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D E M O N S T R A T E MATHEMATICAVol. XLINo 12008Albert I. PetrosyanON WEIGHTED HARMONIC BERGMAN SPACESAbstract. This paper is devoted to the investigation of the weighted Bergman harmonic spaces b^(B) in the unit ball in R n . The reproducing kernel Ra for the ball isconstructed and the integral representation for functions inby means of this kernel is obtained. Besides an linear mapping between the b2a(B) spaces and the ordinaryL 2 -space on the unit sphere, which has an explicit form of integral operator along withits inversion, is established.IntroductionThis paper is devoted to the investigation of the weighted Bergman harmonic spaces tidc(B) in the unit ball in R n . In Section 1 we introduce thespaces t%t(B) and prove some preliminary statements. Section 2 is devotedto the construction of reproducing kernel Ra, to the integral representation of t%,(B) by means of Ra (Theorems 1 and 2) and to the orthogonalprojection from ^(B.dVa) to b^(B) (Theorem 3). Section 3 gives an integral representation of the considered spaces b^(B) over the unit sphere.This leads to an linear mapping between the b\{B) spaces and the ordinaryL 2 -space on the unit sphere, which has an explicit form of
Demonstratio Mathematica – de Gruyter
Published: Jan 1, 2008
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