An extension of Riesz transform

Publication date: October 2019Source: Nonlinear Analysis: Real World Applications, Volume 49Author(s): Huan Yu, Quansen JiuAbstractIn this paper, we consider the following singular integral Tjf(x)=Kj∗f(x),Kj(x)=xj|x|n+1−β,where x∈Rn,0≤β<n,j=1,2,…,n. When β=0, it corresponds to the Riesz transform. Based on the L2 estimate of Tjf in Yu et al. (2019) and making use of the refined Calderon–Zygmund decomposition, we establish an estimate of Tjf in the Lq space for 1<q<2. For 2<q<∞ and q′=qq−1 (the dual number of q), by the duality method, we prove an estimate of Tjf in the (Lq′∩Lp′)∗ space which is the dual space of Lq′∩Lp′ with 1q′=1p′(1−βn). 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.

Source: Nonlinear Analysis: Real World Applications - April 26, 2019 Category: Research Source Type: research