Sharp singular Adams inequalities in high order Sobolev spaces

TL;DR

This paper establishes sharp singular Adams inequalities in high order Sobolev spaces, extending previous results to unbounded domains and critical cases, with implications for exponential integrability in fractional and higher order Sobolev spaces.

Contribution

It proves sharp singular Adams inequalities for high order Sobolev spaces in bounded and unbounded domains, extending prior inequalities to the singular and higher order cases.

Findings

Sharp inequalities for fractional integrals with optimal constants.

Extension of singular Moser-Trudinger inequalities to higher order Sobolev spaces.

Resolution of an open problem for $W^{2,2}(R^4)$ at the critical case.

Abstract

In this paper, we prove a version of weighted inequalities of exponential type for fractional integrals with sharp constants in any domain of finite measure in $\mathbb{R}^{n}$. Using this we prove a sharp singular Adams inequality in high order Sobolev spaces in bounded domain at critical case. Then we prove sharp singular Adams inequalities for high order derivatives on unbounded domains. Our results extend the singular Moser-Trudinger inequalities of first order in \cite{Ad2, R, LR, AdY} to the higher order Sobolev spaces $W^{m,\frac{n}{m}}$ and the results of \cite{RS} on Adams type inequalities in unbounded domains to singular case. Our singular Adams inequality on $W^{2,2}(\mathbb{R}^{4})$ with standard Sobolev norm at the critical case settles a unsolved question remained in \cite{Y}.