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The small ball inequality with restricted coefficients Karslidis, Dimitrios


The main focus of this document is the small ball inequality. The small ball inequality is a functional inequality concerning the lower bound of the supremum norm of a linear combination of Haar functions supported on dyadic rectangles of a fixed volume. The sharp lower bound in this inequality, as yet unproven, is of considerable interest due to the inequality's numerous applications. We prove the optimal lower bound in this inequality under mild assumptions on the coefficients of a linear combination of Haar functions, and further investigate the lower bounds under more general assumptions on the coefficients. We also obtain lower bounds of such linear combinations of Haar functions in alternative function spaces such as exponential Orlicz spaces.

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