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Abstract:
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This thesis has been written under the supervision of my mentor, Prof. dr. Milo s
Arsenovi c at the University of Belgrade academic, and my co-mentor dr. Vladimir
Bo zin in year 2013. The thesis consists of three chapters. In the rst chapter we start
from de nition of harmonic functions (by mean value property) and give some of their
properties. This leads to a brief discussion of homogeneous harmonic polynomials, and
we also introduce subharmonic functions and subharmonic behaviour, which we need
later. In the second chapter we present a simple derivation of the explicit formula for the
harmonic Bergman reproducing kernel on the ball in euclidean space and give a proof that
the harmonic Bergman projection is Lp bounded, for 1 < p < 1, we furthermore discuss
duality results. We then extend some of our previous discussion to the weighted Bergman
spaces. In the last chapter, we investigate the Bergman space for harmonic functions bp,
0 < p < 1 on RnnZn. In the planar case we prove that bp 6= f0g for all 0 < p < 1.
Finally we prove the main result of this thesis bq bp for n=(k + 1) q < p < n=k,
(k = 1; 2; :::). This chapter is based mainly on the published paper [44]. M. Arsenovi c,
D. Ke cki c,[5] gave analogous results for analytic functions in the planar case. In the
plane the logarithmic function log jxj, plays a central role because it makes a di erence
between analytic and harmonic case, but in the space the function jxj2n; n > 2 hints at
the contrast between harmonic function in the plane and in higher dimensions. |