C. R. Acad. Sci. Paris, t. 333, Série I, p. 285–290, 2001 - PXMA????.TEX - Analyse mathématique/Mathematical Analysis

Integral Formulas for a Sub-Hardy Hilbert Space on the Ball with Complete Nevanlinna-Pick Reproducing Kernel

Daniel Alpay^{a}, H. Turgay Kaptanoğlu^{b}

a Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel E-mail: dany@math.bgu.ac.il

b Mathematics Department, Middle East Technical University, Ankara 06531, Turkey E-mail: kaptan@math.metu.edu.tr

(Reçu le 8 mai 2001, accepté le 11 juin 2001)

Abstract. We use the theory of Hilbert spaces of analytic functions on bounded symmetric domains in ℂ^{N} to obtain information on the (1/(N + 1))st power of the Bergman kernel of the ball. This kernel has played recently an important and growing role in operator theory. We present several integral formulas for the Hilbert space generated by this kernel. \begin{matrix}c \\ ◯ \end{matrix} 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

Représentation intégrale pour un sous-espace de Hardy de la boule de noyau reproduisant de type Nevanlinna-Pick.

Résumé. Nous utilisons la théorie des espaces de Hilbert de fonctions analytiques dans des domaines symétriques de ℂ^{N} pour étudier la puissance 1/(N + 1) du noyau de Bergman de la boule unité. Ce noyau joue depuis peu un rôle important en théorie des opérateurs. Nous donnons des représentations intégrales pour l’espace à noyau reproduisant associé. \begin{matrix}c \\ ◯ \end{matrix} 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

Version française abrégée

Soit ℋ l’espace à noyau reproduisant de fonctions analytiques dans la boule unité B de ℂ^{N} de noyau reproduisant k(z, w) = (1−(z_{1}\overline{w}_{1}+⋯+z_{N} \overline{w}_{N}))^{−1}. Les monômes \{z^{α}\} forment un ensemble orthogonal dense de ℋ,

∥z^{α}∥_{ℋ}^{2} = \frac{α!}{|α|!}, et (f, g)_{ℋ} = ∑_{α} f_{α}\overline{g}_{α} \frac{α!}{|α|!},

où f(z) = ∑_{α} f_{α}z^{α} et g(z) = ∑_{α} g_{α}z^{α} sont dans ℋ, et où nous utilisons les multi-indexes. Cet espace joue un rôle important depuis peu en théorie des opérateurs, et en particulier en relation avec le problème

Note présentée par Jean-Pierre KAHANE

S0764-4442(00)0????-?/FLA \begin{matrix}c \\ ◯ \end{matrix} 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés. 1

D. Alpay, H. T. Kaptanoğlu

d’interpolation de Nevanlinna–Pick. L’espace ℋ est inclus contractivement dans l’espace de Hardy. L’inclusion est stricte dès que N > 1.

Soit σ la mesure d’aire sur ∂B, normalisée par σ(∂B) = 1, et soit ν la mesure de volume dans B avec la normalisation ν(B) = 1. Nous dénotons par τ la mesure dτ(z) = k(z, w)^{N+1} dν(z). Cette mesure est invariante sous l’action des automorphismes de B. Si f est une fonction analytique, le symbole Rf(z) désigne sa dérivée radiale au point z définie par (4). Let I be the identity operator, and for integers m, j, put ℛ_{m} = (R + (N − 1)I)(R + (N − 2)I) ⋯ (R + (N − m)I), 𝒟^{m} = (R + I)(R + 2I) ⋯ (R + mI), and 𝒮_{j} = ℛ_{N−1}(R + jI).

The integral formulas for the inner product of ℋ are given in Theorems 2, 4, 5, 7, 9 and 10. These lead to integral representations for functions in ℋ given in Corollaries 3, 6, 8, and 11. Integral representations with area integrals for the Hardy space are also obtained.

