Aspects of thinness in harmonic analysis

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Harmonic analysts seek to develop mathematical theories that help to find solutions to problems that might have arisen from mathematical physics, electrical engineering and other branches of mathematics. My research in harmonic analysis is motivated by the harmonic analysis version of the uncertainty principle. This principle is philosophically related to the Heisenberg uncertainty principle. Roughly speaking, it says information gained about a function has to be "paid for" by a corresponding loss of control on its Fourier transform, or vice versa. This is significant in practical applications. For instance, it implies that one cannot have a signal that is bounded in both the time and frequency domain. In my research, I develop rigorous and quantitative interpretations of this principle. I study functions, operators or measures that are "thin" in some sense and seek to understand the consequences of this thinness. My project has three major parts to it: 1. Lie groups are often used in physics as they describe real-world geometry. Orbital measures are elementary components in this setting and have small support. The first part of my project is to relate the size of their transforms and the associated geometric structures. This is important for analysis on Lie groups. 2. The second part of the project involves the study of thin directional maximal operators. Maximal operators are an important tool, used to reconstruct objects from their elementary components. I am interested in understanding how small the set of directions must be before the maximal operator will be well behaved. 3. Sidon sets can be defined in terms of the analytic properties of the functions whose Fourier transform is supported on the set. Although they are thin or sparse in some intuitive sense, they have a complicated structure. By characterizing Sidon sets in terms of thin sets that are simpler we will have greater understanding of the associated function spaces and underlying groups.