• East Asian mathematical journal
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• v.19 no.2
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• pp.165-171
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• 2003

We show that any F-harmonic map from a compact manifold M to N is necessarily constant if N possesses a strictly-convex function, and prove 'Liouville type theorems' for F-harmonic maps. Finally, when the target manifold is the real line, we get a result for F-subharmonic functions.

• Communications of the Korean Mathematical Society
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• v.23 no.2
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• pp.269-277
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• 2008

Let $f=(f_n)$ be a nonnegative submartingale such that ${\parallel}f{\parallel}{\infty}{\leq}1\;and\;g=(g_n)$ be a martingale, adapted to the same filtration, satisfying $${\mid}d_{gn}{\mid}{\leq}{\mid}df_n{\mid},\;n=0,\;1,\;2,\;....$$ The paper contains the proof of the sharp inequality $$\limits^{sup}_ n\;\mathbb{E}{\Phi}({\mid}g_n{\mid}){\leq}{\Phi}(1)$$ for a class of convex increasing functions ${\Phi}\;on\;[0,\;{\infty}]$, satisfying certain growth condition. As an application, we show a continuous-time version for stochastic integrals and a related estimate for smooth functions on Euclidean domain.