Averaging operators and related topics in harmonic analysis

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This project is in harmonic analysis and approximation theory, areas within the mathematical discipline of analysis. The methods have found many applications in understanding phenomena in the natural sciences and engineering. Harmonic analysis seeks to provide efficient mathematical tools for these disciplines and contributes to the unification of seemingly unrelated areas. One of the main objectives of this project is to expand the current mathematical toolbox in harmonic analysis to contribute towards a deeper theoretical understanding which will ultimately be beneficial for applications. The mentoring of graduate students in research is an important educational component of the project. The principal investigator will work on several projects in harmonic analysis. (i) The first project is concerned with the precise regularity properties of certain averages over sub-manifolds of Euclidean space and the boundedness of associated maximal operators in Lebesgue spaces. The PI will consider mainly non-convolution variants and emphasize various classes of spherical maximal operators on nilpotent groups. (ii) In a second project the PI will study versions of the local smoothing problem for solutions of the wave equation when the dilation set is restricted. New phenomena show up even in the simplified version for radial functions. One expects that the outcomes depend on various notions of dimensions of the dilation sets, the Minkowski dimension, the quasi-Assouad dimension and intermediate scales of dimensions. The PI will also study the related problem concerns the Lp improving bounds for spherical maximal operators with restricted dilation sets, for the open case when the dilation set is not Assouad regular. (iii) The PI will pursue various projects on endpoint estimates in sparse domination, focusing on true multiscale phenomena. Atomic decompositions techniques and sharp Lp improving results for single-scaled operators play a crucial role. (iv) A fourth project is in approximation theory and concerns the characterization of approximation spaces for nonlinear wavelet approximation. The PI and his collaborators will focus on the interesting cases when the order of approximation is high, and the approximation spaces will not be normed spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.