Fractional Derivatives of Bloch Functions, Growth Rate, and Interpolation. Boo Rim Choe and Kyung Soo Rim

Transcription

1 Fractional Derivatives of loch Functions, Growth Rate, and Interpolation oo Rim Choe and Kyung Soo Rim Abstract. loch functions on the ball are usually described by means of a restriction on the growth rate of ordinary derivatives of holomorphic functions. In this paper we first give a characterization of loch functions in terms of fractional derivatives. Then we show that the growth rate suggested by such a characterization is optimal in a certain sense. Also we prove a result concerning interpolating sequences for fractional derivatives of loch functions. 0. Introduction and Results Let be the unit ball of the complex n-space C n with norm z = z, z /2 where, is the usual Hermitian inner product on C n. A holomorphic function f on is said to be a loch function if f(z) ( z 2 ) is bounded on where f denotes the complex gradient of f. The space of loch functions endowed with norm f = f(0) + sup f(z) ( z 2 ) z is called the loch space and denoted by (). If f () satisfies the additional boundary vanishing condition f(z) ( z 2 ) 0 as z, we say f 0 (), the little loch space. In this paper we will investigate some properties of the loch space in terms of fractional derivatives. Let f be a function holomorphic on with homogeneous expansion f = k=0 f k. Following [], we define the fractional derivative D α f of order α > 0 as follows: D α f(z) = (k + ) α f k (z). k=0 Our main results are Theorems A,, and C below. The motivations for Theorems and C were one variable results of Ramey-Ullrich [RU] and Attle [At] for ordinary derivatives of order one, respectively. 99 Mathematics Subject Classification. Primary 32A37. Secondary 30D45. Key words and phrases. Fractional derivatives, loch functions, Growth rate, Interpolation. Research supported in part by KOSEF and GARC of Korea Typeset by AMS-TEX

2 Theorem A. Let f be a function holomorphic on. Then, for each 0 < p < and α > 0, the following conditions are equivalent: (a) f (). (b) sup D α f(z) ( z 2 ) α <. z (c) sup D α f(w) p ( w 2 ) pα Jϕ z (w) dv (w) <. z In condition (c) of the above, ϕ z denotes the standard automorphism of such that ϕ z (0) = z and ϕ z ϕ z = identity (see, for example, [R]) and Jϕ z (w) denotes the real Jacobian of ϕ z at w. For ordinary derivatives, a version of Theorem A has been known. See [S] or [Z]. In view of Theorem A, the fractional derivative of order α of a loch function grows no faster than ( z ) α near the boundary. We show this growth rate is optimal in the following sense. Theorem. For each α > 0, there exist finitely many loch functions f i such that D α f i (z) i ( z ) α (z ) and the number of functions depends only on the dimension n. Next we prove an interpolating property of fractional derivatives of loch functions. Given α > 0, we say that a sequence z m in is an interpolating sequence for the fractional derivative of order α of loch functions or simply an α-interpolating sequence if, given (b m ) l, there exists a loch function f such that D α f(z m )( z m 2 ) α = b m for all m. Here, l denotes the space of bounded sequences of complex numbers. In the following the term separated refers to the property of being separated with respect to the pseudo-hyperbolic distance on (see Section 3). Theorem C. Every α-interpolating sequence is separated. Conversely, for a given α > 0, if a sequence in is sufficiently separated, then it is an α-interpolating sequence. In section we prove a characterization of the loch space in terms of fractional derivatives which is needed in later sections. As one may expect, this characterization carries over to the little loch space. Also we relate it with the so-called ergman-carleson measures. Most part of section 2 is devoted to the proof of Theorem. We construct loch functions satisfying the given property by a lacunary series argument. At the end we indicate that a similar construction can be applied to produce a direct generalization of a result of Ramey-Ullrich [RU]. In section 3 we prove Theorem C. A Word on Constants. We will use the same letter c for constants which are not necessarily the same at each occurance. We will often indicate variables, in the parenthesis or in the subscript, on which c depends. However, c will be always independent of particular measures or functions under consideration. 2

