In 1987, A. Bonami and S. Poornima proved that a non-constant function which is homogeneous of degree zero cannot be a Fourier multiplier on homogeneous Sobolev spaces. In this talk, we will discuss the Fourier multipliers on Heisenberg group $\mathbb{H}^n$ and Weyl multipliers on $\mathbb{C}^n$ acting on Sobolev spaces. We define a notion of homogeneity of degree zero for bounded operators on $L^{2}(\mathbb{R}^n)$ and establish analogous results for Fourier multipliers on Heisenberg group and Weyl multipliers on $\mathbb{C}^n$ acting on Sobolev spaces. This talk is based on the recent work with Rahul Garg and Sundaram Thangavelu.

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Last updated: 23 Jun 2024