1. Sub-Hardy Hilbert Space ℋ with Reproducing Kernel (1 − ⟨z, w⟩)^{−1} Let B be the open unit ball in ℂ^{N} and S its boundary, the unit sphere. The space ℋ is the reproducing

kernel Hilbert space on B with reproducing kernel k(z, w) = (1 − ⟨z, w⟩)^{−1}, where ⟨z, w⟩ = z_{1}\overline{w}_{1} + ⋯ + z_{N} \overline{w}_{N}. This means that for any f ∈ ℋ and w ∈ B, we have f(w) = (f, k(·, w))_{ℋ} , where (·, ·)_{ℋ} is the inner product of ℋ. The functions in ℋ are holomorphic in B. It is shown in [2] that the monomials {z^{α}} form an orthogonal dense subset of ℋ, and that

∥z^{α}∥_{ℋ}^{2} = \frac{α!}{|α|!}  and (f, g)_{ℋ} = ∑_{α} f_{α}\overline{g}_{α} \frac{α!}{|α|!},  (1)

if f(z) = ∑_{α} f_{α}z^{α} and g(z) = ∑_{α} g_{α}z^{α} are in ℋ. Here and throughout we use multi-index notation; so α = (α_{1}, …, α_{N}) is an N-tuple of nonnegative integers, |α| = α_{1} + ⋯ + α_{N}, α! = α_{1}! ⋯ α_{N}!, 0^{0} = 1, and z^{α} = z_{1}^{α_{1}} ⋯ z_{N}^{α_{N}}. We adhere to the convention that an empty product is 1, and for operators is I, the identity operator. In addition, \overline{( )} means the closure of a set or the complex conjugate.

The space ℋ has attracted a lot of attention lately in operator theory because of its importance in Nevanlinna-Pick interpolation; see [1] and [6].

The reproducing kernels of the Bergman space A^{2} and the Hardy space H^{2} of the ball are K(z, w) = k(z, w)^{N+1} and C(z, w) = k(z, w)^{N}. We have the inclusions ℋ ⊂ H^{2} ⊂ A^{2}. All inclusions are contractive and proper as partly shown in [2], except that ℋ and H^{2} coincide when N = 1.

2. Preliminaries Let Aut(B) be the group of all automophisms of B, that is, one-to-one holomorphic maps of B onto B.

We refer the reader to [2] and [9, Chapter 2] for detailed descriptions of members ψ of Aut(B). We just note that ψ^{−1} = ψ and Aut(B) acts transitively on B.

Let σ be the Lebesgue (surface) measure on S normalized so that σ(S) = 1. Let V be the Lebesgue (volume) measure on B, and let ν be its normalized form with ν(B) = 1. Let τ be the measure defined on B by dτ(z) = k(z, z)^{N+1} dν(z). The importance of τ lies in the fact that it is invariant under the automorphisms of B; that is, ∫_{B} f dτ = ∫_{B}(f ∘ ψ) dτ for ψ ∈ Aut(B) and integrable f.

Propositions 1.4.8 and 1.4.9 of [9] state that ∫_{S} ζ^{α} \overline{ζ}^{β} dσ(ζ) = 0 = ∫_{B} z^{α} \overline{z}^{β} dν(z) if α ≠ β,

∫_{S} |ζ^{α}|^{2} dσ(ζ) = \frac{(N − 1)! α!}{(N − 1 + |α|)!},  and  ∫_{B} |z^{α}|^{2} dν(z) = \frac{N! α!}{(N + |α|)!}.  (2)

The polar-coordinate formula that relates ν and σ given in 1.4.3 of [9] is

∫_{B} f dν = 2N ∫_{0}^{1} r^{2N−1} dr ∫_{S} f(rζ) dσ(ζ).  (3)

Integral Formulas for a Nevanlinna-Pick Reproducing Kernel Hilbert Space

Lemma 1. – For λ > N, if α ≠ β, then ∫_{B} z^{α} \overline{z}^{β} (1 − |z|^{2})^{λ} dτ(z) = 0, and

∫_{B} |z^{α}|^{2} (1 − |z|^{2})^{λ} dτ(z) = [\frac{N!}{(λ − N) ⋯ (λ − 2)(λ − 1)}][\frac{α!}{(λ)(λ + 1) ⋯ (λ + |α| − 1)}].

The lemma is proved by applying (3) to the integral on the left and then using the elementary properties of the beta function. The integral is with respect to the volume measure ν if the power on 1 − |z|^{2} is written as λ − N − 1. The second formula in (2) is the case λ = N + 1.

