Riesz potentials and \(p\)-superharmonic functions in Lie groups of Heisenberg type (Q2880385)

A Carnot group of step two is a connected, simply connected Lie group \(\mathbb G\) whose Lie algebra admits a decomposition \({\mathfrak g}=V_1\oplus V_2\) such that \([V_1,V_1]=V_2\), \([V_1,V_2]=\{0\}\) and the layers \(V_1\) and \(V_2\) are orthogonal with respect to an inner product \(\langle\cdot,\cdot\rangle\). The map \(J:V_2\rightarrow\text{End}(V_1)\) is defined by the identity NEWLINENEWLINE\[NEWLINE \big\langle J(t)z,z'\big\rangle=\big\langle[z,z'],t\big\rangle,\quad z,z'\in V_1,t\in V_2\,. NEWLINE\]NEWLINE A Carnot group of step two \(\mathbb G\) is said to be of Heisenberg type, or \(H\)-type, if for every \(t\in V_2\) with \(|t|=1\), the map \(J(t)\) is an orthogonal endomorphism of \(V_1\). Let \(Q\) be the homogeneous dimension of the group \(\mathbb G\). The authors introduce the Riesz potential of order \(\alpha\) on \(\mathbb G\). Given \(\rho\in C_0(\mathbb G)\), set NEWLINENEWLINE\[NEWLINE R_\alpha(\rho)(g)=\int_{\mathbb G}\frac{\rho(g')} {N(g^{-1}g')^{Q-\alpha}}\,dg',\quad 0<\alpha< Q\,, NEWLINE\]NEWLINE NEWLINEwhere \(N(g)=\big(|z|^4+16|t|^2\big)^{1/4}\) is the Folland-Kaplan gauge on \(\mathbb G\).NEWLINENEWLINEThe main result of the present paper is the following theorem. NEWLINENEWLINENEWLINE Theorem 1.3. Let \(\mathbb G\) be a group of Heisenberg type with homogeneous dimension \(Q\). For given \(\rho\in C_0(\mathbb G),\) \(\rho\geq0,\) the following three cases hold. {\parindent=6mm \begin{itemize}\item[(1)] If \(2<p< Q,\) then \(R_{Q-\alpha}(\rho)\) is \(p\)-superharmonic when NEWLINE\[NEWLINE 0<\alpha\leq\frac{Q-p}{p-1}\,. NEWLINE\]NEWLINE \item[(2)] If \(p> Q,\) then \(R_{Q-\alpha}(\rho)\) is \(p\)-subharmonic when NEWLINE\[NEWLINE -\alpha\geq\frac{p-Q}{p-1}\,. NEWLINE\]NEWLINE If \(p=\infty,\) one may take \(-\alpha\geq1\). \item[(3)] If \(p=Q,\) then the function NEWLINE\[NEWLINE R_Q(\rho)(g)=\int_{\mathbb G}\rho(g')\log N(g^{-1}g')\,dg' NEWLINE\]NEWLINE is \(Q\)-subharmonic. NEWLINENEWLINE\end{itemize}}