Our investigations center on harmonic analysis in the context of nilpotent Lie groups and time-frequency methodologies, forging connections between classical Fourier techniques and modern operator-theoretic approaches. Particular attention is given to the analysis of modulation spaces, Fourier multipliers, and twisted convolution operators, especially their boundedness properties between various $L^p$ and $L^q$ function spaces. Our research addresses nonlinear Schrödinger equations involving twisted Laplacians, leveraging spectral analysis and pseudo-differential calculus in non-commutative frameworks. We also investigate maximal functions along hypersurfaces and develop sharp Hardy-Sobolev inequalities that capture the behavior of functions under singular integrals and non-Euclidean geometries. A recurring theme is the use of oscillatory integrals to study dispersive partial differential equations and fine-tuned spectral estimates.
Harmonic Analysis Research