Abstract
In this paper we discuss the Lp-Lq boundedness of both spectral and Fourier multipliers on general locally compact separable unimodular groups G for the range 1<p≤2≤q<∞. As a consequence of the established Fourier multiplier theorem we also derive a spectral multiplier theorem on general locally compact separable unimodular groups. We then apply it to obtain embedding theorems as well as time-asymptotics for the Lp-Lq norms of the heat kernels for general positive unbounded invariant operators on G. We illustrate the obtained results for sub-Laplacians on compact Lie groups and on the Heisenberg group, as well as for higher order operators. We show that our results imply the known results for Lp-Lq multipliers such as Hörmander's Fourier multiplier theorem on Rn or known results for Fourier multipliers on compact Lie groups. The new approach developed in this paper relies on advancing the analysis in the group von Neumann algebra and its application to the derivation of the desired multiplier theorems.
Highlights
In this paper we discuss the Lp-Lq boundedness of both spectral and Fourier multipliers on general locally compact separable unimodular groups G for the range 1 < p ≤ 2 ≤ q < ∞
In this paper we aim at proving the Hormander type theorem expressing conditions in terms of the sharp decay property of the spectral information associated to the operator, on general locally compact separable unimodular groups based on developing a new approach relying on the analysis in the noncommutative Lorentz spaces on the group von Neumann algebra
Let G be a locally compact unimodular group with VNR(G) the group von Neumann algebra generated by the right regular representation πR of G
Summary
The aim of this paper is to give sufficient conditions for the Lp-Lq boundedness of Fourier and spectral multipliers on locally compact separable unimodular groups. In this paper we aim at proving the Hormander type theorem expressing conditions in terms of the sharp decay property of the spectral information associated to the operator, on general locally compact separable unimodular groups based on developing a new approach relying on the analysis in the noncommutative Lorentz spaces on the group von Neumann algebra. This suggested approach seems very effective, implying as special cases known results expressed in terms of symbols, in settings when the symbolic calculus is available. Throughout the paper we will use the notation of the type f X f Y if we have f X ≤ C f Y with the constant C that may depend on the spaces X, Y but not on f
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