Integrable harmonic functions on \(\mathbb{R}^ n\)
The authors consider the following problem, posed by Adam Korányi. Let \(\mu (dx) =c_ne^{-2 \pi|x|} dx\) be an absolutely continuous probability measure on \(\mathbb{R}^n\) and let \(f\in L^1 (\mu)\) satisfy the condition \(f*\mu=f\). Does this imply that \(f\) is a harmonic function? As shown by Korányi, there are non-harmonic solutions of the problem if \(n\geq 9\), and among these solutions there are non-harmonic exponentials. In this paper it is proved that for \(n<9\) the only solutions are the harmonic ones. The main technical result of the paper is Theorem 3.1, which solves the problem in a more general setting. However, the authors' main idea lies in proving (Theorem 2.1) that the radial Fourier transform on \(\mathbb{R}^n\) is a topological isomorphism from a convolutive algebra \(S_{w,r} (R^+)\) of weighted rapidly decreasing functions onto an algebra \(S_e (\Sigma_w)\) (with pointwise multiplication) of rapidly decreasing complex functions, where (1) \(u\in S_{w,r} (R^+)\) if \(u\) is radial on \(\mathbb{R}^n\) and \(\sup_{r>0} (1+r)^k |u^{(m)} (r) |e^{wr}\) is finite for all nonnegative integers \(m,k\); (2) \(U\in S_e (\Sigma_w)\) if \(U\) is an even \(C^\infty\)-function on the strip \(\Sigma_w =\{|\text{Im} z|\leq w\}\), holomorphic in the interior of it, and \(\sup_{z \in \Sigma_w} (1+ |z|^2)^k |U^{(m)} (z)|\) is finite for all nonnegative integers \(m,k\). This isomorphism allows to use methods of complex analysis.
scientific article; zbMATH DE number 1098592
scientific article