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

Fix a positive integer 𝑘.Let 𝑅 𝑘 be a higher order Riesz transform of order 𝑘 on ℝ 𝑑 and let 𝑅 𝑡 𝑘 , 𝑡 > 0, be the corresponding truncated Riesz transform.We study the relation between ∥𝑅 𝑘 𝑓 ∥ 𝐿 𝑝 (ℝ 𝑑 ) and ∥𝑅 𝑡 𝑘 𝑓 ∥ 𝐿 𝑝 (ℝ 𝑑 ) for 𝑝 = 1, 𝑝 = ∞, and 𝑝 = 2.We do this by analyzing the factorization operator 𝑀 𝑡 𝑘 defined by the relationThe operator 𝑀 𝑡 𝑘 is a convolution operator associated with an 𝐿 1 radial kernel 𝑏 𝑡 𝑘,𝑑 (𝑥) = 𝑡 -𝑑 𝑏 𝑘,𝑑 (𝑥/𝑡), where 𝑏 𝑘,𝑑 (𝑥) := 𝑏 1 𝑘,𝑑 (𝑥).We prove that 𝑏 𝑘,𝑑 ⩾ 0 only for 𝑘 = 1, 2. We also show that for fixed 𝑘 ⩾ 3,This contrasts with the cases 𝑘 = 1, 2, where it is known that ∥𝑏 𝑘,𝑑 ∥ 𝐿 1 (ℝ 𝑑 ) = 1.Finally, we show that for any positive integer 𝑘, the Fourier transform of 𝑏 𝑘,𝑑 is bounded in absolute value by 1.This implies the contractive estimateand an analogous estimate for general singular integrals with smooth kernels for radial input functions 𝑓 .