A bilinear fractional integral on compact Lie groups (Q2882465)

The authors consider the bilinear Riesz potential on a (connected and simply connected) compact semisimple Lie group \(G\), that is, the bilinear transform defined by: NEWLINE\[NEWLINE R_\alpha(f,g)=\int_G f(xy^{-1})g(xy)K_\alpha(y) dy \qquad (0<\alpha<n) \;, NEWLINE\]NEWLINE where \(K_\alpha\) is, up to renormalisation, the kernel of the Riesz potential \((-\Delta)^{\frac \alpha 2}\) on the group~\(G\). They show its continuity \(L^q(G)\times L^r (G)\to L^p(G)\) when NEWLINE\[NEWLINE \frac 1p=\frac1q+\frac 1r -\frac \alpha n>0\, , \quad 1\leq q,~r\leq \infty \;, NEWLINE\]NEWLINE with \(L^p(G)\) being replaced by weak-\(L^p(G)\) if \(q\) or \(r\) is equal to 1.NEWLINENEWLINE\textit{C. E. Kenig} and \textit{E. M. Stein} [Math. Res. Lett. 6, No. 1, 1--15 (1999; Zbl 0952.42005); erratum ibid. 6, No. 3-4, 467 (1999)] have shown a similar result on \(\mathbb R^n\).NEWLINENEWLINEThe proof of the present paper mainly follows Stein and Kenig's. The difficulty lies in replacing the arguments using scaling on \(\mathbb R^n\). This is achieved in two ways: working on local charts and using the heat kernel on \(G\) to obtain estimates for \(K_\alpha\).