The Octonionic Bergman Kernel for the Half Space
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Advances in Applied Clifford Algebras
The Octonionic Bergman Kernel for the Half Space Jinxun Wang∗ and Xingmin Li Communicated by Uwe Kaehler Abstract. We obtain the octonionic Bergman kernel for half space in the octonionic analysis setting by two different methods. As a consequence, we unify the kernel forms in both complex analysis and hyper-complex analysis. Mathematics Subject Classification. Primary 30G35, Secondary 30H20. Keywords. Octonions, Octonionic analysis, Bergman kernel.
1. Introduction In a recent paper [18] we derived the octonionic Bergman kernel for the unit ball, based on our newly defined inner product of the octonionic Bergman space. Note that in complex analysis the unit disc and the upper half space are holomorphically equivalent through Cayley transform. In octonionic space there is also a similar Cayley transform mapping the unit ball onto the half space (cf. [17]). So the problems for the octonionic Bergman space on the half space in R8 naturally arise. But unfortunately the octonionic Cayley transform is neither left O-analytic not right O-analytic by our definition, and the octonions are neither commutative nor associative, which bring barriers to the study of the problems in half space through Cayley transform. Thus we need to investigate the case for half space by other ways. First we accordingly have the following definition. Definition 1.1. (The octonionic Bergman space on the half space) Let R8+ := {x ∈ O : Re x > 0} This work was supported by the National Natural Science Foundation of China (No. 11701105). ∗ Corresponding
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J. Wang, X. Li
Adv. Appl. Clifford Algebras
be the half space in R8 , the octonionic Bergman space B 2 (R8+ ) is the class of left octonionic analytic functions f on R8+ satisfying 12 1 2 f B2 (R8+ ) := |f | dV < ∞, ω8 R8+ 4
where ω8 = π3 is the surface area of the unit sphere in R8 and dV is the volume element on R8+ . The inner product is defined as the usual way. Definition 1.2. (Inner product on B 2 (R8+ )) Let f, g ∈ B 2 (R8+ ), we define 1 gf dV. (f, g)R8+ := ω8 R8+ By density argument and limit argument we prove the following main theorem of this paper: Theorem 1.3. The octonionic Bergman kernel of B 2 (R8+ ) exists. Let B(x, a) = −2
∂ E(x + a), ∂x0
where E(x) = |x|x 8 is the octonionic Cauchy kernel, then B(·, a) is the desired octonionic Bergman kernel, i.e., B(·, a) ∈ B 2 (R8+ ), and for any f ∈ B 2 (R8+ ) and any a ∈ R8+ , there holds the following reproducing formula f (a) = (f, B(·, a))R8+ . Moreover, the kernel is unique. The rest of the paper is organized as follows. In Sect. 2 to make the paper self-contained we briefly review the octonion algebra and octonionic analysis. In Sect. 3 we investigate the octonionic Hardy space on the half space in R8 , for which the results are of independent interest and will be useful in the proof of our main theorem (Theorem 1.3). In Sect. 4 we give two proofs of Theorem 1.3. In the last section we point out that the Bergman kernels can be unified in one for
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