The distributional Riesz functional of a subharmonic function is a positive Radon measure

Since $u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$ by [F2], it has a regular distribution $T_u$ on $\mathrm{\Omega }$, and [F1] together with [F4] gives ${\mu }_u(\phi )=\frac{1}{2\pi }{\int }_{\mathrm{\Omega }}u\mathrm{\Delta }\phi dA=\frac{1}{2\pi }⟨\mathrm{\Delta }{T}_u,\phi ⟩$ for every $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$.

By [F5], $\mathrm{D}\mathrm{C}$ yields ${\mathrm{A}\mathrm{C}}_{\omega }$, which discharges the choice hypotheses of [F7], [F8], [F9] (second clause) and [F13] used below.

The open set $\mathrm{\Omega }$, with the subspace topology of ${\mathbb{R}}^2\cong \mathbb{C}$, is an LCH space by [F21]: ${\mathbb{R}}^2$ is locally compact and Hausdorff, open subspaces of locally compact Hausdorff spaces are locally compact, and Hausdorffness is hereditary.

Choose $\rho :=b\mathrm{/}\int b$ from the standard step $b$ of [F6] and put ${\rho }_{\epsilon }(x)={\epsilon }^{-2}\rho (x\mathrm{/}\epsilon )$: then ${\rho }_{\epsilon }\in C_c^{\mathrm{\infty }}({\mathbb{R}}^2)$ is nonnegative, has $\int {\rho }_{\epsilon }=1$ and support in $\bar{B}(0,2\epsilon )$, and under ${\mathrm{A}\mathrm{C}}_{\omega }$ of step 1.2 the family $({\rho }_{\epsilon })$ is an $L^1$ approximate identity.

For $z$ in the open set ${\mathrm{\Omega }}_{\epsilon }:=\{z\in \mathrm{\Omega }:\bar{B}(z,3\epsilon )\subseteq \mathrm{\Omega }\}$ define $u_{\epsilon }(z):={\int }_{{\mathbb{R}}^2}u(z-y){\rho }_{\epsilon }(y)dy$; this set equals $\mathrm{\Omega }$ when $\mathrm{\Omega }=\mathbb{C}$. It is open: for any of its points, [F15] gives a positive margin for the compact ball $\bar{B}(z,3\epsilon )$ inside $\mathrm{\Omega }$, and every sufficiently small translate of that ball stays inside $\mathrm{\Omega }$. In general the integrand lives on the compact set $z-\bar{B}(0,2\epsilon )\subseteq B(z,3\epsilon )\subseteq \mathrm{\Omega }$. For each such $z$, choose a relatively compact open $W\subseteq \mathrm{\Omega }$ containing $z-\bar{B}(0,2\epsilon )$ and all its sufficiently small translates. Replacing $u$ by its product with ${\mathbf{1}}_W$, extended by zero outside $\mathrm{\Omega }$ (locally integrable on ${\mathbb{R}}^2$ by [F2]), [F7] and step 1.2 show that $u_{\epsilon }\in C^{\mathrm{\infty }}({\mathrm{\Omega }}_{\epsilon })$ with every derivative given by the convolution of $u$ against the corresponding derivative of ${\rho }_{\epsilon }$; the values are finite real numbers, since $u\in L^1$ near $z-\bar{B}(0,2\epsilon )$.

