A well-known result of Stein-Weiss in 1971 said that a harmonic function, defined on the upper half-space, is the Poisson integral of a Lebesgue function if and only if it is also a Lebesgue function uniformly in the time variable. Under a metric measure space setting, we show that a solution to the elliptic equation with a non-negative potential, defined on the upper half-space, is in the essentially-bounded-Morrey space with variable exponent if and only if it can be represented as the Poisson integral of a variable Morrey function, where the doubling property, the pointwise upper bound on the heat kernel, the mean value property and the Liouville property are assumed.

Stein-Weiss 在 1971 年的一个著名结果指出，定义在上半空间的调和函数是勒贝格函数的泊松积分，当且仅当它在时间变量上一致也是勒贝格函数时。在度量测度空间设置下，我们证明，在上半空间上定义的具有非负势的椭圆方程的解位于具有可变指数的本质有界莫雷空间中当且仅当它可以是表示为变量 Morrey 函数的泊松积分，其中假定了加倍性质、热核的逐点上限、平均值性质和刘维尔性质。