Bicomplex Bergman and Bloch spaces
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Arabian Journal of Mathematics
L. F. Reséndis O. · L. M. Tovar S.
Bicomplex Bergman and Bloch spaces
Received: 4 September 2019 / Accepted: 3 June 2020 © The Author(s) 2020
Abstract In this article, we define the bicomplex weighted Bergman spaces on the bidisk and their associated weighted Bergman projections, where the respective Bergman kernels are determined. We study also the bicomplex Bergman projection onto the bicomplex Bloch space. Mathematics Subject Classification
30G30 · 30H20 · 30H30 · 30G35
1 Introduction The Bergman space is a classic topic in the Complex Analysis. The last years has received a strong impetus (see [2,5,6,11,12]); on the other hand, in recent years, the theory of bicomplex holomorphic functions has consolidated its development (see [1,3,4,9] and references herein). This theory shows that it is quite adequate to deal with some analogous of the classical holomorphic functions spaces on the unit complex disk, but now p defined in the bidisk. Precisely, in this article, we introduce the bicomplex weighted Bergman spaces BCAα 2 on the bidisk U ⊂ C . A previous work in this direction appears in [8] and [10]. We point out that a frequent tool in the theory of bicomplex holomorphic function is the so-called idempotent decomposition: Although it is ubiquitous in all this theory, it is a mistake to think—as we show in this paper—that everything in the theory can be reduced to the idempotent decomposition. In the preliminaries, we fix notations and some fundamental facts in bicomplex theory; also, we prove some results in the context of bicomplex numbers. In the third section, p we define the bicomplex weighted Bergman spaces BCAα , we prove the decomposition (see Theorem 3.1) BCAαp = Aαp e + Aαp e† , and we determine their respective Bergman kernels [see (3.5)]: K k,α = K α e + K α e . The bicomplex weighted Bergman projection Pk,α is factored with a slight modification (see Theorem 3.9). We will see that the bicomplex Bloch space defined in the bidisk U can be splited in two classical Bloch spaces on the unit disk. Therefore, in the fourth section, we study the bicomplex Bergman projection onto the bicomplex Bloch space, see Theorem 4.1. L. F. Reséndis O. (B) Universidad Autónoma Metropolitana, Unidad Azcapotzalco, C.B.I. Apartado Postal 16-306, CDMX. Area de Análisis Matemático y sus Aplicaciones, 02200 México CDMX, México E-mail: [email protected]
L. M. Tovar S. Escuela Superior de Física y Matemáticas del IPN, Edif. 9, Unidad ALM, Zacatenco del IPN, 07300 México CDMX, México E-mail: [email protected]
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Arab. J. Math.
2 Preliminaries We present several common facts about bicomplex numbers and bicomplex holomorphic functions. We will be free to use results and notations of [9]. The set of bicomplex numbers BC is defined as BC := { z 1 + jz 2 : z 1 , z 2 ∈ C(i), j2 = −1}. Sum and product of bicomplex numbers are presented in the expected way. We write all the bicomplex numbers as Z = z 1 + jz 2 , with zl = xl + iyl ∈ C(i), in theirs C(i)-idempotent form, that is: Z = β1 e + β 2 e † ,
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