Conditions for separately subharmonic functions to be subharmonic |
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| Authors: | D H Armitage S J Gardiner |
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| Institution: | (1) Department of Pure Mathematics, Queen's University, BT7 1NN Belfast, Northern Ireland;(2) Department of Mathematics and Statistics, McGill University, H3A 2K6 Montreal, Quebec, Canada;(3) Present address: Department of Mathematics, University College, Dublin 4, Ireland |
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| Abstract: | Letu be a function on 
m
×
n
, wherem 2 andn2, such thatu(x, .) is subharmonic on
n
for each fixedx in
m
andu(.,y) is subharmonic on
m
for each fixedy in
n
. We give a local integrability condition which ensures the subharmonicity ofu on
m
×
n
, and we show that this condition is close to being sharp. In particular, the local integrability of (log+
u
+)
m+n–2+ is enough to secure the subharmonicity ofu if >0, but not if <0. |
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| Keywords: | 31B05 |
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