The mollified function satisfies the submean inequality on ${\mathrm{\Omega }}_{\epsilon }$: if $a\in {\mathrm{\Omega }}_{\epsilon }$ and $\bar{D}(a,r)\subseteq {\mathrm{\Omega }}_{\epsilon }$, then $\bar{D}(a,r+2\epsilon )\subseteq \mathrm{\Omega }$ — for $\mathrm{\mid }w-a\mathrm{\mid }\le r$ one has $w\in {\mathrm{\Omega }}_{\epsilon }$, and for $r<\mathrm{\mid }w-a\mathrm{\mid }\le r+2\epsilon$ the point $v:=a+r(w-a)\mathrm{/}\mathrm{\mid }w-a\mathrm{\mid }$ lies in $\bar{D}(a,r)\subseteq {\mathrm{\Omega }}_{\epsilon }$, so $B(v,3\epsilon )\subseteq \mathrm{\Omega }$ while $\mathrm{\mid }w-v\mathrm{\mid }=\mathrm{\mid }w-a\mathrm{\mid }-r\le 2\epsilon <3\epsilon$ gives $w\in B(v,3\epsilon )\subseteq \mathrm{\Omega }$ — hence $\bar{D}(a-y,r)\subseteq \bar{D}(a,r+2\epsilon )\subseteq \mathrm{\Omega }$ for every $\mathrm{\mid }y\mathrm{\mid }\le 2\epsilon$; applying the submean inequality of [F3] at the centre $a-y$, multiplying by ${\rho }_{\epsilon }(y)\ge 0$ and integrating over $\mathrm{\mid }y\mathrm{\mid }\le 2\epsilon$ with Tonelli and Fubini [F11] applied to the positive and negative parts (the absolute double integral is at most $2\pi \mathrm{\parallel }{\rho }_{\epsilon }{\mathrm{\parallel }}_{\mathrm{\infty }}{\int }_{\bar{D}(a,r+2\epsilon )}\mathrm{\mid }u\mathrm{\mid }dA<\mathrm{\infty }$ by [F2]) gives $u_{\epsilon }(a)\le \frac{1}{2\pi }{\int }_0^{2\pi }{u}_{\epsilon }(a+r{e}^{it})dt$.

As $\epsilon \to 0^{+}$ the regular distributions of $u_{\epsilon }$ converge weakly to $T_u$ on $\mathrm{\Omega }$: the local convolution of [F8] with $T:=T_u$ is exactly $x\mapsto T_u({\rho }_{\epsilon }(x-\cdot ))={\int }_{\mathrm{\Omega }}u(y){\rho }_{\epsilon }(x-y)dy=u_{\epsilon }(x)$ on $V_{\epsilon }=\{x:x-\mathrm{supp}{\rho }_{\epsilon }\subseteq \mathrm{\Omega }\}$, and $V_{\epsilon }\supseteq {\mathrm{\Omega }}_{\epsilon }$ because $x-\bar{B}(0,2\epsilon )\subseteq B(x,3\epsilon )\subseteq \mathrm{\Omega }$ for $x\in {\mathrm{\Omega }}_{\epsilon }$; hence $⟨T_{u_{\epsilon }},\psi ⟩\to ⟨T_u,\psi ⟩$ for every $\psi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$.

Classical compatibility: for every $\epsilon >0$ and every $\phi \in C_c^{\mathrm{\infty }}({\mathrm{\Omega }}_{\epsilon })$ one has $⟨\mathrm{\Delta }{T}_{u_{\epsilon }},\phi ⟩={\int }_{\mathrm{\Omega }}\phi \mathrm{\Delta }{u}_{\epsilon }dA$. Indeed $u_{\epsilon }\in C^{\mathrm{\infty }}({\mathrm{\Omega }}_{\epsilon })$ by step 2.2, so the ${\mathrm{A}\mathrm{C}}_{\omega }$ clause of [F9] applied on the open set ${\mathrm{\Omega }}_{\epsilon }$ to ${\mathrm{\partial }}_x{\mathrm{\partial }}_x{u}_{\epsilon }$ and ${\mathrm{\partial }}_y{\mathrm{\partial }}_y{u}_{\epsilon }$ and added gives $\mathrm{\Delta }{T}_{u_{\epsilon }}=T_{\mathrm{\Delta }{u}_{\epsilon }}$ there, and only values on ${\mathrm{\Omega }}_{\epsilon }$ are tested.

By step 2.2 the function $u_{\epsilon }$ is continuous and real-valued on ${\mathrm{\Omega }}_{\epsilon }$, and by step 3.1 it satisfies the submean inequality at every closed disc in ${\mathrm{\Omega }}_{\epsilon }$; hence $u_{\epsilon }$ is subharmonic on each connected component of ${\mathrm{\Omega }}_{\epsilon }$ in the sense of [F3].

