The Perron-Bremermann envelope for $q$-plurisubharmonic functions on unbounded domains in $\mathbb{C}^n$

Abstract: Let $D$ be an unbounded domain in $\mathbb{C}<sup>n$, and let $f$ be a bounded continuous function prescribed on the boundary of $D$. We show that, if $D$ has $r$-peak points on its boundary and is of bounded type, $f$ extends to a maximal bounded continuous function $F$ on $\overline{D}$ that is $q$-plurisubharmonic and $(n-q-1)$-plurisuperharmonic (i.e., $(dd^c F)<sup>n=0$) on $D$, and that coincides with the Perron-Bremermann envelope created with respect to bounded $q$-plurisubharmonic functions on $\overline{D}$.

• The paper proves that, under bounded-type and r-peak-point hypotheses, the truncated q-Perron–Bremermann envelope produces the unique continuous q-Bremermann solution for bounded continuous boundary data.
• The exhaustion method solves bounded subdomains and combines comparison, maximality, gluing, and a negative plurisubharmonic exhaustion to establish boundary attainment, interior continuity, and uniqueness.
• The paper shows why truncation matters on unbounded domains, including examples where the untruncated envelope is identically infinite, while identifying unresolved cases involving unbounded data, non-bounded-type domains, and continuity on the q-hull.

Background and problem statement

The Dirichlet–Bremermann problem asks for a maximal plurisubharmonic (psh) extension of prescribed continuous boundary data. For bounded strictly pseudoconvex domains it was solved by Bremermann via the Perron envelope, with continuity established by Walsh and uniqueness in the sense of $(dd^c F)^n=0$ by Bedford–Taylor (2606.16584). The paper under discussion extends this program to $q$-plurisubharmonic functions in the sense of Hunt–Murray on unbounded domains in $\mathbb{C}^n$. A function is $q$-plurisubharmonic ($\psi \in PSH_q$) if its restriction to every complex $(q+1)$-dimensional affine slice satisfies the subharmonic comparison property; smooth such functions are exactly those whose complex Hessian has at most $q$ negative eigenvalues at each point.

The central object is the $q$-Perron–Bremermann envelope

$$P_{f,q,D,M}(z)=\sup\{\psi(z) : \psi \in PSH_q(D)\cap C(D),\ \psi \le f \text{ on } \partial D,\ \psi \le M \text{ on } D\},$$

where $f \in C(\partial D)$ and $q$0 satisfies $q$1. The author's definition requires competitors to be continuous and globally bounded by $q$2; the author notes explicitly that the more common variant $q$3 (supremum over psh functions with upper regularization dominated by $q$4) need not be lower semi-continuous, which breaks several arguments, so the two envelopes are not interchangeable in general — one has only $q$5.

The main result states that if an unbounded domain $q$6 admits $q$7-peak points at every boundary point ($q$8) and is of bounded type, then for bounded continuous boundary data $q$9, the envelope yields a unique continuous maximal extension that is simultaneously $\mathbb{C}^n$0-plurisubharmonic and $\mathbb{C}^n$1-plurisuperharmonic — i.e., a $\mathbb{C}^n$2-Bremermann function satisfying the degenerate complex Monge–Ampère equation $\mathbb{C}^n$3.

Structure of $\mathbb{C}^n$4-Bremermann functions

A real-valued $\mathbb{C}^n$5 is almost $\mathbb{C}^n$6-Bremermann on a set $\mathbb{C}^n$7 if $\mathbb{C}^n$8 and $\mathbb{C}^n$9; it is $q$0-Bremermann if moreover continuous. The paper assembles the standard toolkit: local bounded suprema and decreasing limits of $q$1 functions remain in $q$2; sums satisfy $q$3 for $q$4, $q$5; the maximum principle holds on relatively compact sets; and a glueing lemma (max-patching across boundaries where the limsup condition holds) preserves $q$6-plurisubharmonicity. The characterization via Lemma 3.5.5 of the author's thesis — $q$7 iff $q$8 is subpluriharmonic for all continuous $q$9 — drives most uniqueness arguments.