3 . Fractional Derivatives In this section we describe loch functions in terms of fractional derivatives. Let us recall some known results for a motivation. For a function f holomorphic on, Theorem 4.0 of [T] implies f f(0) + sup Rf(z) ( z 2 ) z where Rf denotes the radial derivative of f defined by Rf(z) = n j= z j f z j (z) (z ). Note that Rf = k kf k if f has the homogeneous expansion f = k F k. Thus D f = Rf + f. Using this, one can verify f sup D f(z) ( z 2 ). () z It is also known ([S], [Z]) that, for a given positive integer m, f β m β f z β (0) + z β =m sup m f z β (z) ( z 2 ) m where we use the conventional multi-index notations. We will establish a similar characterization in terms of fractional derivatives and then relate it with ergman-carleson measures. We begin with an observation that fractional differentiation admits an integral representation in certain cases. Suppose a function f holomorphic on is of the form f(z) = for some ψ L () and t > 0. Then, since ( z, w ) t = ψ(w) dv (w) ( z, w ) t m=0 Γ(t + m) m!γ(t) z, w m, one can easily verify the differentiation under the integral sign: D α f(z) = ψ(w)g t,α (z, w)dv (w) (2) where Γ(t + m)(m + ) α G t,α (z, w) = m!γ(t) m=0 3 z, w m.

4 Note that G n++γ,0 (z, w), γ >, is the well-known reproducing kernel (up to a multiplicative constant factor) for A γ(), the so-called γ-weighted ergman space on. More precisely, if f is holomorphic and integrable against the weighted volume measure ( z 2 ) γ dv (z) on, then f(z) = c γ f(w)g n++γ,0 (z, w)( w 2 ) γ dv (w) (z ) (3) where c γ = c γ (n) is a normalizing constant. Thus, differentiating under the integral sign, one obtains the following integral representation for fractional derivatives of functions in A γ(): D α f(z) = c γ f(w)g n++γ,α (z, w)( w 2 ) γ dv (w) (z ). (4) The following is a restricted version of Corollary 2.4 of []. The notation D means the unit disk. Lemma.. Given t > 0 and α real with t + α > 0, there is a function F holomorphic on D and continuous on D such that Γ(t + m)(m + ) α λ m F (λ) = m!γ(t) ( λ) t+α (λ D). m=0 If, in addition, t + α >, then F C ( D). Note that (2), as well as (4), remains true for any real α if the definitions of D α and G t,α are extended to α 0. Hence, using the above lemma, one can relate the size of the fractional derivative of a certain order to another as follows. Lemma.2. Let α > 0, β > 0, and γ >. If n + + γ + α β > 0, then we have D α D β f(w) ( w 2 ) γ f(z) c dv (w) z, w n++γ+α β for any f holomorphic on and for some constant c = c(n, α, β, γ). Proof. We may assume f A γ(). Then, by (4) and Lemma., we find D α f(z) c γ D β f(w) G n++γ,α β (z, w) ( w 2 ) γ dv (w) This proves the lemma. c γ F D β f(w) ( w 2 ) γ dv (w). z, w n++γ+α β For γ > and t real, define ( w 2 ) γ J t,γ (z) = dv (w) (z ). z, w n++γ+t Note that J t,γ is an increasing function of z. A proof of the following lemma can be found in Proposition.4.0 of [R]. 4

5 Lemma.3. If t > 0, then ( z 2 ) t J t,γ t < 0, then J t,γ is bounded on. has a positive finite limit as z and if Now we characterize the loch space in terms of fractional derivatives. Theorem.4. Let f be a function holomorphic on. Then, for each 0 < p < and α > 0, the following quantities are equivalent: (a) f. (b) sup D α f(z) ( z 2 ) α. z ( /p (c) sup D α f(w) p ( w 2 ) pα Jϕ z (w) dv (w)). z Proof of (a) (b): Let us use a temporary notation f α for the quantity in (b). Then, by (), it suffices to show f α f. Pick γ > 0. y Lemmas.2 and.3, we have D α D f(w) ( w 2 ) γ dv (w) f(z) c z, w n+γ+α ( w 2 ) γ c f dv (w) z, w n+γ+α c f ( z 2 ) α, so that f α c f for some c = c(n, α). To prove the reverse inequality, choose γ > α. Then, by Lemmas.2 and.3, we have D f(z) c c f α D α f(w) ( w 2 ) γ dv (w) z, w n+2+γ α c f α z 2, ( w 2 ) γ α dv (w) z, w n+2+γ α and thus we have f c f α for some c = c(n, α). Proof of (b) (c): Fix 0 < r <. Then, by subharmonicity, we have D α f(z) p r 2n (D α f) ϕ z (w) p dv (w). r The change of variables, w ϕ z (w), yields D α f(z) p r 2n D α f(w) p Jϕ z (w) dv (w). ϕ z (r) 5

