New perspectives in harmonic analysis and fractal geometry

This Thesis aims to develop a deeper connection between Harmonic Analysis and Fractal geometry. The work in this Thesis is based on six research papers I wrote during my time as a PhD student. The first part of this Thesis considers the study of orthogonal projections. Whenever a set supports a measure with maximal polynomial Fourier decay pointwise, all directions satisfy the conclusion of Marstrand's theorem. For this reason, we study how the average Fourier decay of measures supported on fractal sets affects the Hausdorff dimension of their exceptional set for orthogonal projections. We obtain a new exceptional set estimate for fractal sets depending on the average Fourier decay of measures they support, which enables us to understand for which fractal sets the best-known bounds are suboptimal, thereby improving on classical results in certain cases. The rest of the Thesis is concerned with the Fourier restriction problem on fractal sets. We generalise the Stein–Tomas theorem in two ways. First, by considering the average Fourier decay of fractal measures, we show that this improves known estimates for many examples, such as multifractal measures and the graph of additive Brownian motion. Second, we show that further improvement can be achieved by considering the 𝐿𐞥-dimensions, widely used to capture the multifractal behaviour of measures. We then apply this estimate to the Mandelbrot cascade measure. Finally, we consider multilinear Fourier extension estimates, and find sufficient and necessary conditions that do not always coincide and depend on whether the convolution of the measures at play is absolutely continuous or singular. The positive estimate depends on the smoothness of the convolution of the measures, by requiring it to be absolutely continuous and lying in some Lebesgue space. To obtain the necessary conditions, we build random measures that contain transversal 'flat' structures within them.