On Certain Linear and Bilinear Sub-Elliptic Bochner-Riesz Means and Calderón-Zygmund Type Singular Integrals

Singh, Joydwip (2025) On Certain Linear and Bilinear Sub-Elliptic Bochner-Riesz Means and Calderón-Zygmund Type Singular Integrals. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Joydwip Singh (20RS078)) 20RS078.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

A central topic in harmonic analysis is understanding the convergence of Fourier series and Fourier integrals within Lebesgue spaces. The study of the boundedness of Bochner-Riesz means plays a significant role in this context. The Bochner-Riesz problem on $\mathbb{R}^n$ seeks to determine the sharp threshold for which the Bochner-Riesz operator acts boundedly on $L^p$-spaces. Although substantial progress has been made over the last several decades, the problem remains unresolved for dimensions $n\geq 3$. Moreover, the Bochner-Riesz problem is closely connected with other major conjectures in harmonic analysis, such as the Kakeya conjecture, the Fourier restriction conjecture, and the local smoothing conjecture. The aim of this thesis is to further investigate in this direction for some specific classes of differential operators, namely sub-Laplacians on M\'etivier groups and for Grushin operators. This thesis addresses two distinct problems concerning Bochner-Riesz multipliers associated with sub-Laplacians on Métivier groups and Grushin operators. In the second and third chapters, we focus on the boundedness of bilinear Bochner-Riesz operators, seeking to determine the minimal smoothness parameter, that ensures the boundedness of the associated bilinear operators. In the fourth and fifth chapters, we further investigate the boundedness and compactness properties of the Bochner-Riesz commutators associated with the sub-Laplacians on M\'etivier groups and for Grushin operators, with smoothness conditions framed in terms of the topological dimensions. Finally in the sixth chapter, outside the Bochner-Riesz setup, we discuss about the Lebesgue space and Hardy space boundedness of an extension of Calder\'on-Zygmund type singular integral and their commutators on $\mathbb{R}^n$.

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