6 Since ϕ z is an involution, we have for w ϕ z (r) r 2 ϕ z (w) 2 = ( z 2 )( w 2 ) z, w 2 2 w 2 z 2. Thus we have D α f(z) p c ( z 2 ) pα D α f(w) p ( w 2 ) pα Jϕ z (w) dv (w) for some c = c(r, p, α). In other words, we have ( f α c sup z The other direction of the inequalities is clear. D α f(w) p ( w 2 ) pα Jϕ z (w) dv (w)) /p. It is not hard to see that the equivalences of the above theorem carry over to the little loch space. Theorem.5. Let f be a function holomorphic on. Then, for each 0 < p < and α > 0, the following conditions are equivalent: (a) f 0 (). (b) D α f(z) ( z 2 ) α 0 as z. (c) D α f(w) p ( w 2 ) pα Jϕ z (w) dv (w) 0 as z. For ζ, and δ > 0, let Ω δ (ζ) = {z : z, ζ < δ}. A positive measure µ is called a ergman-carleson measure if If µ satisfies the additional condition sup µ(ω δ (ζ)) = O(δ n+ ). ζ sup µ(ω δ (ζ)) = o(δ n+ ) (δ 0), ζ then µ is called a vanishing ergman-carleson measure. It is known that µ is a ergman- Carleson measure if and only if Jϕ z (w) dµ(w) <. sup z Similarly, µ is a vanishing ergman-carleson measure if and only if Jϕ z (w) dµ(w) 0 as z. See [Z2] for this and several other characterizations of (vanishing) ergman-carleson measures. Hence the following is a consequence of Theorems.4 and.5. Corollary.6. Let f be a function holomorphic on. Then, for each 0 < p < and α > 0, the following hold. (a) f () if and only if D α f(z) p ( z 2 ) pα dv (z) is a ergman-carleson measure. (b) f 0 () if and only if D α f(z) p ( z 2 ) pα dv (z) is a vanishing ergman- Carleson measure. 6

7 2. Growth Rate y Theorem.4 the growth rate of the fractional derivative of order α of loch functions is governed by ( z ) α. Considering the optimality of this growth rate, one may ask whether there is a loch function f such that D α f(z) ( z ) α (z ) (5) for a given α > 0. The answer is simply no and, in fact, no holomorphic function can satisfy (5). The proof is easy. Suppose (5) holds. Then, since D α f has no zero on, one can write D α f = e g for some holomorphic function g on. Thus, α log( z ) Re g(z). Integrating this over spheres, one obtains log Re g(0) (0 < r < ) ( r) α which is impossible. Nevertheless, the growth rate is optimal in the following sense: Theorem 2.. There exists a positive integer M = M(n) with the following property: for each α > 0, there exist loch functions f i ( i M) such that M D α f i (z) i= ( z ) α (z ). efore proceeding to the proof, a bit of preliminary material is required. As in [Al], define d(ζ, ξ) = ( ζ, ξ 2 ) /2 (ζ, ξ ). Then d satisfies the triangle inequality. Let us write E δ (ζ) for the d-ball with radius δ and center at ζ : E δ (ζ) = {ξ : d(ζ, ξ) < δ}. One easily computes σ(e δ (ζ)) = δ 2n 2 (0 δ ) (6) where σ is the normalized surface area measure on. We shall say that a subset Γ of is d-separated by δ > 0, if d-balls with radius δ and centers at points of Γ are pairwise disjoint. We begin with a couple of lemmas. To see the proof of the following lemma, refer to [U]. Lemma 2.2. For each a > 0, there exists a positive integer M = M n (a) with the following property: if δ > 0, and if Γ is d-separated by aδ, then Γ can be decomposed into Γ = M j= Γ j in such a way that each Γ j is d-separated by δ. (In fact any M (2/a+) 2n 2 will suffice here.) 7

