In the first part of this thesis, we construct a function that lies in (L^p(mathbb{R}^d)) for every (p in (1,infty]) and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We apply Kovač’s maximal restriction principle to show that the same full-dimensional set is avoided by any Borel measure satisfying a nontrivial Fourier restriction theorem. As a consequence of a near-optimal fractal restriction theorem of Łaba and Wang, we hence prove that no previously unknown relations hold between the Hausdorff dimension of a set and the range of valid Fourier restriction exponents for measures supported in the set.
In the second part, we prove sharp local and global variation bounds for the centred Hardy–Littlewood maximal functions of indicator functions in one dimension, establishing that they are variation diminishing. We characterise maximisers, treat both the continuous and discrete settings and extend our results to a larger class of functions. This is partial progress towards proving a conjecture of Kurka and Bober, Carneiro, Hughes and Pierce.