Marcinkiewicz multipliers on the Heisenberg group

The Marcinkiewicz multiplier theorem in {dollar}IRsp{lcub}n{rcub}{dollar} establishes the boundedness on {dollar}Lsp{lcub}p{rcub}{dollar} of multiplier operators, for a class of multipliers which is invariant under multiparameter dilations of {dollar}IRsp{lcub}n{rcub}{dollar}. The first part of this thesis establishes an analogous result on the Heisenberg group H{dollar}sp{lcub}n{rcub}(= doubcsp{lcub}n{rcub}times IR{dollar}). Just as the Marcinkiewicz theorem in {dollar}IRsp{lcub}n{rcub}{dollar} deals with spectral multipliers of {dollar}i{lcub}partialover{lcub}partial xsb1{rcub}{rcub},..., i{lcub}partialover{lcub}partial xsb{lcub}n{rcub}{rcub}{rcub}{dollar}, the operators considered here on the Heisenberg group are functions of iT and the partial sub-Laplacians {dollar}{lcub}cal L{rcub}sb1,...,{lcub}cal L{rcub}sb{lcub}n{rcub}{dollar}. The boundedness on {dollar}Lsp{lcub}p{rcub}{dollar}(H{dollar}sp{lcub}n{rcub}), 1 < p < infty{dollar}, is proved for operators {dollar}m({lcub}cal L{rcub}sb1,...,{lcub}cal L{rcub}sb{lcub}n{rcub}, iT{dollar}), where m satisfies an (n + 1)-fold Marcinkiewicz-type condition.; A characterization is also given for the convolution kernels of these Marcinkiewicz multipliers {dollar}m({lcub}cal L{rcub})sb1,...,{lcub}cal L{rcub}sb{lcub}n{rcub}, iT{dollar}). They are precisely those distributions which are polyradial, singular only along the {dollar}zsb{lcub}i{rcub}{dollar} = 0 planes, and satisfy certain regularity and cancellation conditions.; The second part of this thesis deals with standard radial kernels on H{dollar}sp{lcub}n{rcub}{dollar}. These are kernels radial in the z variable, singular only at the origin, and satisfying the standard Calderon-Zygmund-type regularity and cancellation conditions associated to the automorphic 1-parameter dilations of H{dollar}sp{lcub}n{rcub}{dollar}. Convolution operators with standard radial kernels form a sub-class of Marcinkiewicz multiplier operators of the form {dollar}m({lcub}cal L{rcub}, iT).{dollar} We establish a stricter condition than the Marcinkiewicz condition on the multiplier m, which characterizes this sub-class.