8 Lemma 2.3. Suppose that Γ is d-separated by δ and let k be a positive integer. If P (z) = ζ Γ z, ζ k (z ), then P (z) + (m + 2) 2n 2 e m2 δ 2 k/2. m= Proof. It suffices to show that the lemma is true on. m = 0,, 2,..., define Fix η and, for each y the separation assumption on Γ we have H m (η) = {ζ Γ : mδ d(η, ζ) < (m + )δ}. σ(e δ ) (the cardinality of H m (η) ) σ(e (m+2)δ ). So this and (6) show the cardinality of H m (η) is not more than (m+2) 2n 2. Let ζ H m (η). Then d(η, ζ) mδ, so η, ζ 2 m 2 δ 2 e m2 δ 2. Thus η, ζ k e m2 δ 2 k/2. If ζ H 0 (η), then d(η, ζ) < δ and thus η = ζ by the separation assumption. So H 0 (η) contains at most one point. Consequently, we have P (η) + Therefore the proof is complete. (m + 2) 2n 2 e m2 δ 2 k/2. m= We say that a holomorphic function f on is a lacunary series if it has a homogeneous expansion of the form f(z) = P k (z) k= with inf k (m k+ /m k ) > where m k denotes the degree of P k. Proposition 2.4([T]). Let P k be a sequence of holomorphic homogeneous polynomials uniformly bounded on and let f(z) = P k (z) k= (z ). If f is a lacunary series, then f is a loch function. We now turn to the proof of Theorem 2.. 8

9 Proof of Theorem 2.. We will prove the theorem by constructing loch functions satisfying the given property only near the boundary (then, by adding a suitable constant, one obtains the given property on all of ). For a small positive number A < such that (m + 2) 2n 2 e m2 /2A 2 3 3, (7) m= let M = M n (A) be a positive integer provided by Lemma 2.2 with A/2 in place of a. Let p be a sufficiently large positive integer so that and p α + For each positive integer j M, take δ j,0 so small that and inductively choose δ j,ν such that ( ) p p 3, (8) p α < e p, (9) p α e p p α 3 5. (0) 22α+ A 2 p j δ 2 j,0 = () p M δ 2 j,ν = δ 2 j,ν (ν =, 2, ). (2) From () and (2), we obtain A 2 p νm+j δ 2 j,ν =. (3) For each fixed j and ν, let Γ j,ν be a maximal subset of subject to the condition that Γ j,ν is d-separated by Aδ j,ν /2. Then, by Lemma 2.2, we can write Γ j,ν = M Γ j,νm+l (4) l= in such a way that each Γ j,νm+l is d-separated by δ j,ν. For each i, j =, 2,, M, and ν 0, put P i,νm+j (z) = ζ Γ j,νm+τ i (j) z, ζ p νm+j where τ i is the i th iterate of the permutation τ on {, 2,, M} defined by { j + if j < M τ(j) = if j = M. 9

10 Then, since p νm+j δ 2 j,k = /A2 by (3), we see from Lemma 2.3 and (7) that for all i, j and ν. Define P i,νm+j (z) 2 (z ) g i,j (z) = P i,νm+j (z) ν=0 (z ). eing a lacunary series with uniformly bounded homogeneous terms, each g i,j is a loch function by Proposition 2.4. We will show that for every ν 0, j M and z such that there exists an index i such that p νm+j z, (5) pνm+j+/2 D α g i,j (z) c ( z ) α for some constant c = c(α). So, fix ν, j and z for which (5) holds. Let z = z η where η. Then, since d-balls with radius Aδ j,ν and centers at points of Γ j,ν cover by maximality, there is some ξ Γ j,ν such that η E Aδj,ν (ξ). Note that ξ Γ j,νm+l for some l M by (4) and therefore ξ Γ j,νm+τ i (j) for some i M. We now estimate D α g i,j (z). It follows from Lemma 2.3 and (7) that D α g i,j (z) ( p νm+j + ) α Pi,νM+j (z) ( p km+j + ) α Pi,kM+j (z) k ν p α(νm+j) Pi,νM+j (η) νm+j z p νm+j 2 (p k + ) α 2 k=νm+j+ = I II III, say. (p k + ) α z pk k=0 y (8) and (5), we obtain z pνm+j ( ) p νm+j p νm+j 3 0

