On homogeneous polynomials on a complex ball

Abstract

We prove that there exist «-homogeneous polynomials p" on a complex d-dimensional ball such that II /»" II "o = 1 and Ily7"ll2 * fñ2~d. This enables us to answer some questions about Hp and Bloch spaces on a complex ball. We also investigate interpolation by «-homogeneous polynomials on a 2-dimensional complex ball. Introduction. The starting point of our investigation was a question asked by S. Waigner: Is the identity map from Hx(Bd) into Hx(Bd) (Bd is a unit ball in Cd), d > 1, a compact linear map. This question has a connection with the well-known open problem (cf. [5]): does there exist a nonconstant inner function on Bd, d > l?1 The existence of such an inner function would imply our Corollary 1.5, namely that this operator is not compact. We obtain this result by exhibiting «-homogeneous polynomials pn which in some respects resemble the monomials z" in the one-dimensional case (Theorem 1.2). We give two proofs of this theorem, one in §1 and the other at the end of §2. Those polynomials enable us to also answer a question of R. Timoney (Corollary 1.9). We hope that they will find some other applications. In §2 we investigate interpolating «-homogeneous polynomials on the unit sphere in two-dimensional complex space. The motivation for this study is the following well-known open problem (cf. [6]): does there exist a function <p(t), t > 1, such that for every finite-dimensional Banach space X we have d(X, l^nX) < y(X(X)) (for the definitions, see below). To the best of our knowledge, the spaces W™(S2), of all «-homogeneous polynomials on the unit ball in C2, are the first spaces known for which X(W™(S2)) is bounded independently of n, while d(W™(S2), /£+1) is not known to be bounded. Thus the following problem naturally arises. Problem. Compute or estimate d(W™(Sd), /*), k = dim W™(Sd). In our opinion, the results of §2 indicate that this problem may not be trivial.2 Apart from this Banach space motivation our results may be interesting as interpolation results. They are very different from corresponding results for trigonometric polynomials on the unit circle, of degree at most n (cf. [8, Chapter X]). Now let us fix some notations. We will work with Cd, the complex ^-dimensional space equipped with the usual scalar product denoted by (• , •). Bd will stand for the

On Functions in the Ball Algebra

“…It was shown in [4 (x depends only on the dimension of the ball, it can be taken x = 2d/Vñ-) Those polynomials will be crucial in our further considerations. In particular, we will use the following inequality (cf.…”

“…It was observed in [4,Remark 1.10], that using Theorem 1.2 of [4] such examples can be constructed in arbitrary dimension. In the present note we further pursue this idea.…”

A property of series of holomorphic homogeneous polynomials with Hadamard gaps

“…In order to state our main result, we require the so-called Ryll-Wajtaszczyk polynomials. The sequence of such polynomials that we shall need here is slightly different from the original one [7] and was obtained by Ullrich [10,Corollary 1]. We state the existence of those polynomials as a lemma.…”

The spaces 𝐻^{𝑝}(𝐵_{𝑛}),0<𝑝<1, and 𝐵_{𝑝𝑞}(𝐵_{𝑛}),0<𝑝<𝑞<1, are not locally convex

“…In this note we will prove that Hp(Bn), 0 < p < 1, and Bpq(Bn), 0 < p < q < 1, are not locally convex. Combining (3) and (4) we obtain (5) \\fh<A\\f\\i.…”