Two structural results organize the theory. First, the uniqueness property: two almost $\psi \in PSH_q$0-Bremermann functions on $\psi \in PSH_q$1 agreeing continuously on $\psi \in PSH_q$2 coincide, obtained by applying the maximum principle to the $\psi \in PSH_q$3-plurisubharmonic differences $\psi \in PSH_q$4 and $\psi \in PSH_q$5. Second, an equivalence between maximality and Bremermann regularity: a continuous $\psi \in PSH_q$6 is $\psi \in PSH_q$7-Bremermann on $\psi \in PSH_q$8 iff it is $\psi \in PSH_q$9-maximal on bounded open subsets of $(q+1)$0. The converse direction proceeds by contradiction using the glueing lemma against a continuous $(q+1)$1-psh test function violating the Hessian eigenvalue condition. This equivalence means that solving the Dirichlet problem reduces to exhibiting a $(q+1)$2-maximal competitor.

Properties of the envelope

The envelope $(q+1)$3 is lower semi-continuous, dominated by $(q+1)$4 on $(q+1)$5, and almost $(q+1)$6-Bremermann off the exceptional set $(q+1)$7, where $(q+1)$8 and $(q+1)$9 is the discontinuity set of $q$0. Since $q$1 is continuous, $q$2; if $q$3 or $q$4 is bounded, then $q$5. The proof that $q$6 is $q$7-plurisuperharmonic is a glueing-lemma contradiction argument identical in spirit to the classical Bedford–Taylor scheme.

The unbounded setting introduces genuine pathologies absent in the bounded case. An explicit example shows that on the strip-type domain $q$8 with zero boundary data, the untruncated envelope satisfies $q$9 identically (witnessed by the functions $q$0), whereas $q$1 for every finite $q$2; hence $q$3. This demonstrates that finiteness of the envelope on unbounded domains genuinely depends on the truncation parameter, and motivates the bounded-data hypothesis throughout. A further example from Shcherbina–Tomassini shows that even strictly convex unbounded domains can carry nonpositive continuous boundary data forcing every competing psh function to be identically $q$4; accordingly, the paper restricts to boundary data bounded below.

Peak points supply the boundary contact mechanism. If $q$5 is a $q$6-peak point, then $q$7 for all admissible $q$8: the proof glues scaled peak functions against the data to construct competitors attaining $q$9 near $$P_{f,q,D,M}(z)=\sup\{\psi(z) : \psi \in PSH_q(D)\cap C(D),\ \psi \le f \text{ on } \partial D,\ \psi \le M \text{ on } D\},$$0. Strict Levi $$P_{f,q,D,M}(z)=\sup\{\psi(z) : \psi \in PSH_q(D)\cap C(D),\ \psi \le f \text{ on } \partial D,\ \psi \le M \text{ on } D\},$$1-pseudoconvexity at $$P_{f,q,D,M}(z)=\sup\{\psi(z) : \psi \in PSH_q(D)\cap C(D),\ \psi \le f \text{ on } \partial D,\ \psi \le M \text{ on } D\},$$2 implies a local $$P_{f,q,D,M}(z)=\sup\{\psi(z) : \psi \in PSH_q(D)\cap C(D),\ \psi \le f \text{ on } \partial D,\ \psi \le M \text{ on } D\},$$3-peak point via $$P_{f,q,D,M}(z)=\sup\{\psi(z) : \psi \in PSH_q(D)\cap C(D),\ \psi \le f \text{ on } \partial D,\ \psi \le M \text{ on } D\},$$4, so strictly $$P_{f,q,D,M}(z)=\sup\{\psi(z) : \psi \in PSH_q(D)\cap C(D),\ \psi \le f \text{ on } \partial D,\ \psi \le M \text{ on } D\},$$5-pseudoconvex domains fall within the scope of the theorem. On bounded domains admitting $$P_{f,q,D,M}(z)=\sup\{\psi(z) : \psi \in PSH_q(D)\cap C(D),\ \psi \le f \text{ on } \partial D,\ \psi \le M \text{ on } D\},$$6-peak points, the known theorem (Hunt–Murray for $$P_{f,q,D,M}(z)=\sup\{\psi(z) : \psi \in PSH_q(D)\cap C(D),\ \psi \le f \text{ on } \partial D,\ \psi \le M \text{ on } D\},$$7, Kalka in general, uniqueness by Słodkowski) gives continuity of $$P_{f,q,D,M}(z)=\sup\{\psi(z) : \psi \in PSH_q(D)\cap C(D),\ \psi \le f \text{ on } \partial D,\ \psi \le M \text{ on } D\},$$8 and equality $$P_{f,q,D,M}(z)=\sup\{\psi(z) : \psi \in PSH_q(D)\cap C(D),\ \psi \le f \text{ on } \partial D,\ \psi \le M \text{ on } D\},$$9; the paper also records a useful local characterization: $f \in C(\partial D)$0 is $f \in C(\partial D)$1-Bremermann iff $f \in C(\partial D)$2 for every ball $f \in C(\partial D)$3.

