Harmonic analysis on negatively curved manifolds. I
The author states several results on the harmonic analysis on a complete, simply connected Riemannian manifold M with sectional curvature satisfying \(-b^ 2\leq K\leq -a^ 2\) \((a>0,b>0)\). For instance he presents a generalization of Hardy spaces and BMO in this context and he states the following theorems: (1) If \(f\in L^ 2\), \(g\in BMO\), \(| \int fg d\omega | \leq C \| f\|_{H^ 1} \| g\|_*\) where C depends only on dim M and on the curvature bounds and \(\omega\) is the harmonic measure with respect to a fixed point 0. (2) If M satisfies a technical condition (for instance if M is rotationally symmetric to 0) then every continuous linear functional on \(H^ 1\) is of the form \(F(f)=\int fg d\omega\) \((f\in L^ 2)\) for some \(g\in BMO.\) Detailed proofs of the results will appear in a subsequent paper. It is pointed out that the proofs depend strongly on recent results by \textit{M. Anderson} and \textit{R. Schoen} [Ann. Math., II. Ser. 121, 429-461 (1985; Zbl 0587.53045)] and \textit{D. Sullivan} [J. Differ. Geom. 18, 723-732 (1983; Zbl 0541.53037)].
scientific article; zbMATH DE number 4067904