On the dimension-free control of higher order truncated Riesz transforms by higher order Riesz transforms

Fix a positive integer k. Let Rₖ be a higher order Riesz transform of order k on Rᵈ and let Rₖᵗ, $t>0,$ be the corresponding truncated Riesz transform. We study the relation between \|Rₖ f\|Lᵖ(Rᵈ) and \|Rₖᵗ f\|Lᵖ(Rᵈ) for $p=1$, p=∞, and $p=2.$ This is performed via analysis of the factorization operator Mₖᵗ defined by the relation Rₖᵗ=Mₖᵗ Rₖ. The operator Mₖᵗ is a convolution operator associated with an L¹ radial kernel bk,dᵗ(x)=t⁻ᵈbk,d(x/t), where bk,d(x):=bk,d¹(x). We prove that bk,d ≥ 0 only for $k=1,2.$ We also justify that for fixed k≥ 3 it holds \[ limd→ ∞\|bk,d\|L^1(R^d)=∞. \] This is contrary to the cases $k=1,2$ where it is known that \|bk,d\|L¹(Rᵈ)=1. Finally, we show that for any positive integer k the Fourier transform of bk,d is bounded in absolute value by $1.$ This implies the contractive estimate \[ \|R_k^t f\|L^2(R^d)≤ \|R_k f\|L^2(R^d) \] and an analogous estimate for general singular integrals with smooth kernels for radial input functions $f.$