# Embeddings of homogeneous Sobolev spaces on the entire space

## TL;DR

This paper characterizes when certain Sobolev space inequalities hold for rearrangement-invariant spaces on the entire space, simplifying the problem to one-dimensional inequalities and identifying optimal spaces.

## Contribution

It provides a complete characterization of Sobolev space inequalities involving rearrangement-invariant spaces, including Orlicz spaces, and determines optimal spaces in these inequalities.

## Key findings

- Reduction of multidimensional inequalities to one-dimensional inequalities
- Complete description of optimal rearrangement-invariant spaces for the inequalities
- Application to Orlicz spaces and common function spaces

## Abstract

We completely characterize the validity of the inequality $\|u\|_{Y(\mathbb{R}^n)}\leq C \|\nabla^m u\|_{X(\mathbb{R}^n)}$, where $X$ and $Y$ are rearrangement-invariant spaces, by reducing it to a considerably simpler one-dimensional inequality. Furthermore, we fully describe the optimal rearrangement-invariant space on either side of the inequality when the space on the other side is fixed. We also solve the same problem within the environment in which the competing spaces are Orlicz spaces. A variety of examples involving customary function spaces suitable for applications is also provided.