Grushin Multipliers and Toeplitz Operators

Abstract

This work investigates advanced concepts in functional analysis and partial differential equations, focusing on Sobolev spaces and their weighted variants. The study explores properties of functions in Lp(Rn+1)L^p(\mathbb{R}^{n+1})Lp(Rn+1) and L2(Tm)L^2(T^m)L2(Tm), along with exponential weight adjustments such as e??L2(Tm)e^{-\lambda} L^2(T^m)e??L2(Tm). Special attention is given to harmonic analysis on compact sets, represented by HL2(Kc)???c\mathcal{H} L^2(K_c)^{\lambda - \kappa_c}HL2(Kc?)???c?, and the role of multi-index derivatives in reproducing polynomial spaces. These formulations are essential for understanding regularity, approximation, and stability in solutions to elliptic and parabolic PDEs. Applications include spectral theory, variational methods, and numerical schemes for high-dimensional problems.