For any fixed $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, the compact support of $\phi$ and of $\mathrm{\Delta }\phi$ lies inside ${\mathrm{\Omega }}_{\epsilon }$ for all sufficiently small $\epsilon$ by [F15]. On those open domains, the definition of distributional derivatives gives $⟨\mathrm{\Delta }{T}_{u_{\epsilon }},\phi ⟩=\int u_{\epsilon }\mathrm{\Delta }\phi dA$. Step 3.2 applied to the fixed test $\mathrm{\Delta }\phi$ shows that this tends to $⟨T_u,\mathrm{\Delta }\phi ⟩=⟨\mathrm{\Delta }{T}_u,\phi ⟩$. These pairings are local for each $\epsilon$; no distribution on all of $\mathrm{\Omega }$ is asserted for a locally defined $u_{\epsilon }$.

By [F10] applied on the components of the open set ${\mathrm{\Omega }}_{\epsilon }$, step 4.1 gives $\mathrm{\Delta }{u}_{\epsilon }\ge 0$ pointwise on ${\mathrm{\Omega }}_{\epsilon }$.

Positivity on nonnegative tests: let $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ be real with $\phi \ge 0$. Since $\mathrm{supp}\phi \subseteq \mathrm{\Omega }$ is compact in the open set $\mathrm{\Omega }$, the positive-margin fact [F15] gives $\eta >0$ with $\mathrm{supp}\phi +\bar{B}(0,\eta )\subseteq \mathrm{\Omega }$; then any ${\epsilon }_0>0$ with $3{\epsilon }_0<\eta$ satisfies $\mathrm{supp}\phi \subseteq {\mathrm{\Omega }}_{{\epsilon }_0}$, because $\bar{B}(x,3{\epsilon }_0)\subseteq B(x,\eta )\subseteq \mathrm{supp}\phi +\bar{B}(0,\eta )\subseteq \mathrm{\Omega }$ for $x\in \mathrm{supp}\phi$. Fix such an ${\epsilon }_0$, so that $\mathrm{supp}\phi \subseteq {\mathrm{\Omega }}_{\epsilon }$ and $\phi \ge 0$ for every $0<\epsilon \le {\epsilon }_0$. Steps 1.1, 4.2 and 3.3 give ${\mu }_u(\phi )={\lim }_{n\to \mathrm{\infty }}\frac{1}{2\pi }⟨\mathrm{\Delta }{T}_{u_{{\epsilon }_0\mathrm{/}(n+1)}},\phi ⟩={\lim }_{n\to \mathrm{\infty }}\frac{1}{2\pi }{\int }_{\mathrm{\Omega }}\phi \mathrm{\Delta }{u}_{{\epsilon }_0\mathrm{/}(n+1)}dA$, and each integrand is nonnegative by step 5.1; [F14] therefore gives ${\mu }_u(\phi )\ge 0$.

Monotonicity on smooth tests: if $\phi \le \psi$ are real-valued compactly supported smooth functions on $\mathrm{\Omega }$, then $\psi -\phi \ge 0$ is a test function with ${\mu }_u(\psi )-{\mu }_u(\phi )={\mu }_u(\psi -\phi )\ge 0$ by step 6.1 and the linearity of [F1]; hence ${\mu }_u(\phi )\le {\mu }_u(\psi )$.

For $f\in C_c(\mathrm{\Omega };\mathbb{R})$ set $S_f:=\{{\mu }_u(\phi ):\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R}),\phi \le f\}$ and $T_f:=\{{\mu }_u(\psi ):\psi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R}),\psi \ge f\}$. If $f\mathrm{\ne }0$, put $M:=\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$ and use [F12] to choose $\chi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ with $0\le \chi \le 1$ and $\chi =1$ on a neighbourhood of $\mathrm{supp}f$; then $-M\chi \le f\le M\chi$ pointwise, so $S_f$ and $T_f$ are nonempty; if $f=0$, then $0\in S_f\cap T_f$. By step 7.1 every element of $S_f$ is at most every element of $T_f$, so $S_f$ is bounded above and $T_f$ bounded below; [F17] makes $\sup S_f$ and $\inf T_f$ well-defined real numbers with $\sup S_f\le \inf T_f$.

