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

Abstract

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 $(T_{M}f{)}^{\hat{}}=M\hat{f}$ on the $(2n+1)$-dimensional Heisenberg group ${\mathbb{H}}^n$. Here “$\hat{}$” 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 $T_M$. $\Vert T_{M}f{\Vert }_{L^1}\le C\Vert f{\Vert }_{H^1}$, for ${\mathbb{H}}^n$ under the same hypotheses of $L^p$-boundedness.