We show that, under certain geometric conditions, there are no nonconstant quasiminimizers with finite p th power energy in a (not necessarily complete) metric measure space equipped with a globally doubling measure supporting a global $$p$$ p -Poincaré inequality. The geometric conditions are that either (a) the measure has a sufficiently strong volume growth at infinity, or (b) the metric space is annularly quasiconvex (or its discrete version, annularly chainable) around some point in the space. Moreover, on the weighted real line $$\mathbf {R}$$ R , we characterize all locally doubling measures, supporting a local $$p$$ p -Poincaré inequality, for which there exist nonconstant quasiminimizers of finite $$p$$ p -energy, and show that a quasiminimizer is of finite $$p$$ p -energy if and only if it is bounded. As $$p$$ p -harmonic functions are quasiminimizers they are covered by these results.

中文翻译：

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$$p$$-调和函数和具有有限能量的拟最小化器的 Liouville 定理

我们表明，在某些几何条件下，在配备了支持全局 $$p$$ p-Poincaré 不等式的全局倍增测度的（不一定是完整的）度量空间中不存在具有有限 p 次幂能量的非常数拟最小化器。几何条件是（a）度量在无穷远处具有足够强的体积增长，或者（b）度量空间在空间中的某个点周围是环状拟凸（或其离散版本，环状可链接）。此外，在加权实数线 $$\mathbf {R}$$ R 上，我们刻画了所有局部倍增测度，支持局部 $$p$$ p -Poincaré 不等式，其中存在有限 $$p$ 的非常量拟最小化$ p -energy，并证明一个拟最小化器是有限的 $$p$$ p -energy 当且仅当它是有界的。