Local Riesz transforms characterization of local Hardy spaces

For 0 < p ≤ 1, let hp (Rn) denote the local Hardy space. Let Θ̂ be a smooth, compactly supported function, which is identically one in a neighborhood of the origin. For k = 1,..., n, let (rkf)̂(Ζ) = -i(1 - Θ̂(Ζ)) Ζk/|Ζ|f̂(Ζ) be the local Riesz transform and define (rof)̂(Ζ) = (1 - Θ̂(Ζ))f̂(Ζ). Let Ψ be a fixed Schwartz function with f Ψ dx = 1, let M > 0 be an integer and suppose (n-1)/(n + M -1) < p ≤ 1. We show that a tempered distribution f which is restricted at infinity belongs to hp(R n) if and only if Θ *f ε hp (R n) and there exists a constant A > 0 such that for all ε with 0 < ε ≤ 1 we have ΣM≤|α|≤M+1∥r α(f)*Ψε∥LP(Rn) ≤ A Here, Ψε(x) = ε-nΨ(x/ε), α = (α0,..., αn)ε Nn+1, rα, as usual, denotes the composition r0α0 O...O r nαn. This result extends to the local Hardy spaces the analogous characterization of the classical Hardy spaces H p(Rn) (see e.g. [9, Chapter III.5.16]).