Singular Integrals with Modulation or Rotational Symmetry

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Harmonic Analysis is the branch of Mathematics concerned with the representation and reconstruction of signals (functions) as a superposition of basic harmonics--signals of well-specified duration, intensity and frequency--as well as the study of how suitable operations (filtering, denoising, compression, etc.) affect the reconstructed signal. Concrete versions of this decomposition/filtering/reconstruction process, sometimes referred to as the "time-frequency" method, are performed in a broad range of real-world applications, such as audio or image compression, image pattern or facial recognition, data assimilation, compressed sensing and many others. A similar procedure is employed in tomographic imaging, where the shape of a solid body is reconstructed by means of sampling the body's density along penetrating waves, which can be mathematically described as lines in three dimensional space. The first main component of this mathematics research project deals with toy mathematical models of sampling three and higher dimensional objects (for instance, solid bodies) along lower dimensional sets such as lines or planes. The second, deeply related component of this project is concerned with extending the time-frequency decomposition method to suitable vector-valued signals. Integral components of the project are the training of graduate and undergraduate students within the active research group in Harmonic Analysis and Partial Differential Equation at University of Virginia, as well as the mentoring and research start-up of undergraduates, graduate students and researchers coming from underrepresented groups in the profession.This Harmonic Analysis research project deals with singular integral operators exhibiting further invariance properties, such as modulation or rotational symmetries, in addition to those (translation and dilation invariance) characterizing Calderon-Zygmund operators: a fundamental example is the Carleson maximal operator dictating pointwise convergence of the Fourier series of square-integrable functions.The first part of this research project deals with rotation invariant singular integrals: in particular, with the Hilbert transform along Lipschitz vector fields. The PI will work on a novel characterization of those vector fields giving rise to a bounded directional Hilbert transform, in terms of boundedness of the related directional maximal function. The PI also proposes an array of model problems, of independent interest, obtained by constraining the range of the vector field. A novelty is that questions set up in higher dimensional ambient spaces are considered. The intrinsic multi-parameter nature of directional operators leads naturally to connected outstanding questions on the theory of double Fourier series: the parabolic and the polygonal summation problems. The second, related circle of problems investigated in this project concerns linear and multilinear singular integrals acting on Banach space valued functions: among other questions, the PI will investigate T(1)-type operator valued theorems in the multilinear setting, and fully noncommutative analogues of Carleson's theorem. The strength and relevance of operator-valued type theorems are that they self-improve to their multi-parameter analogues, which are of interest for applications and are often not attainable with direct techniques.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.