Embedding Theorems for a Certain Function Class of Sobolev Type on Metric Spaces

Given a metric space with a Borel measure μ, 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 μ. We prove some analogs of the classical theorems on embedding Sobolev function classes into Lebesgue spaces.