The main theorem on unbounded domains

The proof of the main result proceeds by exhaustion. Writing $f \in C(\partial D)$4, the author solves the bounded $f \in C(\partial D)$5-Dirichlet problem on each $f \in C(\partial D)$6 with data $f \in C(\partial D)$7 equal to $f \in C(\partial D)$8 on the inner boundary portion and $f \in C(\partial D)$9 on the spherical part $q$00. The comparison principle forces $q$01 on $q$02, so $q$03 is well-defined, upper semi-continuous, almost $q$04-Bremermann, equals $q$05 on $q$06, and lies in $q$07. Maximality follows since any competitor $q$08 restricts to each $q$09 and is dominated there by $q$10. Lower semi-continuity on $q$11 uses a global continuous $q$12-psh extension $q$13 of $q$14 constructed by a locally finite partition-of-unity-style glueing over ball coverings of $q$15 — a construction that itself relies on the bounded-domain theorem applied to the pieces $q$16.

Continuity in the interior is the delicate step and requires $q$17 to be of bounded type, i.e., the existence of a continuous negative psh function tending to $q$18 at infinity. The argument adapts Lemma 2 of Simioniuc–Tomassini: given $q$19 and $q$20, the perturbed function $q$21 is glued into a $q$22-psh competitor (using uniform continuity of $q$23 on compact portions of $q$24 and the decay of $q$25 to control the far field), and $q$26-maximality of $q$27 forces $q$28 for small $q$29, establishing lower semi-continuity at $q$30. Uniqueness among $q$31-Bremermann extensions again exploits the bounded-type function: adding $q$32 to a hypothetical second solution $q$33 makes it dominate $q$34 at infinity, and local $q$35-maximality propagates the inequality inward.

The conclusion is strong: on an unbounded domain of bounded type with $q$36-peak boundary points, the truncated envelope $q$37 with $q$38 is independent of the choice of $q$39 (a consequence of the maximum principle on bounded-type domains), is continuous, and is the unique $q$40-Bremermann solution. This extends the Hunt–Murray–Kalka–Słodkowski theory from bounded to a substantial class of unbounded domains, and recovers the Simioniuc–Tomassini results for strictly convex domains as a special case.

Limitations and open questions

Several hypotheses are essential and their removal is left open. The boundedness of $q$41 is used throughout; the case $q$42 is acknowledged to be "more delicate." The final remark proposes the $q$43-hull $q$44 and its union $q$45 over exhaustions as the natural locus where the untruncated envelope stays finite — indeed $q$46 — but continuity of $q$47 on $q$48 is not established, even for $q$49, as Simioniuc–Tomassini already observed. Domains containing a copy of $q$50 are excluded by the bounded-type assumption (Liouville), so e.g. the strictly pseudoconvex domain $q$51 lies outside the theorem's scope. Finally, the requirement of $q$52-peak points at every boundary point, while satisfied by strictly $q$53-pseudoconvex domains, leaves open the treatment of weaker regularity assumptions such as the $q$54-regularity used by Nguyen–Hung in the $q$55 case.

Conclusion

The paper completes a coherent extension of the Perron–Bremermann method to $q$56-plurisubharmonic functions on unbounded domains: it isolates the two obstructions specific to the unbounded case — possible blow-up of the untruncated envelope and failure of continuity — and removes both under the bounded-type and peak-point hypotheses. The resulting solution is characterized intrinsically as the unique continuous function on $q$57 that is $q$58-plurisubharmonic, $q$59-plurisuperharmonic, and agrees with the data on the boundary. The remaining cases — unbounded boundary data, domains without bounded type, and continuity on the $q$60-hull — are identified precisely and remain unresolved.