Harmonik Bloch spaces

We study the properties of one-parameter family of harmonic Bloch (ba) and harmoııic little Bloch (ba0) spaces on the unit ball of Rn in a dctailcd and sys- tematic way. Firstly the theorems which enable to define harmonic Bloch and harmonic little Bloch spaces for the whole range a E M are proved. Thus it is shown that sufhciently high-order derivatives give the same space. The equiva-lence of defining these spaces using either partial derivatives, radial derivatives or radial differential operators Dls is also shown. The fundamental properties (completeness, seperability, ete.) of the spaces ba and baG are obtained. Non- trivial and non-polynomial harmonic funetion examples are obtained and as a consequence it is shown that ali ba and bao are distinet. The projeetion operators from L™ onto ba are defined by using reproducing kerııels of Bergman-Bcsov spaces. The projeetions provide iııtegral representa- tions for the funetions in these spaces. Similarly, the projeetion operators from Ca and Cao onto bao are defined. Using iııtegral representations it is shown that the dual of Bergman-Besov space b)} (for every q E R) is ba and its pre-dual is bao under suitable pairings. Fiııally, for ali a E R, the Gleasoıı problems in ba and 6a0 are solved and atomic decompositions of the funetions in these spaces are obtained. Oscillatory clıaracterizations of ba and ba0 for a > —1 are also given. Anahtar Sözcükler: Harmonic Bloch and little Bloch spaces, Bergmaıı pro¬jeetions, Atomic decomposition, Gleason problem, Oscillatory characterization