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On highly nonintegrable functions and homogeneous polynomials. (English) Zbl 0872.32001

Let \(\mathbb S\) (resp. \(\mathbb B\)) denote the unit sphere (resp. the unit ball) in the complex space \(\mathbb C^d\).
The author shows that there exists an integer \(k = k(d)\) and a sequence \(\{p_n\}\) of homogeneous polynomials of \(d\) complex variables with \(\deg p_n = n\) such that (i) \(|p_n(z) \leq 2\) for all \(z \in \mathbb S\), (ii) for each \(s\) (large enough), \(\sum_{n=ks}^{k(n+1)-1} |p_n(z)|\geq 0,5\) for all \(z\in \mathbb S\), (iii) the function \(f(z) := \sum_n n^{\ln n} p_n (z)\) is holomorphic in the ball \(\mathbb B\) such that for each complex line \(L\) in \(\mathbb C^d\) and any \(p> 0\) one has \(\int_{L\cap \mathbb B} |f(z)|^p d\nu (z) = \infty\), where \(\nu\) is the volume measure on \(\mathbb B\).
The last result improves Theorem 1 of P. Jakóbczak [“Highly nonintegrable functions in the unit ball”, Isr. J. Math. 97, 175-181 (1997)].
Reviewer: J.Siciak (Kraków)

MSC:

32A05 Power series, series of functions of several complex variables
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
32A40 Boundary behavior of holomorphic functions of several complex variables
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