in \mathbb R^+$ in $\mathbb C$ with arc length measure $\sigma$, and let $s\colon \mathbb R^+\to \mathbb R^+\setminus 0$ be an arbitrary function. Then each $\delta$-subharmonic function $U\not\equiv \pm\infty$ on $\mathbb C$ is locally integrable with respect to the arc length measure $\sigma$ on $l_y(O)$, and for each Borel set $E\subset l_y(O)$ and any $r\in \mathbb R^+$, Proof. Consider the open set $O_r :=l_y^{-1}(l(O)\cap D(r))$ and the bi-Lipschitz curve $l_y^r\colon O_r\to D(r)\subset \overline D(r)$, which is the restriction of $l_y$ to $O_r$. By Corollary 7 applied to $l_y^r$, from (6.22) for $R:=r+s(r)$, taking (6.25) and (6.26) into account, we obtain the inequality required in Corollary 8. 6.5.2. The case of a hypersurface in $\mathbb R^n$Let $O\neq \varnothing$ be an open subset of $\mathbb R^{n-1}$ and let $s\colon O\to \mathbb R^n$ be an injective Lipschitz function with finite Lipschitz constant $\operatorname{Lip}(l)$ defined as in (6.18). We will also call it a Lipschitz hypersurface (without boundary or self-intersections) in $\mathbb R^n$. By Rademacher’s theorem, the modulus of Jacobian $ |Jl|(x)$, which is equal to the (arithmetic) square root of the sum of squares of all the $n$ $(n-1)$-minors of the Jacobian matrix at $x$, is defined for almost all $x\in O$ with respect to the Lebesgue measure $\lambda_{\mathbb R^{n-1}}$. In particular, the essential supremum of the modulus of Jacobian is defined almost everywhere on $O$ with respect to the Lebesgue measure $\lambda_{\mathbb R^{n-1}}$ and
$$
\begin{equation}
\|Jl\|_{\infty}\leqslant \sqrt{(n-1)!\,n}\,(\operatorname{Lip}(l))^{n-1}\in \mathbb R^+.
\end{equation}
\tag{6.27}
$$
For a Borel set $E\subset l(O)$ its $(n-1)$-dimensional area
$$
\begin{equation}
\sigma(E):=\int_{l^{-1}(E)} |Jl|\,\mathrm{d} \lambda_{\mathbb R^{n-1}}
\end{equation}
\tag{6.28}
$$
is equal to its $(n - 1)$-dimensional Hausdorff measure $(n - 1)\text{-}{\mathfrak m}^{0}(E)$. In particular, the surface area measure $\sigma$ on $l(O)$ is a regular Borel measure on $\mathbb R^n$. We say that a Lipschitz hypersurface is bi-Lipschitz if the quantity $\operatorname{Lip}(l^{-1})$ in (6.21) is finite. Corollary 9. Let $l \colon O\to \mathbb R^n$ be a bi-Lipschitz hypersurface in $\mathbb R^n$ and let ${l(O)\subset \overline D(r)}$ for some $r\in \mathbb R^+$. Then each $\delta$-subharmonic function $U\not\equiv\pm\infty$ on a ball $\overline B(R)\subset \mathbb R^n$ of radius $R>r$ is summable with respect to the surface area measure $\sigma$ on $l(O)$ and for each Borel set $E\subset l(O)$ Proof. By (6.28) for arbitrary $x\in \mathbb R^n$ the area of the intersection of $\overline B_x(t)$ with a Borel set $E\subset l(O)$ is
$$
\begin{equation}
\int_{l^{-1}(E\cap \overline B_x(t) )}|Jl|\,\mathrm{d} \lambda_{\mathbb R^{n-1}} \stackrel{(6.27)}{\leqslant} \sqrt{(n-1)!\,n}\,(\operatorname{Lip}(l))^{n-1} \lambda_{\mathbb R^{n-1}}\bigl(l^{-1}(\overline B_x(t))\bigr).
