Littlewood-Paley square functions and the local Hardy space for Multi-Norm Structures on R^d

Multi-norm singular integrals and Fourier multipliers were introduced in 29, and one application of these notions was a precise description of the composition of convolution operators with Calderón-Zygmund kernels adapted to n different families of dilations. The description of the resulting operators was given in terms of differential inequalities specified by a matrix E, and in terms of dyadic decompositions of the kernels and multipliers. In this paper we extend the analysis of multi-norm structures on Rᵈ by studying the induced Littlewood-Paley decomposition of the frequency space and various associated square functions. After establishing their L¹-equivalence, we use these square functions to define a local multi-norm Hardy space h^1₄ (Rᵈ). We give several equivalent descriptions of this space, including an atomic characterization. There has been recent work, limited to the 2-dilation case, by other authors. The general n-dilation case treated here is considerably harder and requires new ideas and a more systematic approach.