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Integral operators on function spaces
Peša, Dalimil ; Pick, Luboš (advisor) ; Nekvinda, Aleš (referee)
In this thesis, we consider Lorentz-Karamata spaces with slowly varying fuc- tions and study their properties. We first provide simpler definition of slowly varying functions and derive some of their properties. We then consider Lorentz-Karamata functionals over an arbi- trary sigma-finite measure space equipped with a non-atomic measure and corre- sponding Lorentz-Karamata spaces. We characterise non-triviality of said spaces, then study when they are equivalent to a Banach function space and obtain mul- titude of conditions, either sufficient or necessary. We further study embeddings between Lorentz-Karamata spaces and provide a partial characterisation. At last, we try to describe the associate spaces of Lorentz-Karamata spaces and succeed even in some of the limiting cases. Our contribution is mainly the characterisation of non-triviality, the partial characterisation of embeddings, the description of associate spaces in some lim- iting cases and all the results concerning Lorentz-Karamata spaces with the sec- ondary parameter q smaller than one. Those results are, as far as we are aware, new. 1
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