Sharp Adams type inequalities in Sobolev spaces $W^{m,\frac{n}{m}}(\mathbb{R}^{n})$ for arbitrary integer $m$

TL;DR

This paper establishes sharp Adams-type inequalities in Sobolev spaces on unbounded domains for any positive integer order less than the dimension, extending and strengthening previous results for specific cases.

Contribution

It generalizes Adams-type inequalities to all positive integers less than the dimension, improving upon prior work limited to even integers and specific cases.

Findings

Proved sharp inequalities for all positive integers m less than n.

Extended previous results to broader Sobolev norms.

Strengthened inequalities for the case n=2m, m integer.

Abstract

The main purpose of our paper is to prove sharp Adams-type inequalities in unbounded domains of $\mathbb{R}^{n}$ for the Sobolev space $W^{m,\frac{n}{m}}\left(\mathbb{R} ^{n}\right)$ for any positive integer $m$ less than $n$. Our results complement those of Ruf and Sani \cite{RS} where such inequalities are only established for even integer $m$. Our inequalities are also a generalization of the Adams-type inequalities in the special case $n=2m=4$ proved in \cite{Y} and stronger than those in \cite{RS} when $n=2m$ for all positive integer $m$ by using different Sobolev norms.