Density of smooth tests for a fixed $f\mathrm{\ne }0$: let $K:=\mathrm{supp}f$, $M:=\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$, choose $\chi$ as in step 8.1, and use [F15] to fix $\eta >0$ with $L:=\mathrm{supp}\chi +\bar{B}(0,\eta )\subseteq \mathrm{\Omega }$; use [F12] again to choose ${\chi }_1\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ with $0\le {\chi }_1\le 1$ and ${\chi }_1=1$ on the compact set $L$. For all large $n$ put $f_n:=(f\chi )\ast {\rho }_{1\mathrm{/}n}$: each $f_n$ is smooth by [F7], supported in $\mathrm{supp}\chi +\bar{B}(0,2\mathrm{/}n)\subseteq L\subseteq \mathrm{\Omega }$, and $f_n\to f\chi =f$ uniformly on the compact $L$ by [F13], and hence on ${\mathbb{R}}^2$ since both functions vanish outside $L$, because $f\chi$ is continuous with compact support and $f\chi =f$ on $\mathrm{supp}f$.

Sandwich for the sets of step 8.1: keep $f\mathrm{\ne }0$ and $\chi ,{\chi }_1,L$ of steps 8.1 and 9.1, and let ${\delta }_n:=\mathrm{\parallel }{f}_n-f{\mathrm{\parallel }}_{\mathrm{\infty }}\to 0$. Since $\mathrm{\mid }{f}_n-f\mathrm{\mid }\le {\delta }_n$ everywhere and $\mathrm{\mid }{f}_n-f\mathrm{\mid }=0$ outside $L$, one has $f_n-{\delta }_n{\chi }_1\le f\le f_n+{\delta }_n{\chi }_1$ pointwise, and both bounds are smooth test functions of the kinds defining $S_f$ and $T_f$; applying ${\mu }_u$ and using step 7.1 gives ${\mu }_u(f_n)-{\delta }_n{\mu }_u({\chi }_1)\le \sup S_f\le \inf T_f\le {\mu }_u(f_n)+{\delta }_n{\mu }_u({\chi }_1)$. Hence $({\mu }_u(f_n))$ is Cauchy, and with $\mathrm{\Lambda }(f):=\sup S_f=\inf T_f$ one has $\mathrm{\mid }\mathrm{\Lambda }(f)-{\mu }_u(f_n)\mathrm{\mid }\le {\delta }_n{\mu }_u({\chi }_1)\to 0$ for every admissible sequence $(f_n)$ of smooth functions converging uniformly to $f$ with supports in a fixed compact subset of $\mathrm{\Omega }$. For $f=0$ set $\mathrm{\Lambda }(0):=0$, consistently with step 8.1.

Positivity of $\mathrm{\Lambda }$: if $f\ge 0$ in $C_c(\mathrm{\Omega };\mathbb{R})$, then the zero test function satisfies $0\le f$, so $0={\mu }_u(0)\in S_f$ and $\mathrm{\Lambda }(f)=\sup S_f\ge 0$. If $f=0$ this is step 10.1.

Homogeneity of $\mathrm{\Lambda }$: for $c>0$ one has $S_{cf}=c{S}_f$, so $\mathrm{\Lambda }(cf)=c\mathrm{\Lambda }(f)$; for $c<0$ one has $S_{cf}=c{T}_f$, so by [F17] $\mathrm{\Lambda }(cf)=\sup (c{T}_f)=c\inf T_f=c\mathrm{\Lambda }(f)$; and $\mathrm{\Lambda }(0)=0$. Thus $\mathrm{\Lambda }$ is positively homogeneous and $\mathrm{\Lambda }(-f)=-\mathrm{\Lambda }(f)$.

Extension: if $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R})$ then $\phi \in S_{\phi }$ and $\phi \in T_{\phi }$, so step 7.1 gives ${\mu }_u(\phi )\le \mathrm{\Lambda }(\phi )\le {\mu }_u(\phi )$; hence $\mathrm{\Lambda }(\phi )={\mu }_u(\phi )$ for every smooth test function.

