Research Projects

This project aims to develop a comprehensive framework for establishing the sharp version of the Heisenberg Uncertainty Principle (HUP) in the fractional Sobolev space. We further aim to identify extremal functions (extremals) in the fractional setting, and to analyze the stability of the HUP in this framework. Alongside this line of investigation, the project also seeks to determine the sharp constants in Hardy and Trudinger-Moser type inequalities within fractional Sobolev spaces on bounded Lipschitz domains. Furthermore, we intend to extend this analysis to establish sharp constant in Hardy-type inequalities involving singularities on smooth submanifolds. We also aim to derive singular Trudinger-Moser inequalities on weighted fractional Sobolev spaces. Establishing the sharpness of such constants is essential in the study of nonlocal PDEs, particularly in proving existence and regularity results. For the Sobolev space (i.e., the local case), the HUP has been extensively studied. Over the past few years, researchers have contributed to improvements, sharp constant estimates, extremals, and stability results, both in Euclidean and hyperbolic settings. Determining the fractional version of sharp HUP, finding its extremals and the stability has not been studied so far. The present project aims to study the HUP in the fractional framework, establish the sharp HUP in fractional Sobolev space, identify extremals, and perform stability analysis. In a similar direction, fractional Hardy inequalities have also been widely explored over the past two and a half decades, leading to several extensions and refinements. However, the determination of the optimal constant in fractional Hardy inequalities with boundary singularities on bounded Lipschitz domains remains an open and challenging problem. On such domains, fractional Trudinger-Moser inequalities have been established, and existence of upper bounds, as well as thresholds beyond which the inequality fails, have been identified. Nevertheless, the problem of determining the supremum over all such upper bounds remains unresolved. In our prior research, we addressed several key challenges in this area, establishing fractional Hardy inequalities for the critical cases with an optimal logarithmic weight function, including those with singularities on smooth submanifolds. We also considered the remaining cases and established fractional Hardy inequalities with singularities on smooth submanifolds. Building upon this foundation, the present project also seeks to determining the sharp constant in fractional Hardy inequalities with singularities on boundary of Lipschitz domains and also singularities on smooth submanifolds. Subsequently, we aim to determine the sharp upper bounds in fractional Trudinger-Moser inequalities on bounded Lipschitz domains. These results will advance nonlocal analysis and strengthen the theoretical framework of functional inequalities, with broader impacts on geometric analysis.