Abstract
Geometric inequalities have been hot topics in analysis, geometry, PDE, probability and combinatorial theory.Among so many geometric inequalities, fractional integral inequalities exceptionally attract attention of analysts,which play an important role in Analysis.The reason is due to their diverse connections to questions regardingthe restriction of the Fourier transform, Radon transform and the $k$-plane transform.In this paper, we simply review a series of fractional integral inequalities,some geometric extremal inequalities and symmetric decreasing rearrangement inequalitieswhich are extremely useful analytic tools to deal with many classic extremal problems.Together with competing symmetries, we focus on presenting rearrangement methodto determine the optimisers of some fractional integral inequalities.We also introduce some properties of Lebesgue spaces with mixed normsand consider some fractional integral inequalities in mixed norm spaces as well.
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