Embedding theorems for a certain function class of Sobolev type on metric spaces

Given a metric space with a Borel measure $\mu$, we consider a class of functions whose increment is controlled by the measure of a ball containing the corresponding points and a nonnegative function $p$-summable with respect to $\mu$. We prove some analogs of the classical theorems on embedding Sobolev function classes into Lebesgue spaces.