The radial derivative at z of a holomorphic function f and its most useful applications are

Rf(z) = ∑_{j=1}^{N} z_{j} \frac{∂f}{∂z_{j}} (z),  R(z^{α}) = |α| z^{α},  and R(k(z, w)^{m}) = m ⟨z, w⟩ k(z, w)^{m+1}. (4)

The presentation below mainly originates in work on analysis on bounded symmetric domains that depend heavily on Lie-group machinery. Readable accounts are [5] and [4]. What is interesting is that the space ℋ and its kernel k(z, w) is the special case λ = 1 in B of a whole family of spaces and kernels on bounded symmetric domains indexed by λ, and the range of values of λ, called the Wallach set, depend on the type of the domain Ω and its parameters such as genus g. The Wallach set W (Ω) is the set of all λ for which K_{λ}(z, w) = K_{Ω}(z, w)^{λ/g} is nonnegative definite, where K_{Ω} is the Bergman kernel of Ω. Then reproducing kernel Hilbert spaces ℋ_{λ}(Ω) are defined on Ω that admit K_{λ}(z, w) as reproducing kernels. These spaces include the Bergman and Hardy spaces as special cases. It would be interesting to investigate Nevanlinna-Pick-type and operator-theoretic problems on such spaces. There is a lot of room even in the unit ball, the simplest bounded symmetric domain, where g = N + 1 and W (B) = [0, ∞). So we have a Hilbert space ℋ_{λ} on B with reproducing kernel K_{λ}(z, w) = (1 − ⟨z, w⟩)^{−λ} for each λ ⩾ 0.

Each of the unitary operators (U_{ψ}f)(z) = f(ψ(z)) [(Jψ)(z)]^{1/(N+1)} maps ℋ onto itself, where ψ ∈ Aut(B), (Jψ)(z) is its complex Jacobian, and the principal branch is used for the root. In fact, ℋ is the unique Hilbert space of holomorphic functions on B with this property in which point evaluations are continuous. Further, the inner product of ℋ is invariant under each U_{ψ}, that is, (U_{ψ}f, U_{ψ}g)_{ℋ} = (f, g)_{ℋ} for f, g ∈ ℋ. These are in [5].

3. Integral Representation with Surface Measure

Let ℛ_{0} = I and ℛ_{m} = (R + (N − 1)I)(R + (N − 2)I) ⋯ (R + (N − m)I) for m = 1, 2, . . .. These linear differential operators are introduced in [3]. We need only the cases m = N − 1 and m = N.

Theorem 2. – For f, g ∈ ℋ, we have

(f, g)_{ℋ} = lim_{r→1^{−}} \frac{1}{(N − 1)!} ∫_{S}(ℛ_{N−1}f)(rζ) \overline{g(rζ)} dσ(ζ).

This theorem is an adaptation of a result in [3] to functions holomorphic in B, but not in a neighborhood of \overline{B}. It suffices to check the formula on the monomials z^{α}. This is an easy matter by (4) and (2). Since the monomials are dense in ℋ, the Hilbert space completion of {z^{α}} is just ℋ.

Let’s give some applications of this formula. By the reproducing property of the kernel k(z, w), if f ∈ ℋ and w ∈ B, we have

f(w) = \overline{(k(·, w), f)_{ℋ}} = \frac{1}{(N − 1)!} ∫_{S} \overline{(ℛ_{N−1}k)(ζ, w)} f(ζ) dσ(ζ),  (5)

where no limit as r → 1^{−} is needed, because k(z, w) is holomorphic in a neighborhood of \overline{B}, and f has boundary values almost everywhere since ℋ ⊂ H^{2}. Now suppose f(0) is real, put u = Ref, define

h(z, w) = 2k(z, w) − 1 = \frac{1 + ⟨z, w⟩}{1 − ⟨z, w⟩},

and consider the pairing (u, h(·, w))_{ℋ} . This pairing does not make sense as it stands since u is not in ℋ unless it is constant. However, it does make sense when the integral form of the inner product is used. Then

(u, h(·, w))_{ℋ} = \frac{1}{(N − 1)!} ∫_{S} \overline{(ℛ_{N−1}h)(ζ, w)} u(ζ) dσ(ζ)  (6)
= \frac{1}{2(N − 1)!}\overline{∫_{S}[2(ℛ_{N−1}k)(ζ, w) − (N − 1)!] [\overline{f(ζ)} + f(ζ)] dσ(ζ)}
= f(w) + \overline{f(0)} − Ref(0) = f(w).

Above, f(w) follows from (5); the product of (ℛ_{N−1}k)(ζ, w) and f(ζ) give \overline{f(0)} as seen by expanding them into power series and using (2); the mean-value property of the harmonic function u yields Ref(0). Thus we obtain a Herglotz-type representation with a Herglotz-type kernel, and the functions in ℋ are recovered from the boundary values of their real parts using the integral (6).

Corollary 3. – For f ∈ ℋ, we have

f(w) = ∫_{S} \frac{f(ζ)}{(1 − ⟨w, ζ⟩)^{N}} dσ(ζ).

