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Abstract:
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In this thesis, subjects of consideration are the embeddings theorems of weighted
Bergman spaces in Lp-spaces, as well as embeddings theorems of harmonic mixed
norm spaces.
The first part of the thesis generalizes the theorems of embeddings Bergman spaces
into Lp(μ)-spaces, where μ is a Borel measure on a given domain. They have been
earlier studied on domains such as unit ball and upper half-space. Generalization
refers to bounded domains Ω ⊂ Rn with C1 boundary. This embedding will be
valid to any p > 0, whenever the measure of the spaces Lp satisfies the Carledon
condition. Reverse the direction will be valid only in case if p > 1 + α+2
n−2 .
The second part of the dissertation also generalizes the embeddings theorems of
mixed norm spaces of harmonic functions on a unit ball, where the generalization
is applied to the domain Ω ⊂ Rn with C1 boundary. However, in addition we
are obtaining another important result relating to the limitation of the maximum
operators in the mixed norm on the general domain for the class of QNS functions. |