Grant: Discovery Grants Program - Individual

Work on the various Hardy operators in the 1970's and 80's set the standard for a new level of understanding of positive operators. Decades of intensive investigation has advanced the study of positive operators greatly. My recent work on normal form Hardy operators unifies and extends the theory by identifying the strong internal connections between different operators. This, in essence, reduces the study of a large collection of complicated Hardy operators to the study of a small collection of simple Hardy operators. The benefits of the normal form approach can be achieved for many other types of positive operators; this is one of the two main objectives of my research proposal. It also feeds into the second objective. The other objective involves the Fourier transform, specifically, the spaces of functions between which the Fourier transform operates continuously. The Fourier transform is the single most important operator in Mathematics, first in both theoretical and practical applications. But it is not a positive operator. Despite centuries of scrutiny, there are still fundamental questions about it that remain unsolved. One of these asks for a complete description of the weighted Lebesgue spaces for which the Fourier transform maps one space into the other. A weighted Lebesgue space is set of functions along with a standard for measuring each function's "size". The standard is determined by the space's index, a number, and its weight, a function. Achieving a complete description is a very difficult problem, but work and progress continue. There is an array of partial descriptions available due to the work of many researchers. My approach is to estimate the Fourier transform by carefully constructed positive operators and apply techniques from that theory. In past work, my team has answered the above question for weighted Lorentz spaces, which is a class close to the weighted Lebesgue. It has long been known that results for Lorentz spaces give results for Lebesgue spaces. However, it turns out that our complete answer for Lorentz spaces gives only a partial answer to the Lebesgue space question. The work on weighted Lorentz spaces will continue. In addition, it is time for a synthesis of the Lorentz space results and progress made by other researchers, with the goal of advancing the state of the art on the elusive Lebesgue space problem. This is fundamental mathematics, part of a tradition with proven value to human life worldwide for thousands of years. The research on Hardy operators and other positive operators will provide new tools to researchers in Functional Analysis, Differential Equations, Statistical Methods, and more. Basic work on the Fourier Transform affects too many areas to list, but Harmonic Analysis and Partial Differential Equations stand out. As an example, past work on the solution to the Lorentz space question above has been cited in the 2017 book, Advances in Quantum Mechanics, edited by Paul Bracken.