Some functional inequalities on polynomial volume growth Lie groups (Q2888921)

The author studies improved Sobolev inequalities on polynomial volume growth Lie groups. The results are as follows.NEWLINENEWLINETheorem. Let \(G\) be a polynomial volume growth Lie group.NEWLINENEWLINE\(\cdot\) Strong inequalities \(p > 1\):\ \ If \(f \in \dot{W}^{s_{1},p}(G)\) and \(f \in \dot{B}^{-\beta, \infty}_{\infty}(G)\), then we have NEWLINE\[NEWLINE ||f||_{\dot{W}^{s,q}} \leq C||f||_{\dot{W}^{s_{1},p}}^{\theta} ||f||_{\dot{B}^{-\beta, \infty}_{\infty}}^{1-\theta} NEWLINE\]NEWLINE where \(1<p<q<\infty,\;\theta = p/q,\;s=\theta s_{1}-(1-\theta)\beta\), and \( -\beta <s<s_{1}\).NEWLINENEWLINE\(\cdot\) Strong inequalities \(p=1\): If \(\nabla f \in L^1(G)\) and \(f \in \dot{B}^{-\beta, \infty}_{\infty}(G)\), then we have NEWLINE\[NEWLINE ||f||_{L^q} \leq C||\nabla f||_{L^1}^{\theta} ||f||_{\dot{B}^{-\beta, \infty}_{\infty}}^{1-\theta} NEWLINE\]NEWLINE where \(1<q<\infty,\;\theta=1/q\) and \(\beta=\theta/(1-\theta)\).NEWLINENEWLINE\(\cdot\) Weak inequalities \(p=1\): If \(\nabla f \in L^1(G)\) and \(f \in \dot{B}^{-\beta,\infty}_{\infty}(G)\), then we have NEWLINE\[NEWLINE ||f||_{\dot{W}^{s,q}_{\infty}} \leq C||\nabla f||_{L^1}^{\theta} ||f||_{\dot{B}^{-\beta, \infty}_{\infty}}^{1-\theta} NEWLINE\]NEWLINE where \(1<q<\infty,\;0< s< 1/q<1,\;\theta=1/q\) and \(\beta = \frac{1-sq}{q-1}\).