# Interpolating sequences on analytic Besov type spaces

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INTERPOLATING SEQUENCES ON ANALYTIC BESOV TYPE SPACES NICOLA ARCOZZI, DANIEL BLASI, AND JORDI PAU Abstract. We characterize the interpolating sequences for the weighted an- alytic Besov spaces Bp(s), defined by the norm ‖f‖p Bp(s) = |f(0)|p + ∫ D |(1− |z|2)f ′(z)|p(1− |z|2)s dA(z) (1− |z|2)2 , 1 < p < ∞ and 0 < s < 1, and for the corresponding multiplier spaces M(Bp(s)). Contents 1. Introduction and main results 1 2. Background and preliminaries 4 3. Carleson measures and multipliers for Bp(s) 6 4. Boundary values 9 5. Necessity of condition (CS) 15 6. A ∂¯-problem with estimates and the corona theorem 19 7. Sufficiency of condition (CS) 23 8. Concluding remarks 27 References 27 1. Introduction and main results In this paper we characterize the interpolating sequences for the multiplier spaces of the family of weighted Besov spaces Bp(s), 1 < p <∞, 0 < s < 1, defined below. Throughout the paper, we denote byM(X) the multiplier space of the Banach space X. The problem of finding the interpolating sequences for spaces of holomorphic functions (and their multiplier spaces) is an old one. The prototype of all such results is Carleson’s celebrated Interpolation Theorem [8] for H∞ = M(Hp), later extended to the spaces Hp themselves, 0 < p <∞ in [24],[17]. All known proofs of Carleson’s Theorem in the Hardy class make an essential use of Blaschke products, and this has been so far the main obstruction to an extension of the theorem to 2000 Mathematics Subject Classification. 30H05, 31C25, 46J15. Key words and phrases. Besov spaces, interpolating sequences, Carleson measures, corona problems. The first author is partially supported by the COFIN project Analisi Armonica, funded by the Italian Minister for Research. The second and third authors are supported by the grant 2005SGR00774. Also, the second author is partially supported by the DGICYT grant MTM2005- 00544, while the third author is partially supported by the DGICYT grant MTM2005-08984-C02- 02. 1 2 NICOLA ARCOZZI, DANIE

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