论文标题
$ l^p $ - $ l^q $频谱乘数在单模型lie组上的更新
An Update On The $L^p$-$L^q$ Norms of Spectral Multipliers on Unimodular Lie Groups
论文作者
论文摘要
该注释给出了Akylzhanov和第二作者的乘数定理的广泛更新[J.功能。肛门，278（2020），108324]。后者的证据至关重要地依赖于$ l^p $ - $ l^Q $规范估算值，用于剩余的加权副驾驶员的频谱投影仪，对单模型的谎言组（例如Laplacians，sublaplacians and Rockland Operators）。通过将光谱投影仪与加热内核联系起来，可以立即利用$ l^p $ - $ l^Q $规范的明确估计值，以比以前已知的（连接的单模型）谎言组和操作员更广泛地利用。作者与先前建立的边界的比较表明，热核估计值很清晰。作为一个应用程序，显示出乘数定理的几个后果，例如$ l^p $ - $ -L^q $ n Norms the Heat-norms and Sobolev-type嵌入的时间渐近，然后是自动的。
This note gives a wide-ranging update on the multiplier theorems by Akylzhanov and the second author [J. Funct. Anal., 278 (2020), 108324]. The proofs of the latter crucially rely on $L^p$-$L^q$ norm estimates for spectral projectors of left-invariant weighted subcoercive operators on unimodular Lie groups, such as Laplacians, sub-Laplacians and Rockland operators. By relating spectral projectors to heat kernels, explicit estimates of the $L^p$-$L^q$ norms can be immediately exploited for a much wider range of (connected unimodular) Lie groups and operators than previously known. The comparison with previously established bounds by authors show that the heat kernel estimates are sharp. As an application, it is shown that several consequences of the multiplier theorems, such as time asymptotics for the $L^p$-$L^q$ norms of the heat kernels and Sobolev-type embeddings, are then automatic for the considered operators.
