On a problem of CHirka

Abstract

We observe that a slight adjustment of a method of Caffarelli, Li, and Nirenberg yields that plurisubharmonic functions extend across subharmonic singularities as long as the singularities form a closed set of measure zero. This solves a problem posed by Chirka.

Radó- type theorem for subharmonic and plurisubharmonic functions

“…In this section we slightly generalize the main result of [5] that subharmonic functions which are plurisubharmonic outside a small set are actually plurisubharmonic.…”

“…If E 1 and hence E is of Lebesgue measure zero, then Theorem 3.1 is a corollary of the main result in [5]. However, even for smooth u the critical set of u may be quite big, and the Morse-Sard theorem specifies the minimal regularity of u that guarantees that at least the image u(E 1 ) is small.…”

“…In the next section we discuss various technical results of independent interest related to weaker notions of differentiability that we shall use later on. In Section 3 we recall and slightly generalize our findings from [5]. In Section 4 we thoroughly discuss the result of Gardiner and Södin from [8] providing yet another slight generalization.…”

“…As in [5], we recall (see [11], Proposition 3.2.10') that subharmonicity and plurisubharmonicity is equivalent to subharmonicity and respectively plurisubharmonicity in the viscosity sense.…”

“…In fact we shall use a recent result of Gardiner and Södin [8] about extensions of subharmonic functions past critical sets. Another key tool in our approach comes from our previous work [5] where we proved that plurisubharmonic functions with subharmonic singularities across a Lebesgue measure zero set extend plurisubharmonically. Also throughout the paper we make use of the following simple but fundamental fact from calculus:…”