Local growth envelopes and optimal embeddings of generalized Sobolev spaces

The order-sharp estimates for the local growth envelopes of functions from the generalized Sobolev spaces are obtained, and an explicit description of the rearrangement-invariant hulls of the generalized Lorentz-Sobolev spaces, is also presented. It was assumed that the norm of any function from a rearrangement-invariant space (RIS) can be represented in terms of its rearrangement. Such a representation is known as the Luxemburg representation. A positive function is said to be essentially decreasing if it satisfies the monotonically inequality with some positive constant, not necessarily equal to 1. The interpretation of the notion of a RIS followed the axiomatics suggested by Bennet and Sharpley.