Third Order Commutator and Product Bmo
نویسندگان
چکیده
Let Mb denote the operation of multiplication by b. Let Hj denote the Hilbert transform performed in the jth coordinate, for j = 1, 2, . . .. Consider the jth order iterated commutator Cb,j = [H1, · · · , [Hj , Mb] · · · ], j = 1, 2, . . . We show that the operator norm of Cb,3 on L (R) is comparable to the norm of b in Chang–Fefferman product BMO(⊗31R 2 +). This fact has some well known equivalences, in particular the weak factorization result for the Hardy space H1(⊗31R 2 +) = H2(⊗31R 2 +)⊗̂H 2(⊗31R 2 +). The corresponding fact for Cb,1 is classical. The corresponding fact for Cb,2 is a Theorem of Ferguson and Lacey [7]. In this paper, we follow the strategy of Ferguson and Lacey, and find that a new ingredient is needed at a particular juncture of the argument.
منابع مشابه
Weighted estimates of higher order commutators generated by BMO-functions and the fractional integral operator on Morrey spaces
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