This follows by computing the radial derivatives in (5) explicitly using (4). Thus we obtain the Cauchy-Szegö representation in B [9, 3.2.4]. This is not surprising because of the last paragraph of Section 1. Correspondingly, the explicit version of (6) is the Herglotz formula in B [9, 3.2.5].

4. Integral Representations with Volume Measure

Theorem 4. – For f, g ∈ ℋ, we have

(f, g)_{ℋ} = f(0)\overline{g(0)} + \frac{1}{N!} ∫_{B}(ℛ_{N}f)(z) \overline{g(z)} \frac{1}{|z|^{2N}} dν(z)

This formula also appears in [3]. Again we check only the norms of the monomials using (4) and (3). Other families of linear differential operators on bounded symmetric domains are introduced in [10], but

explicit expressions for most of them apparently are not known in general; see [4, p. 55]. However, in B only one such operator is needed and it is computed in [4, p. 55] for a different kind of domain; yet it works in B too. It turns out that these operators are also relatives of the radial derivative in B. Put 𝒟^{0} = I and 𝒟^{m} = (R + I)(R + 2I) ⋯ (R + mI) for m = 1, 2, . . ..

Theorem 5. – For f, g ∈ ℋ and for any integer m ⩾ N, we have

(f, g)_{ℋ} = \frac{1}{N! (m − N)!} ∫_{B}(𝒟^{m}f)(z) \overline{g(z)} k(z, z)^{N−m} dν(z).

As usual, it is sufficient to check only the monomials. This is done by applying (4) and then Lemma 1 with λ = m + 1. Note that the power on k(z, z) is never positive.

Corollary 6. – For integers m ⩾ N and f ∈ ℋ, we have

f(w) = \binom{m}{N} ∫_{B} \frac{(1 − |z|^{2})^{m−N}}{(1 − ⟨w, z⟩)^{m+1}} f(z) dν(z).

The proof is similar to that of Corollary 3 using the operators 𝒟^{m} in place of ℛ_{N−1}. What we obtain now are those cases with integer powers of the Bergman-type projections discussed in [9, Section 7.1]. The case m = N, in particular, is the Bergman projection. This is not surprising either because of the last paragraph of Section 1. The corresponding Herglotz-type formulas look complicated and are not given.

There is another description of ℋ in [10]. Let m be an integer satisfying 2m ⩾ N, and put

[f, g] = \binom{2m}{N} ∫_{B}(𝒟^{m}f)(z) \overline{(𝒟^{m}g)(z)} k(z, z)^{N−2m} dν(z).

Then there are positive constants c_{1} and c_{2} such that c_{1}[f, f] ⩽ (f, f)_{ℋ} ⩽ c_{2}[f, f], and therefore ℋ is the space of functions holomorphic in B for which [f, f] < ∞.

We now introduce some new integral formulas for the inner product of ℋ that are intermediate between those in Theorems 4 and 5. First let 𝒮_{j} = ℛ_{N−j}(R + jI)(R + (j − 1)I) ⋯ (R + I) = ℛ_{N−1}(R + jI) for j = 1, …, N so that 𝒮_{N} = 𝒟^{N}. Let also 𝒮_{0} = ℛ_{N}.

Theorem 7. – For f, g ∈ ℋ and for j = 1, 2, …, N, we have

(f, g)_{ℋ} = \frac{1}{N!} ∫_{B}(𝒮_{j}f)(z) \overline{g(z)} \frac{1}{|z|^{2(N−j)}} dν(z).

The formula holds also for j = 0 provided we add to the integral the term f(0)\overline{g(0)}.

The proof of this follows the same lines as that of Theorem 4. Clearly, the case j = 0 is in Theorem 4, and the case j = N is the case m = N of Theorem 5. The case j = 1 also appears in [3].

Corollary 8. – For j = 1, …, N and f ∈ ℋ, we have

f(w) = \frac{1}{N} ∫_{B} \frac{j + (N − j)⟨w, z⟩}{(1 − ⟨w, z⟩)^{N+1} |z|^{2(N−j)}} f(z) dν(z).

The formula holds also for j = 0 provided we add to the integral the term f(0).

We follow the proof of Corollary 3 again using the operators 𝒮_{j} in place of ℛ_{N−1}. The case j = N gives the Bergman formula again. The only simple-looking Herglotz-type formula is that of j = 0.

5. Integral Representations with Invariant Measure Now we obtain a new and simpler integral formula that describes ℋ as a weighted-Bergman-type space.

