Revista Matemática Iberoamericana

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Volume 11, Issue 2, 1995, pp. 269–308
DOI: 10.4171/RMI/173

Published online: 1995-08-31

$L^p$ multipliers and their $H^1$-$L^1$ estimates on the Heisenberg group

Chincheng Lin[1]

(1) National Central University, Chung-Li, Taiwan

We give a Hörmander-type sufficient condition on an operator-valued function $M$ that implies the $L^p$-boundedness result for the operator $T_M$ defined by ($T_Mf$^ = $M\hat{f}$ on the $(2n + 1)$- dimensional Heisenberg group $\mathbb H^n$. Here "^" denotes the Fourier transform on $\mathbb H^n$ defined in terms of the Fock representations. We also show the $H^1$-$L^1$-boundedness of $T_M$. $\|T_Mf\|_{L^1} ≤C \|f\|_{H^1}$, for $\mathbb H^n$ under the same hypotheses of $L^p$-boundedness.

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Lin Chincheng: $L^p$ multipliers and their $H^1$-$L^1$ estimates on the Heisenberg group. Rev. Mat. Iberoam. 11 (1995), 269-308. doi: 10.4171/RMI/173