11 and therefore I 3 pα(νm+j) ( η, ξ pνm+j ζ Γ j,νm+τ i (j) ζ ξ η, ζ pνm+j ). Recall that d(η, ξ) < Aδ j,ν < δ j,ν. Thus, the separation property shows d(η, ζ) δ j,ν for any ζ Γ j,νm+τ i (j), ζ ξ. Thus, the proof of Lemma 3.3 shows that the summation in the parenthesis of the above is less than /3 2 by (7) and (3). Also, the inequality d(η, ξ) < Aδ j,ν, together with (3), yields η, ξ pνm+j η, ξ 2pνM+j Accordingly, we have an estimate for I: For II, we have an estimate: νm+j II 2 k= I pα(νm+j). νm+j (p k + ) α 2 α+ ( ) p νm+j p νm+j k= 3. p αk α+ pα(νm+j) 2 p α. Note that and hence z pνm+j+ z e p pνm+j+/2 νm j /2 e p. Hence we have an estimate for III: III 2 α+ 2 α+ k=νm+j+ k=νm+j+ 2 α+ p α(νm+j) p αk z pk p αk( z pνm+j+ ) k νm j k= ( p α e p ) k = 2 α+ p α(νm+j) p α e p p α, where the last equality comes from (9). Since z /p νm+j+/2, or equivalently, p νm+j+/2 /( z ) by (5), combining the estimates for I, II and III, we now have D α g i,j (z) pα(νm+j) 3 4 pα(νm+j+/2) 3 4 p α/2 3 4 p α/2 ( z ) α

12 by (0). In summary, we have M i= j= M D α g i,j (z) 3 4 p α/2 ( z ) α, for all z such that p k z p k /2 (k =, 2, ). Next, pick a sequence of positive integers q k such that 0 q k p k+/2 < and, for each j M, a sequence of positive numbers ɛ j,ν such that A 2 q νm+j ɛ 2 j,ν =. Choose a sequence of subsets Γ j,k of with the same property as before: for each nonnegative integer ν, the set M l= Γ j,νm+l is a maximal subset which is d-separated by Aɛ j,ν /2 and each Γ j,νm+l is d-separated by ɛ j,ν. For each i, j =, 2,, M and ν 0, put Q i,νm+j (z) = z, ζ q νm+j ζ Γ j,νm+τ i (j) and define h i,j (z) = Q i,νm+j (z). ν=0 Then each h i,j is a lacunary series because q νm+j /q (ν )M+j p M /2 > and homogeneous polynomials Q i,νm+j are uniformly bounded by 2 as before. Thus each h i,j is a loch function by Proposition 2.4 and an easy modification of the previous argument yields M i= j= M D α h i,j (z) 3 5 p α/2 ( z ) α for all z such that p k /2 z p k (k =, 2, ). Consequently, we finally have M M i= j= ( ) D α g i,j (z) + D α h i,j (z) c ( z ) α, for all z sufficiently close to the boundary and for some constant c = c(α). Therefore the proof is complete. Since Rf = k kf k for a function f holomorphic on with homogeneous expansion f = k f k, a slight modification of the proof of Theorem 2. yields the following: Corollary 2.5. There exist finitely many loch functions f i such that Rf i (z) i z z (z ). In case of the unit disk, Corollary 2.5 (with two loch functions) is proved by Ramey and Ullrich [RU]. Our proof of Theorem 2. is modeled on their proof. 2