Following the method of [8], consider the pairing

⟨⟨f, g⟩⟩_{λ} = \frac{(λ − N) ⋯ (λ − 2)(λ − 1)}{N!} ∫_{B} f(z) \overline{g(z)} k(z, z)^{−λ} dτ(z).

Theorem 9. – For f, g ∈ ℋ, we have (f, g)_{ℋ} = lim_{λ→1}⟨⟨f, g⟩⟩_{λ}, and ℋ is the space of all holomorphic

functions f on B that satisfy lim_{λ→1}⟨⟨f, f⟩⟩_{λ} < ∞.

The advantage of this formula is that there is no derivative on either function. The weight in the integral is essentially (1 − |z|^{2})^{−N} with respect to the measure dν(z).

Proof . – We start with the monomials once again. If λ > N, then by Lemma 1,

⟨⟨z^{α}, z^{α}⟩⟩_{λ} = \frac{α!}{(λ)(λ + 1) ⋯ (λ + |α| − 1)}.

Thus the function λ ↦ ⟨⟨z^{α}, z^{α}⟩⟩_{λ} extends holomorphically at least to the region {Reλ > 0}. Moreover, lim_{λ→1}⟨⟨z^{α}, z^{α}⟩⟩_{λ} = α!/|α|!. It follows that the Hilbert space completion of {z^{α}} coincides with ℋ. □

We again introduce some new integral formulas for the inner product of ℋ that are intermediate this time between those in Theorems 5 and 9.

Theorem 10. – For f, g ∈ ℋ and m = 0, 1, …, N, we have

(f, g)_{ℋ} = lim_{λ→1} \frac{(λ − N + m) ⋯ (λ − 2)(λ − 1)}{N!} ∫_{B}(𝒟^{m}f)(z) \overline{g(z)} k(z, z)^{−(m+λ)} dτ(z).

The proof of this follows the same lines as that of Theorem 9. Clearly, the case m = 0 is Theorem 9, and the case m = N is the case m = N of Theorem 5.

Corollary 11. – For m = 0, 1, …, N and f ∈ ℋ, we have

f(w) = lim_{λ→1} \frac{m! (λ − N + m) ⋯ (λ − 2)(λ − 1)}{N!} ∫_{B} \frac{(1 − |z|^{2})^{m−N+λ−1}}{(1 − ⟨w, z⟩)^{m+1}} f(z) dν(z).

The proof is as above. The formulas obtained are the extensions of the Bergman-type projections in [9, Section 7.1] to certain negative integer values of m−N for the space ℋ (see Corollary 6). The case m = N gives the Bergman integral once again and requires no limit. The Herglotz-type formula for m = 0 is

f(w) = lim_{λ→1} \frac{(λ − N) ⋯ (λ − 2)(λ − 1)}{N!} ∫_{B} \frac{1 + ⟨w, z⟩}{1 − ⟨w, z⟩} (1 − |z|^{2})^{λ} u(z) dτ(z).

6. Hardy Space Revisited The inner products in Theorems 4, 5, 7, 9, and 10 written explicitly when N = 1 give new and unusual

formulas for the inner product of H^{2}. Here is a short list.

(f, g)_{H^{2}} = f(0)\overline{g(0)} + \frac{1}{π} ∫_{0}^{2π} ∫_{0}^{1} f′(re^{iθ}) \overline{g(re^{iθ})} e^{iθ} dr dθ

= \frac{1}{π} ∫_{0}^{2π} ∫_{0}^{1}(f(re^{iθ}) + f′(re^{iθ}) re^{iθ}) \overline{g(re^{iθ})} r dr dθ

= lim_{λ→1} \frac{λ − 1}{π} ∫_{0}^{2π} ∫_{0}^{1} f(re^{iθ}) \overline{g(re^{iθ})} (1 − r^{2})^{λ−2} r dr dθ.

The last formula is actually identical to the classical form of the inner product of H^{2}, because the weak*- limit as λ → 1 of the measures dμ_{λ}(r) = (λ − 1) (1 − r^{2})^{λ−2} 2r dr on [0, 1] is the unit point mass at r = 1. By Theorem 2.3 of [7], this is seen by computing the pointwise limit as λ → 1 of their distribution functions, which are F_{λ}(x) = μ_{λ}([0, x]) = 1 − (1 − x^{2})^{λ−1} for 0 ⩽ x < 1 and F_{λ}(1) = μ_{λ}([0, 1]) = 1.

Acknowledgements. The research of the first author was supported by a grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel, and by the Israeli Academy of Sciences. The authors also thank N. C. Heldermann for providing them with a copy of [10].

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