Additivity of $\mathrm{\Lambda }$: given $f,g\in C_c(\mathrm{\Omega };\mathbb{R})$ and $\eta >0$, choose by the definition of the supremum $\phi \in S_f$, $\psi \in S_g$ with ${\mu }_u(\phi )>\mathrm{\Lambda }(f)-\eta \mathrm{/}2$ and ${\mu }_u(\psi )>\mathrm{\Lambda }(g)-\eta \mathrm{/}2$; then $\phi +\psi \le f+g$, so $\mathrm{\Lambda }(f+g)\ge {\mu }_u(\phi )+{\mu }_u(\psi )>\mathrm{\Lambda }(f)+\mathrm{\Lambda }(g)-\eta$, and $\eta \downarrow 0$ gives $\mathrm{\Lambda }(f+g)\ge \mathrm{\Lambda }(f)+\mathrm{\Lambda }(g)$. Dually, choose ${\psi }_f\in T_f$, ${\psi }_g\in T_g$ with ${\mu }_u({\psi }_f)<\mathrm{\Lambda }(f)+\eta \mathrm{/}2$ and ${\mu }_u({\psi }_g)<\mathrm{\Lambda }(g)+\eta \mathrm{/}2$; then ${\psi }_f+{\psi }_g\ge f+g$, so $\mathrm{\Lambda }(f+g)\le \mathrm{\Lambda }(f)+\mathrm{\Lambda }(g)+\eta$ and hence $\mathrm{\Lambda }(f+g)\le \mathrm{\Lambda }(f)+\mathrm{\Lambda }(g)$. Therefore $\mathrm{\Lambda }$ is additive; it is real-linear together with the homogeneity of step 11.2.

By steps 11.1, 12.1 and 1.3 the map $\mathrm{\Lambda }:C_c(\mathrm{\Omega };\mathbb{R})\to \mathbb{R}$ is a positive real-linear functional on the LCH space $\mathrm{\Omega }$ in the sense of [F16]; the RMK construction of [F18] therefore produces a Radon measure $\nu$ on $\mathrm{\Omega }$ with $\mathrm{\Lambda }(w)={\int }_{\mathrm{\Omega }}wd\nu$ for every $w\in C_c(\mathrm{\Omega };\mathbb{R})$.

Representing smooth tests: combining steps 11.3 and 13.1, for every $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R})$ one has ${\mu }_u(\phi )=\mathrm{\Lambda }(\phi )={\int }_{\mathrm{\Omega }}\phi d\nu$; since ${\mu }_u$ is complex-linear, the same identity holds for complex test functions, so $\nu$ represents ${\mu }_u$.

Uniqueness: let ${\nu }^{\mathrm{\prime }}$ be a Radon measure on $\mathrm{\Omega }$ with ${\mu }_u(\phi )={\int }_{\mathrm{\Omega }}\phi d{\nu }^{\mathrm{\prime }}$ for every $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R})$. For $f\in C_c(\mathrm{\Omega };\mathbb{R})$ and an admissible sequence $(f_n)$ as in step 10.1 with common support in a compact $L^{\mathrm{\prime }}\subseteq \mathrm{\Omega }$, one has ${\int }_{\mathrm{\Omega }}{f}_{n}d{\nu }^{\mathrm{\prime }}={\mu }_u(f_n)\to \mathrm{\Lambda }(f)$ by step 10.1, while ${\int }_{\mathrm{\Omega }}{f}_{n}d{\nu }^{\mathrm{\prime }}\to {\int }_{\mathrm{\Omega }}fd{\nu }^{\mathrm{\prime }}$ by [F20], since $f_n\to f$ pointwise and $\mathrm{\mid }{f}_n\mathrm{\mid }\le \mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}+1$ on the compact set $L^{\mathrm{\prime }}$ of finite ${\nu }^{\mathrm{\prime }}$-measure. Hence ${\int }_{\mathrm{\Omega }}fd{\nu }^{\mathrm{\prime }}=\mathrm{\Lambda }(f)={\int }_{\mathrm{\Omega }}fd\nu$ for every $f\in C_c(\mathrm{\Omega };\mathbb{R})$, and [F19] gives ${\nu }^{\mathrm{\prime }}=\nu$.

Conclusion: clause 1 is step 6.1, and clause 2 is the existence of $\nu$ in steps 13.1 and 14.1 together with the uniqueness in step 14.2.