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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Removable sets for pointwise subharmonic functions


Author: Victor L. Shapiro
Journal: Trans. Amer. Math. Soc. 159 (1971), 369-380
MSC: Primary 31A05
DOI: https://doi.org/10.1090/S0002-9947-1971-0390252-5
MathSciNet review: 0390252
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0390252-5
Keywords: Subharmonic functions, $ {L^1}$ total differential at a point, removable set, integral mean, generalized Laplacian, convex function
Article copyright: © Copyright 1971 American Mathematical Society

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