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Carleson Measures and Interpolating Sequences for Besov Spaces on Complex Balls

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Carleson Measures and Interpolating Sequences for Besov Spaces on Complex Balls Carleson Measures and Interpolating Sequences for Besov Spaces on Complex Balls Nicola Arcozzi Dipartimento do Matematica Universita di Bologna 40127 Bologna, ITALY Richard Rochberg Department of Mathematics Washington University St. Louis, MO 63130, U.S.A. Eric Sawyer Department of Mathematics & Statistics McMaster University Hamilton, Ontairo, L8S 4K1, CANADA Contents CARLESON MEASURES AND INTERPOLATING SEQUENCES 1 1. Introduction 1 n 2. A tree structure for the unit ball Bn in C 11 3. Carleson measures 21 4. Pointwise multipliers 34 5. Interpolating sequences 36 6. An almost invariant holomorphic derivative 80 7. Besov spaces on trees 90 8. Holomorphic Besov spaces on Bergman trees 108 9. Completing the multiplier interpolation loop 155 10. Appendix 160 Bibliography 163 v Abstract We characterize Carleson measures for the analytic Besov spaces Bp on the n unit ball Bn in C in terms of a discrete tree condition on the associated Bergman tree Tn. We also characterize the pointwise multipliers on Bp in terms of Carleson measures. We then apply these results to characterize the interpolating sequences in Bn for Bp and their multiplier spaces MBp , generalizing a theorem of B¨oein one dimension. The interpolating sequences for Bp and for MBp are precisely those sequences satisfying a separation condition and a Carleson embedding condition. 1 These results hold for 1 < p < ∞ with the exceptions that for 2 + n−1 ≤ p < ∞, the necessity of the tree condition for the Carleson embedding is left open, 1 and for 2 + n−1 ≤ p ≤ 2n, the sufficiency of the separation condition and the Carleson embedding for multiplier interpolation is left open; the separation and tree conditions are however sufficient for multiplier interpolation. Novel features of our proof of the interpolation theorem for MBp include the crucial use of the discrete tree condition for sufficiency, and a new notion of holomorphic Besov space on a Bergman tree, one suited to modeling spaces of holomorphic functions defined by the size of higher order derivatives, for necessity. Received by the editor 22 October 2004. 2000 Mathematics Subject Classification. Primary 46E15; Secondary 30H05, 32A25, 32A37, 46E22, 46E35. Key words and phrases. Carleson measure, interpolating sequence, Besov space, multiplier algebra, complex ball, tree model. (Arcozzi) Partially supported by the COFIN project Analisi Armonica, funded by the Italian Minister for Research. (Rochberg) This material is based upon work supported by the National Science Foundation under Grant No. 0400962. (Sawyer) This material based upon work supported by the National Science and Engineering Council of Canada. vi CARLESON MEASURES AND INTERPOLATING SEQUENCES 1. Introduction In this paper we consider the analytic Besov spaces Bp (Bn) on the unit ball n Bn in C , consisting of those holomorphic functions f on the ball such that Z m p 2 (m) 1 − |z| f (z) dλn (z) < ∞, Bn −n−1 n 2 where m > p , dλn (z) = 1 − |z| dz is invariant measure on the ball with dz Lebesgue measure on Cn, and f (m) is the mth order complex derivative of f. We characterize their Carleson measures (except for p in an exceptional range h 1 2 + n−1 , ∞ ), pointwise multipliers and interpolating sequences. We also char- acterize interpolating sequences for the corresponding pointwise multiplier spaces h i M (except for p in the smaller exceptional range 2 + 1 , 2n ). Finally, in Bp(Bn) n−1 order to obtain the characterization of interpolating sequences for M in the Bp(Bn) 1 difficult range 1+ n−1 ≤ p < 2, we introduce “holomorphic” Besov spaces HBp (Tn) on the Bergman trees Tn, and develop the necessary part of the analogous theory of Carleson measures, pointwise multipliers and interpolating sequences. The main feature of these holomorphic Besov spaces on Bergman trees is that they provide a martingale-like analogue of the analytic Besov space on the ball. They also en- joy properties not found in the actual Besov spaces, such as reproducing kernels with a positivity property for all p in the range 1 < p < ∞ (Lemma 8.11 below). Various solutions to the problems mentioned here in one dimension can be found in [Car], [MaSu], [Wu], [Boe] and our earlier paper [ArRoSa]. We remark that the one-dimensional methods for characterizing multiplier interpolation generalize 1 for n > 1 to prove necessity at most in the ranges 1 < p < 1 + n−1 and p > 2, and sufficiency at most in the range p > 2n, resulting in a common range of only p > 2n for all n > 1. 