Weighted Hölder continuity of hyperbolic harmonic Bloch functions.
A smooth function \(f \) on the unit ball of \(\mathbb{R}^ n \) is said to be hyperbolic harmonic if its invariant Laplacian vanishes in the ball. If in addition its invariant gradient is bounded, \(f \) is said to belong to the hyperbolic harmonic Bloch space. This is an \(\mathbb{R}^ n \) version of the standard harmonic Bloch space of the disk and an analogue of the Bloch space of holomorphic functions in the disk. The little Bloch space can also be defined in analogy with the one variable case. The (diagonal) Besov \(p\)-space is defined by requiring that the \(p\)-th power of the invariant gradient be integrable with respect to the invariant volume measure on the ball. There are three main results in the paper: (1) membership in the harmonic Bloch space is equivalent to a Lipschitz type condition, as well as to a Hölder type condition; (2) a Hölder type condition also characterizes the functions in the little Bloch spaces; (3) a weighted \(L^ p \) variant of the latter condition describes the functions in the Besov space, for sufficiently large exponents that depend on the dimension. These results are analogous to the ones for the disk proved by \textit{F. Holland} and \textit{D. Walsh} [Math. Ann. 273, 317--335 (1986; Zbl 0561.30025)] and by \textit{K. Stroethoff} [Bull. Aust. Math. Soc. 54, No. 2, 211--219 (1996; Zbl 0865.30051)], possibly also independently by other authors. The main tools used are the standard symmetry lemma for the ball and some properties of certain special functions that come up in computations, especially the Gegenbauer polynomials and the Bateman integral formula.
scientific article; zbMATH DE number 1827600
scientific article