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Abstract: We consider some second order quasilinear partial differential inequalitiesfor real valued functions on the unit ball and find conditions under whichthere is a lower bound for the supremum of nonnegative solutions that do notvanish at the origin. As a consequence, for complex valued functions $fz$satisfying $\partial f-\partial\bar z=|f|^\alpha$, $0<\alpha<1$, and$f0 e0$, there is also a lower bound for $\sup|f|$ on the unit disk. Foreach $\alpha$, we construct a manifold with an $\alpha$-H\-older continuousalmost complex structure where the Kobayashi-Royden pseudonorm is not uppersemicontinuous.



Author: Adam Coffman, Yifei Pan

Source: https://arxiv.org/



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