Harmonic Bergman Kernel for Some Balls

We treat the complex harmonic function on the Np–ball which is defined by the Np–norm related to the Lie norm. As a subspace, we treat Hardy spaces and consider the Bergman kernel on those spaces. Then, we try to construct the Bergman kernel in a concrete form in 2–dimensional Euclidean space. Introduction. In [2], [4], [6] and [7], we studied holomorphic functions and analytic functionals on the Np–ball in the complex Euclidean space C, n ≥ 2, and in [2], we expressed the Bergman kernel for a Hardy space on the Np–ball by a double series by using of homogeneous harmonic extended Legendre polynomials. The closed form is known only for p = 2 and∞. In the 2–dimensional case, we can calculate the coefficients of the double series expansion ([3]). However, even if we restrict our consideration to the 2–dimensional case, it is hard to express the Bergman kernel in a closed form. In this paper, we mainly treat complex harmonic functions on the Np–balls and determine the “harmonic” Bergman kernel by an infinite sum (Theorem 3.1). Then we represent the harmonic Bergman kernel more explicitly for the 2–dimensional Np–ball (Theorem 3.2) and represent it in a concrete form for p = 1, 2 and ∞. The author would like to express her thanks to Professor Józef Siciak for his useful advice. 1. Np–ball. 1.1. Np–norm. First we review the definition of the Np–balls in Cn+1, n = 0, 1, 2, · · · . For z = (z1, z2, · · · , zn+1), let L(z) = √ ‖z‖2 + √ ‖z‖4 − |z2|2