Removable sets for pointwise subharmonic functions

Abstract:

Pointwise subharmonic is defined in terms of the pointwise ${L^1}$ total derivative of order 2. The class $\mathcal {A}({x^ \ast },{r_ \ast })$ is introduced for the ball $B({x^ \ast },{r_ \ast })$, and the following theorem is established: Let $Q$ be a Borel set of Lebesgue measure zero contained in $B({x^ \ast },{r_ \ast })$. Then a necessary and sufficient condition that $Q$ be removable for pointwise subharmonic functions with respect to the class $\mathcal {A}({x^ \ast },{r_ \ast })$ is that $Q$ be countable. It is also shown that the class $\mathcal {A}({x^ \ast },{r_ \ast })$ is in a certain sense best possible for the sufficiency of the above theorem.