Bounded harmonic functions on some Lie groups (Q2722493)

On a solvable connected Lie group \(G\) with abelian nilradical a bounded measurable function \(h\) satisfying the equation \(h=h*\mu\) for a probability measure \(\mu\) is called \(\mu\)-harmonic. If some convolution power of \(\mu\) is nonsingular with respect to Haar measure a subgroup \(H\) of \(G\) is described such that a \(\mu\)-harmonic function \(h\) admits an integral representation NEWLINE\[NEWLINEh(g)=\int_{G/H}\hat h(g\cdot x) \nu(dx)NEWLINE\]NEWLINE with probability measure \(\nu\) on the coset space \(G/H\). This result without description of \(H\) via a.s. convergence of projections of random variables on \((G,\mu)\) has previously been obtained by \textit{W. Jaworski} [J. Anal. Math. 68, 183-208 (1996; Zbl 0865.43008)] and under stronger conditions on \(G\) and \(\mu\) by \textit{A. Raugi} [Bull. Soc. Math. Fr., Suppl., Mém. 54, 5-18 (1977; Zbl 0389.60003)].NEWLINENEWLINEFor the entire collection see [Zbl 0958.00034].