13 3. Interpolation Recall that ϕ z denotes the standard automorphism of such that ϕ z (0) = z and ϕ z ϕ z = identity. The pseudo-hyperbolic distance between two points z and w in is defined by ρ(z, w) = ϕ w (z). As is well-known, the pseudo-hyperbolic distance is automorphism-invariant: ρ(z, w) = ρ ( ϕ a (z), ϕ a (w) ) for all a. Let Q δ (w) denote the pseudo-hyperbolic ball with center at w and radius δ > 0. We say that a sequence z m in is ρ-separated by δ if pseudo-hyperbolic balls Q δ (z m ) are pairwise disjoint. In this section we prove Theorem C. We first prove the necessity. The hard part of the proof is to show the following inequality of Schwarz-Pick type. Proposition 3.. Given α > 0, there exists a constant c = c(n, α) such that D α f(z)( z 2 ) α D α f(w) ( w 2 ) α c f ρ(z, w) (6) for all z, w and for all f (). Proof. Since the pseudo-hyperbolic distance is automorphism-invariant, (6) is equivalent to ( D α f ) (ϕ w (z))( ϕ w (z) 2 ) α D α f(w) ( w 2 ) α c f z (7) We now show (7). Note that we need prove (7) only for z staying away from boundary by Theorem.4. So assume z /2. Fix a loch function f. Then f is a ergman integral of some bounded orel function ψ on (see, for example, [C] and references therein): f(z) = Differentiating under the integral sign (see (2)),we have D α f(z) = ψ(λ)g n+,α (z, λ) dv (λ). ψ(λ) dv (λ). (8) ( z, λ ) n+ Let F be the function provided by Lemma. with t = n +. Note that F is continuously differentiable up to the boundary. Since ψ in (8) can be chosen so that ψ is comparable with the loch norm of f, it suffices to show that F (ϕ w (z), λ)( ϕ w (z) 2 ) α ( ϕ w (z), λ ) n++α F (w, λ)( w 2 ) α ( w, λ ) n++α dv (λ) c z (9) where we abuse the notation F (z, w) = F ( z, w ) for simplicity. The change of variables, λ ϕ w (λ), turns the left hand side of (9) into F (ϕw (z), ϕ w (λ))h(z, w, λ) F (w, ϕ w (λ))h(0, w, λ) Jϕw (λ) dv (λ) (20) where H(z, w, λ) = ( ϕ w (z) 2 ) α ( ϕ w (z), ϕ w (λ) ) n++α 3

14 Since (see, for example, [R]) and a straightforward calculation yields ( w 2 Jϕ w (λ) = w, λ 2 ) n+ ϕ w (z), ϕ w (λ) = ( w 2 )( z, λ ) ( z, w )( w, λ ), H(z, w, λ)jϕ w (λ) = ( w, λ )n++α w, λ 2n+2 ( z 2 z, w 2 ) α ( ) n++α z, w. z, λ Thus, the integral in (20) is bounded by where Since h(z, w, λ) h(0, w, λ) w, λ n+ α dv (λ) ( z 2 h(z, w, λ) = F (ϕ w (z), ϕ w (λ)) z, w 2 sup w by Lemma.3, it remains to prove that dv (λ) w, λ n+ α < h(z, w, λ) h(0, w, λ) c z ) α ( ) n++α z, w. z, λ for all z /2 and for all λ, w. Since z /2 and F is continuously differentiable up to the boundary, we can see that the left hand side of the above is bounded by c F ( z, w ) n++α ( z, λ ) n++α + c F ( z 2 ) α z, w 2α + F w ϕw (z). Now, each term of the above is easily seen to be bounded by c z for some constant c = c(n, α), as desired. This completes the proof. As a corollary of Proposition 3., we derive a necessary condition for a sequence to be an α-interpolating sequence. Theorem 3.2. Every α-interpolating sequence is ρ-separated. Proof. Suppose z m is an α-interpolating sequence. Define a linear operator T : () l by T f = ( D α f(z m )( z m ) α). 4

15 y Theorem.4, the operator T is bounded. Moreover, since z m is an α-interpolating sequence, T is onto. Thus, the open mapping theorem gives a constant c > 0 with the following property: given a sequence (a m ) l, there exists a loch function f such that T f = (a m ) and f c (a m ). Now, a consideration of sequences in l whose components are all 0 except for just one component shows that the sequence z m must be ρ-separated. For the proof of the sufficiency in Theorem C, we need a duality lemma. It is well-known that the dual of the ergman space A () = A 0() can be identified with the loch space. We reprove this by using a pairing which is useful for our purposes. Lemma 3.3. For each α > 0, the dual of A () can be identified with the loch space under the following integral pairing: ( ) f, g = f(z)d α g(z)( z 2 ) α dv (z) for g () and holomorphic polynomial f. Proof. Since holomorphic polynomials are dense in A (), it is clear from Theorem.4 that the given pairing defines a bounded linear functional on A () for each fixed loch function g. Now let Λ be a bounded linear functional on A (). Then, by the Hahn- anach theorem, there exists a bounded orel function ψ on such that ψ = Λ and Λf = f(z)ψ(z) dv (z) for all holomorphic polynomials f. Insert the integral representation (3) (with γ = α) into the above and then interchange the order of integration. The result is Λf = f(z) D α g(z)( z 2 ) α dv (z) where g(z) = c α ψ(w)g n++α, α (z, w) dv (w). This, together with Lemma.3, shows that D α g(z)( z 2 ) α is bounded on and thus g is a loch function by Theorem.4. Therefore Λ is induced by g (). The uniqueness of g follows from the fact that if ( f, g ) = 0 for all holomorphic homogeneous polynomials f, then Taylor coefficients of g are all 0 and thus g = 0. The proof is complete. Remark. y a similar proof to that of Lemma 3.3 one can verify that the predual of A () can be identified with the little loch space under the same pairing. We now turn to the sufficiency in Theorem C. Theorem 3.4. Given α > 0, there exists a positive number δ 0 < such that every sequence ρ-separated by δ > δ 0 is an α-interpolating sequence. Since Lemma 3.3 is known to hold, the proof is now an easy modification of the argument in [At]. The proof below is included for the sake of completeness. 5

