On homogeneous polynomials on a complex ball

We prove that there exist «-homogeneous polynomials p„ on a complex d-dimensional ball such that II /»„ II „o = 1 and Ily7„ll2 * fn2~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 1, such that for every finite-dimensional Banach space X we have d(X, l^nX) = 1}. The point (1,0,. ,0) G Sd will be denoted by 1. On Sd we have the natural rotation-invariant probability measure a. Wn(Sd) will denote the space of all «-homogeneous polynomials on Cd restricted to Sd. On Wn(Sd) we will consider various norms. For f E Wn(Sd) we put 11/11^ = ilsd\fWd°iS)r/p if 1 " we obtain A/(l) = (P„f)(l)=f(i), so X = 1. This means that Pn is an orthogonal projection onto W2(Sd). Its norm as an operator on LJSd) equals aid, n)a(d,2n)-x xfn2~d. Before we start the proof of Theorem 1.2 let us recall some well-known notions concerning finite-dimensional Banach spaces. Let X, Y be finite-dimensional Banach spaces. We put X(*) = infill Til ■ \\S\\:X^Lxtx, TS = Id*, Lx is an arbitrary Ljuy-spacej and d(X, Y) = inf{||71 • lir-'ll: T: X -> Yis 1-1 and onto}. License or copyright restrictions may apply to redistribution; see http://www.ams. org/journal-terms-of-use HOMOGENEOUS POLYNOMIALS ON A COMPLEX BALL 109 The number A(A') is called a projection constant of X and d(X, Y) is called the Banach-Mazur distance between X and Y. It is well known and easy to see that (1) X(X) ^/«~. More information on this can be found in [6] and [4,22.1 and 28.1]. We will also use the following two lemmas. Lemma 1.3. inf{\\p\\2: p E W„(Sd) and H/HI«, = 1) = ya(d,2n). Proof. Since the spaces under consideration are finite dimensional, there exists a polynomial p0 realising the minimum. We can assume p0ii) — 1 = ll/Poll«,It is easily seen that Pin = -^r---f%o(h,ei9iz2,.,ea^zd)der-'ded_l = (S,ir. (2ir) •'o Jo Moreover, ||p\\2 « ||p0\\2 and \\p\\x = IIpQIIx = 1. Sincep0 was minimal we infer Il PII2 = II Po II2This proves the lemma. Lemma 1.4. dimW„(Sd) = a(d,2n)~x. Proof. It follows from Proposition 1.1 that dim Wn(Sd) = trace Pn. Since Pn is a projection its trace equals the square of its Hubert-Schmidt norm, which, as is well known, equals a(d,2n)-2f j \(S,V)\2"da(! ;)da(Ti) = a(d,2n)-1. Now we are ready to prove the theorem. Proof of Theorem 1.2. We apply (1) for X = W2(Sd) and Y = Wn°°(Sd). Using (2), Proposition 1.1 and Lemma 1.4 we obtain (3) d{W2(Sd), W? iSd)) ^ X{W2(Sd))XiW-(Sd))-X > 2x~d(fr/2)]ia(d,2ny] . Let / denote the identity map from WTM(Sd) into W2(Sd). By the definition of the Banach-Mazur distance we have 11/II ■ III~x\\ > d(W2(Sd),WTM(Sd)). Lemma 1.3 gives || 7"11| = ]Ja(d, 2«)~ so by (3) we have \\I\\^2x'd^/2. This inequality proves the theorem. Remark. Actually d(W2(Sd), W„°°(Sd)) = \\I\\ ■ \\rx\\. This follows from [1, Lemma 4.6]. Corollary 1.5. id: Hx(Bd) -» Hx(Bd) is not a compact operator. Proof. Polynomials pn exhibited in Theorem 1.2 are orthogonal. Moreover by the Schwartz inequality, fW2~d tr4-d. This implies that id: Hx(Bd) -» Hx(Bd) is not a compact operator. License or copyright restrictions may apply to redistribution; see http://www.ams. org/journal-terms-of-use 110 J. RYLL AND P. WOJTASZCZYK Proposition 1.6. Let (nk) be a lacunary sequence of natural numbers and let p„k E W„k(Sd) be such that \\p„k\\x = 1 and II/>„J|, > c, k = 0,1,2. Then there exist constants A and B such that (4) A-X(i\ak\2)l/1 <\\lakP„kj