\end{equation}
\tag{6.30}
$$
Since $\operatorname{Lip}(l^{-1})<+\infty$, the set $l^{-1}(\overline B_x(t))=l^{-1}(l(O)\cap \overline B_x(t))$ has diameter at most $\operatorname{Lip}(l^{-1}) \cdot 2t$. Hence the set $l^{-1}(\overline B_x(t))\subset \mathbb R^{n-1}$ lies in a ball of radius $2\,\operatorname{Lip}(l^{-1})\,t$ in $\mathbb R^{n-1}$ and
$$
\begin{equation*}
\begin{aligned} \, \lambda_{\mathbb R^{n-1}}\bigl(l^{-1}(\overline B_x(t))\bigr) &\leqslant \frac{\pi^{n/2}}{\Gamma(n/2+1)}(2\,\operatorname{Lip}(l^{-1})t)^{n-1} \\ &\stackrel{(6.15)}{\leqslant}6\cdot 2^{n-1}(\operatorname{Lip}(l^{-1}))^{n-1}t^{n-1}\quad\text{for } n>2 \end{aligned}
\end{equation*}
\notag
$$
for any $x\in \mathbb R^n$ and $t\in \mathbb R^+$. By (6.30) this means that for $n>2$ the modulus of continuity ${\mathrm h}_{\sigma_E}$ of the restriction $\sigma_E:=\sigma{\lfloor}_E$ of $\sigma$ to $E\subset l(O)$ satisfies
$$
\begin{equation*}
\begin{aligned} \, {\mathrm h}_{\sigma_E}(t) &\leqslant {\mathrm h}_{\sigma}(t) \leqslant 6\cdot 2^{n-1}(\operatorname{Lip}(l^{-1}))^{n-1}t^{n-1} \sqrt{(n-1)!\,n}\, (\operatorname{Lip}(l))^{n-1} \\ &\leqslant 3n^n (\operatorname{Lip}(l)\operatorname{Lip}(l^{-1}))^{n-1} t^{n-1} \quad\text{for all } t\in \mathbb R^+. \end{aligned}
\end{equation*}
\notag
$$
Hence for $\mu:=\sigma_E$ condition (6.1) in Theorem 13 holds for
$$
\begin{equation*}
p:={n-1}\quad\text{and} \quad b:=3n^n \bigl(\operatorname{Lip}(l)\,\operatorname{Lip}(l^{-1})\bigr)^{n-1}.
\end{equation*}
\notag
$$
By Theorem 13 every $\delta$-subharmonic function $U\not\equiv \pm \infty$ on a ball $\overline B(R)$ of radius $R>r$ is $\sigma_E$-summable for each Borel set $E\subset l(O)$. By (6.3) with $\mu:=\sigma_E$ we obtain
$$
\begin{equation*}
\begin{aligned} \, \int_{E} U^+\,\mathrm{d} \sigma &=\int_{\overline B(r)} U^+\,\mathrm{d} \sigma_E \leqslant 3n^{2n} \bigl(\operatorname{Lip}(l)\,\operatorname{Lip}(l^{-1})\bigr)^{n-1} A_n(r,R) {\boldsymbol T}_U(r,R) \\ &\qquad \times (n-1)\text{-}{\mathfrak m}^{0}(E) \biggl(1+\frac{1}{((n-1)\text{-}{\mathfrak m}^{0}(E))^{(n-2)/(n-1)}}\biggr) \quad\text{for } n>2. \end{aligned}
\end{equation*}
\notag
$$
However, as we mentioned after (6.28), we have $\sigma(E)=(n-1)$-${\mathfrak m}^{0}(E)$. Making this substitution on the right-hand side we obtain (6.29) as required.
The proof is complete.
Citation in format AMSBIB
\Bibitem{Kha22} |
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
Contact us: |
Terms of Use
|
Registration to the website |
Logotypes |
|