An extension of Riesz transform. (October 2019) - Yu, Huan; Jiu, Quansen
Abstract: In this paper, we consider the following singular integral T j f ( x ) = K j ∗ f ( x ), K j ( x ) = x j | x | n + 1 − β, where x ∈ R n, 0 ≤ β < n, j = 1, 2, …, n . When β = 0, it corresponds to the Riesz transform. Based on the L 2 estimate of T j f in Yu et al. (2019) and making use of the refined Calderon–Zygmund decomposition, we establish an estimate of T j f in the L q space for 1 < q < 2 . For 2 < q < ∞ and q ′ = q q − 1 (the dual number of q ), by the duality method, we prove an estimate of T j f in the ( L q ′ ∩ L p ′ ) ∗ space which is the dual space of L q ′ ∩ L p ′ with 1 q ′ = 1 p ′ ( 1 − β n ) . The obtained estimates hold uniformly on β > 0 when β is appropriately small. As a result, the strong ( q, q ) estimate with 1 < q < ∞ and the weak ( 1, 1 ) estimate of the Riesz transform can be recovered from the obtained estimates as β → 0, respectively.