A new type of Littlewood-Paley partition

Abstract

We define a partition of Z into intervals {I_j} and prove the Littlewood-Paley inequality ‖f‖_p≦C_p‖Sf‖_p, 2≦p<∞. Here f is a function on [o, 2π) and $Sf = (\sum |\Delta _j |^2 )^{1/2} , \hat \Delta j = \hat f\chi _{Ij} $ . This is a new example of a partition having the Littlewood-Paley property since the {I_j} are not of the type obtained by iterating lacunary partitions finitely many times.