On the $L^2\to L^{\infty}$ norms of spectral multipliers of "quasi-homogeneous" operators on homogeneous group

Research Report MRR97-001

On the $L^2 \to L^{\infty}$ norms of spectral multipliers of "quasi-homogeneous" operators on homogeneous group

Adam Sikora

Abstract: We study the $L^2 \to L^{\infty}$ norms of spectral projections and spectral multipliers of left-invariant elliptic and subelliptic second-order differential operators on homogeneous Lie groups. We obtain a precise description of the $L^2 \to L^{\infty}$ norms of spectral multipliers for some class of operators which we call quasi-homogeneous. As an application we prove a stronger version of Alexopoulos' spectral multiplier theorem for this class of operators.

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