A central block is a function supported in such that . A central -atom is a central -block having integral zero. The block space consists of those distributions that can be expressed as superpositions of central -blocks such that and is -normed by , with infimum taken over block representations of . The space is defined in exactly the same way with blocks replaced by atoms. The -norm then makes sense when .
The author introduces a notion intermediate to that of a block and an atom, namely, a -block is a -block supported in for some such that . This notion allows the author to define a new block space just as above, but in terms of blocks, and to extend boundedness of certain singular integrals to these spaces.
Specifically, a -central singular integral is a linear operator that is bounded on and has integral kernel satisfying
The author’s main result says the following: Suppose that , , , and is a -central singular integral with . If then is bounded from to .
Here, the finite central oscillation space consists of those such that
The result extends previous work of J. Alvarez, J. Lakey and M. Guzmán-Partida [Collect. Math. 51, No. 1, 1–47 (2000; Zbl 0948.42013)] concerning boundedness of operators from block spaces into Herz-Hardy spaces. The author also corrects a minor error in that work.