科研之友
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摘要
In this paper, we study the duality theory of the multi-parameter Triebel-Lizorkin spaces (F)over dot(p)(alpha,q) (R-m) associated with the composition of two singular integral operators on R-m of different homogeneities. Such composition of two singular operators was considered by Phong and Stein in 1982. For gular integral operators on R-m of different homogeneities. Such composition of two singular operators was considered by Phong and Stein in 1982. For 1 < p < infinity, we establish the dual spaces of such spaces as (F)over dot(p)(alpha,q) (R-m))* = (F)over dot(p)(-alpha,q') (R-m), and for 0 < p <= 1 we prove (F)over dot(p)(alpha,q) (R-m))* = CMOp-alpha,q' (R-m).We then prove the boundedness of the composition of two Calderon-Zygmund singular integral operators with different homogeneities on the spaces CMOp-alpha,q'. Surprisingly, such dual spaces are substantially different from those for the classical one-parameter Triebel-Lizorkin spaces (F)over dot(p)(-alpha,q') (R-m). Our work requires more complicated analysis associated with the underlying geometry generated by the multi-parameter structures of the composition of two singular integral operators with different homogeneities. Therefore, it is more difficult to deal with than the duality result of the Triebel-Lizorkin spaces in the one-paramter settings. We note that for 0 < p <= 1, q = 2 and alpha = 0, (F)over dot(p)(-alpha,q') (R-m) is the Hardy space associated with the composition of two singular operators considered in Rev. Mat. Iberoam. 29 (2013), 1127-1157. Our work appears to be the first effort on duality for Triebel-Lizorkin spaces in the multi-parameter setting.
