Reproducing kernels for harmonic Besov spaces on the ball
The authors consider Besov spaces of harmonic functions on the unit ball \(\mathbb{B\subset R}^{n}\). They form a two parameter Sobolev type subfamily of harmonic functions on the unit ball \(\mathbb{B}\), extending Bergman spaces. Earlier such functions were studied by \textit{E. Ligocka} [Stud. Math. 84, 79--87 (1986; Zbl 0627.46033)], [Stud. Math. 86, 255--271 (1987; Zbl 0642.46035)], [Stud. Math. 87, 23--32 (1987; Zbl 0658.31006)], and [Stud. Math. 87, 223--238 (1987; Zbl 0657.31006)]. In the main result, the reproducing kernels of those Besov spaces which are Hilbert spaces are computed. They are weighted sums of zonal harmonics and natural radial fractional derivatives of the Poisson kernel, extending results of \textit{R. R. Coifman} and \textit{R. Rochberg} [Astérisque 77, 11--66 (1980; Zbl 0472.46040)] and \textit{S. Axler, P. Bourdon} and \textit{W. Ramey} [Harmonic function theory. Graduate Texts in Mathematics. 137. New York: Springer-Verlag. xii, 231 p. (1992; Zbl 0765.31001)].
scientific article; zbMATH DE number 5571954
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