16 Proof. Let z m be a sequence which is ρ-separated by δ. Define a linear operator T : A () l by T f = ( f(z m )( z m 2 ) n+). Since f(w) dv (w) = Q δ (z) w <δ f ϕ z (w) Jϕ z (w) dv (w) 4 (n+) ( z 2 ) n+ w <δ f ϕ z (w) dv (w) 4 (n+) δ 2n ( z 2 ) n+ f(z) (2) for functions f holomorphic on, we have T f l 4 n+ δ 2n f(z) dv (z) m Q δ (z m ) 4 n+ δ 2n f(z) dv (z) and thus T is bounded. Let T : l () denote the adjoint of T induced by the pairing in Lemma 3.3. Then, by the integral representation (3), one has D α T (a m )(z) = m a m ( z m 2 ) n+ ( z, z m ) n++α. Let I be the identity operator on l and define an operator S on l by Then the k th term of (S I)(a m ) is S(a m ) = ( D α T (a m )(z k )( z k 2 ) α). m k a m ( z m 2 ) n+ ( z k 2 ) α ( z k, z m ) n++α, and thus, applying (2) to the function ( w, z k ) n α, one obtains S I 4 n+ δ 2n sup( z k 2 ) α k \Q δ (z k ) = 4 n+ δ 2n sup k = 4 n+ δ 2n w >δ w >δ dv (w) w, z k n+ α dv (w) w, ζ n+ α 6 dv (w) w, z k ) n++α

17 where ζ is an arbitrary point of by Lemma.3. Thus, for δ sufficiently close to, the operator S is invertible on l. Now, given (b m ) l, choose (a m ) l such that S(a m ) = (b m ) and put f = T (a m ). Then we have f () and for all m. This completes the proof. D α f(z m )( z m 2 ) α = b m We now close the paper with a few remarks. Remarks.. The proof of the above theorem actually provides some more informations on the function f () selected as an interpolating function. We have f = T S (b m ) T S (b m ) T S I (b m) so that f c(δ) ( c(δ)c α (δ) ) (bm ) where c(δ) = 4 n+ δ 2n and c α (δ) 0 as δ. 2. y a result of Amar [Am], a ρ-separated sequence can be decomposed into finitely many sequences which are sufficiently ρ-separated. Thus, we have Given α > 0, every ρ-separated sequence is a finite union of α-interpolating sequences. 3. Let c 0 denote the space of all sequences which converge to 0. Let us say that a sequence z m in is an α-interpolating sequence for 0 () if, given (b m ) c 0, there exists a function f 0 () such that D α f(z m )( z m 2 ) α = b m for all m. Having seen Theorem 3.4, one can easily go a little bit further: Given α > 0, every sufficiently ρ-separated sequence is an α-interpolating sequence for 0 (). 4. Since sufficiently ρ-separated sequences are interpolating ones, a sequence close enough to a given sequence, which is sufficiently ρ-separated, is also an interpolating sequence. More is true: Given an α-interpolating sequence z m, there exists a number γ > 0 such that any sequence w m with ρ(z m, w m ) < γ for all m is also an α-interpolating sequence. The last two remarks of the above are consequences of some well-known results of functional analysis. See [At] for details. 7

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