On uniformly bounded bases in spaces of holomorphic functions (Q2804447)

Let \(S_d\) denote the unit sphere in \(\mathbb{C}^d\), and let \(L^2(S_d)\) be the Hilbert space of square-integrable functions on \(S_d\) with respect to the area measure on the sphere. For every \(N\), let \(\mathcal{P}_N\) denote the space of homogeneous analytic polynomials of degree \(N\). This paper treats two problems.NEWLINENEWLINEProblem 1. Do the spaces \(\mathcal{P}_N\) admit orthonormal bases that are uniformly bounded in \(L^\infty(S_d)\)?NEWLINENEWLINEProblem 2. Does the space of analytic polynomials in \(L^2(S_d)\) admit an orthonormal basis consisting of functions that are uniformly bounded in \(L^\infty(S_d)\)?NEWLINENEWLINENote that the ``natural'' choice of an orthonormal basis consisting of monomials does not serve as a solution to these problems.NEWLINENEWLINEThe first result in this paper is a positive solution to Problem 1. In fact, a positive solution is given with the unit sphere replaced by the boundary of a smooth strictly pseudo convex domain. The proof is a quick application of a theorem of \textit{R. I. Ovsepian} and \textit{A. Pełczyński} [in: Sémin. Maurey-Schwartz 1973-1974, Espaces \(L^p\), Appl. radonif., Géom. Espaces de Banach, Exposé XX, 15 p. (1974; Zbl 0302.46008)], which characterises when a linear subspace of \(L^2(\mu)\) has a uniformly bounded orthonormal basis.NEWLINENEWLINEThe author next turns to Problem 2 (a positive solution to which obviously implies a positive solution to Problem 1). The author provides a positive solution for the case where \(d=3\) (in a previous paper [Proc. Am. Math. Soc. 93, 277--283 (1985; Zbl 0584.46038)] the author solved the case \(d=2\)). This is the main part of the paper.