Research Projects

Introduction to Theme 1. The study of bilinear rough singular integrals have attracted a great deal of research activities in the past two decades due to their importance in Harmonic analysis and PDEs. Despite that, the end-point estimates for these operators remain a mystery till date. As the first part of the project we propose to develop suitable trilinear smoothing inequalities to address the issue of Lp boundedness of bilinear rough singular integral operators. Christ, Durcik and Roos, Adv. Math. 2021, Christ and Zhou, J Funct. Anal. 2024, Chen and Guo, Math Ann. 2024 have proved effectiveness of trilinear smoothing inequalities in the context of trianglular bilinear Hilbert transform, bilinear spherical maximal function and twisted bilinear Hilbert transform along polynomial curves respectively. We believe that appropriate versions of these ideas would lead us to address the long standing open problems for many important bilinear operators. In particular, we propose to develop this line of thoughts for bilinear rough singular integral operators. This method would be a new tool in this context with a huge scope of further applications. Introduction to theme 2. Magyar, Rev. Math. Iber. 1997 and Magyar, Stein and Wainger, Ann. of Math. 2002 introduced the study of discrete spherical maximal function and developed a bridge connecting various ideas from harmonic analysis to number theory. These ideas are fundamental in nature and have been proved effective inn many other important research works. Under the second theme of the project, we propose to strengthen these ideas by establishing variational norm estimates for discrete spherical maximal functions. Note that variational norm estimates are stronger than the corresponding estimates of maximal functions. Moreover, we propose to develop the bilinear theory of spherical maximal functions involving suitable discrete analogues of trilinear smoothing inequalities. These methods will provide us with new tools in the discrete setting. Once established, they would prove to be applicable in many other situations. This project aims to develop modern analytical techniques involving trilinear smoothing inequalities and the Hardy–Littlewood–Ramanujan circle method. These powerful tools have been extensively utilized by leading mathematicians to achieve significant advances in harmonic analysis and number theory. Despite their wide-ranging applications, there is currently a lack of expertise in these methods within the Indian mathematical research community. Through this project, we seek to systematically develop these techniques and apply them to investigate a new class of operators. We hope to contribute to the underlying subject significantly through this proposed study. We aim to develop methods for the themes mentioned above with respect to the following research problems. 1. Trilinear smoothing inequalities in the context of bilinear rough singular operators 2. Applications of trilinear smoothing inequalities in the context of Roth type theorems in continuous setting. 3. End-point L^p-estimates for bilinear rough singular operators and Bochner Riesz means. 4. Develop the discrete analogue of trilinear smoothing inequalities 5. Applications of trilinear smoothing inequalities in the context of discrete bilinear spherical maximal functions 6. Obtain variational norm Lp-estimates for discrete spherical maximal function. Note-Please refer to the proposal draft for technical details as math symbols could not be typed here.