\(L^ p\) multipliers and their \(H^ 1\)-\(L^ 1\) estimates on the Heisenberg group
Summary: 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 \((T_MF) \widehat{ }=M\widehat f\) on the \((2n+1)\)-dimensional Heisenberg group \(\mathbb{H}^n\). Here ``\( \widehat{ } \)'' 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 \(T_M\), \(|T_Mf|_{L^1}\leq C|f|_{H^1}\), for \(\mathbb{H}^n\) under the same hypotheses of \(L^p\)-boundedness.
scientific article; zbMATH DE number 813823