1.1. History. We begin with an informal discussion of the context in which our results can be viewed. For more details on this background (at least the part having to do with Hilbert spaces) we refer to the beautiful recent monograph of K. Seip [Sei]. The theory of Carleson measures and interpolating sequences has its roots in Lennart Carleson’s 1958 paper [Car], the first of his papers motivated by the corona problem for the Banach algebra H∞ (D) of bounded holomorphic functions in the 1 2 N. ARCOZZI, R. ROCHBERG, AND E. SAWYER J ∞ unit disk D: if {fj}j=1 is a finite set of functions in H (D) satisfying J X |fj (z)| ≥ c > 0, z ∈ D, j=1 J ∞ are there are functions {gj}j=1 in H (D) with J X fj (z) gj (z) = 1, z ∈ D, j=1 i.e., is every multiplicative linear functional on H∞ (D) in the closure of the point evaluations, so that there is no “corona”? In [Car], Carleson observed the following connection between the corona problem and interpolating sequences. A Blaschke product B0 has the “baby corona” property, ∞ For all f1 ∈ H (D) satisfying inf {|B0 (z)| + |f1 (z)|} > 0,(1.1) z∈D ∞ there are g0, g1 ∈ H (D) with B0g0 + f1g1 ≡ 1, if the zero set ∞ Z0 = {z ∈ D : B0 (z) = 0} = {zj}j=1 ∞ of B0 is an interpolating sequence for H (D): (1.2) ∞ ∞ ∞ The map f → {f (zj)}j=1 takes H (D) boundedly into and onto ` (Z0). Carleson solved this latter problem completely by showing that a sequence Z = ∞ ∞ {zj}j=1 is an interpolating sequence for H (D) if and only if Y zj − zk (1.3) ≥ c > 0, k = 1, 2, 3, ... 1 − zkzj j:j6=k The proof made crucial use of Blaschke products and duality. In the same paper he showed implicitly that the characterizing condition (1.3) can be rephrased in modern language as zj − zk ≥ c > 0 for j 6= k, and 1 − zkzj ∞ X 2 p 1 − |zj| δzj is a Carleson measure for H (D), j=1 where a positive Borel measure µ on the disk D is now said to be a Carleson measure for Hp (D) if the embedding Hp (D) ⊂ Lp (dµ) holds. Carleson later solved the corona problem affirmatively in [Car2] by demonstrating the absence of a corona in the maximal ideal space of H∞ (D). In 1961, H. Shapiro and A. Shields [ShSh] demonstrated that the interpolation property (1.2) is equivalent to weighted interpolation for Hardy spaces Hp (D); that is, the map 1 ∞ 2 p f → 1 − |zj| f (zj) j=1 1. INTRODUCTION 3 1 p p 2 p maps H (D) boundedly into and onto ` (Zf ) .The factor 1 − |zj| forces the ∞ map to be into ` (Zf ); the Carleson measure condition ensures that it is into p ` (Zf ). We now recast the case p = 2 of this result in a way that will emphasize the analogy with what comes later. The Hardy space H2 (D) is a Hilbert space with 2 reproducing kernel. This means that for each z ∈ D, there is kz ∈ H (D), the reproducing kernel for z, which is characterized by the fact that for any f ∈ H2 (D) ∞ we have f (z) = hf, kzi. A sequence Z = {zj}j=1 is an interpolating sequence for H2 (D) if one can freely assign the values of an H2 (D) function on Z, subject only to the natural size restriction. More precisely, Hilbert space basics ensure that if f ∈ H2 (D), then the function 1 f (zi) 2 2 zi → = 1 − |zi| f (zi) kkzi k is a bounded function on Z. The sequence Z is called an interpolating sequence for H2 (D) if all the functions on Z which are obtained in this way are in `2 (Z), and if furthermore, every function in `2 (Z) can be obtained in this way. For any z ∈ D, kz set kz = , and note that by the Cauchy-Schwarz inequality, e kkz k D E kez, kfw ≤ 1, z, w ∈ D. If Z is an interpolating sequence, then it must be possible, given any i and j, to find f ∈ H2 (D) so that 1 1 2 2 2 2 1 − |zi| f (zi) = 0, 1 − |zj| f (zj) = 1; and to do this with control on the size of kfk. This implies a weak separation con- dition on the points of Z which is necessary for Z to be an interpolating sequence: D E there is ε > 0 so that for all i 6= j, k , k ≤ 1 − ε. An equivalent geomet- fzi fzj ric statement is that there is a uniform lower bound on the hyperbolic distances β (zi, zj). The result of Shapiro and Shields can now be restated as saying that in- terpolating sequences for H2 (D) are characterized by the following two conditions: D E (1.4) There is ε > 0 so that k , k ≤ 1 − ε for all i 6= j, fzi fzj and ∞ X −2 2 (1.5) kzj δzj is a Carleson measure for H (D) . j=1 Interpolation problems, multiplier questions and Carleson measure characteri- zations were then studied by various authors in other classical function spaces on α the disk, including certain of the spaces Bp (D) normed by m−1 1 Z m+α p p X (k) 2 (m) f (0) + 1 − |z| f (z) dλ (z) , k=0 D −2 where dλ (z) = 1 − |z|2 dz is invariant measure on the disk, and for fixed α and p, the norms are equivalent for (m + α) p > 1.
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