Sharp higher order Admas' inequality with exact growth condition on weighted Sobolev spaces

TL;DR

This paper establishes a new higher order Adams inequality with precise growth conditions in weighted Sobolev spaces, proving its optimality and applying it to analyze solutions and regularity of certain ODEs.

Contribution

It introduces a novel higher order Adams inequality with exact growth conditions and explores its implications for ODE solutions and regularity theory.

Findings

Validated the new inequality with rigorous proof.

Identified the optimal critical constant and exponent.

Applied the inequality to derive solution concepts and regularity results for ODEs.

Abstract

This paper introduces a novel higher order Adams inequality that incorporates an exact growth condition for a class of weighted Sobolev spaces. Our rigorous proof confirms the validity of this inequality and provides insights into the optimal nature of the critical constant and the exponent within the denominator. Furthermore, we apply this inequality to study a class of ordinary differential equations (ODEs), where we successfully derive both a concept of the weak solution and a comprehensive regularity theory.