Sharp singular Adams inequalities in high order Sobolev spaces (Q351061)

Let \(\Omega\subset \mathbb{R}^n\) be a bounded domain, and denote by \(W_0^{k,p}(\Omega)\) the Sobolev space which is the completion of \(C_0^{\infty}(\Omega)\). The Moser-Trudinger inequality says that there exists a constant \(C_n>0\) depending only on \(n\) such that NEWLINE\[NEWLINE \sup_{u\in W_0^{1,n}(\Omega), \|\nabla u\|_{L^n(\Omega)}\leq1}\frac{1}{|\Omega|}\int_{\Omega}\exp\big(\beta|u|^{\frac{n}{n-1}}\big)\,dx<C_n NEWLINE\]NEWLINE holds for any \(\beta\leq\beta(n)\), where \(\beta(n)=n\,\omega_{n-1}^{1/{n-1}}\) and \(\omega_{n-1}\) is the surface area of the unit sphere in \(\mathbb{R}^n\). The inequality is sharp in the sense that if \(\beta>\beta_n\), the supremum above is infinite. \textit{D. R. Adams} [Ann. Math. (2) 128, No. 2, 385--398 (1988; Zbl 0672.31008)] extended such an inequality to higher order spaces \(W_0^{m,\frac{n}{m}}(\Omega)\), \(m>1\).NEWLINENEWLINEIn this paper, the authors prove a sharp singular Adams inequality on bounded domains, namely, NEWLINE\[NEWLINE\sup_{u\in W_0^{m,\frac{n}{m}}(\Omega), \|\nabla^m u\|_{L^{\frac{n}{m}}(\Omega)}\leq1}\frac{1}{|\Omega|}\int_{\Omega}\frac{\exp\big(\beta|u|^{\frac{n}{n-m}}\big)}{|x|^\alpha}\,dx< \infty. \tag{1}NEWLINE\]NEWLINE This result is achieved by first showing an inequality of exponential type with weights for fractional integrals. As an application of (1), the following version of the sharp singular Adams inequality in unbounded domains is established: NEWLINE\[NEWLINE\sup_{u\in W^{m,\frac{n}{m}}(\mathbb{R}^n), \|(-\Delta+I)^{\frac{m}{2}}u\|_{L^{\frac{n}{m}}(\mathbb{R}^n)}\leq1}\int_{\mathbb{R}^n}\frac{\phi\big(\beta|u|^{\frac{n}{n-m}}\big)}{|x|^\alpha}\,dx<\infty, \tag{2}NEWLINE\]NEWLINE where \(\phi(t)=e^t-\sum_{j=0}^{j_{\frac{n}{m}}-2}\frac{t^j}{j!}\) and \(j_{\frac{n}{m}}=\min\{j\in \mathbb{N}:j\geq\frac{n}{m}\}\). Inequalities (1) and (2) extend, respectively, known results concerning singular Moser-Trudinger inequalities to higher-order Sobolev spaces, and results on Adams-type inequalities in unbounded domains to the singular case.NEWLINENEWLINEBesides, a sharp singular Adams inequality on \(W^{2,2}(\mathbb{R}^4)\) with the standard Sobolev norm is attained.