Characterization of the central Campanato space via the commutator operator of Hardy type

Let 1≤p<∞1≤p<∞ and −1/p<λ<1/n−1/p<λ<1/n. Then a function f∈Llocp(Rn) is said to belong to the central Campanato space C˙p,λ(Rn) if‖f‖C˙p,λ(Rn)=supr>0⁡1|B(0,r)|λ(1|B(0,r)|∫B(0,r)|f−fB(0,r)|pdx)1/p<∞, where B(0,r)B(0,r) denotes the ball centered at 0 with radius r>0r>0 and fB(0,r)=1|B(0,r)|∫B(0,r)f(y)dy. There are many known characterizations of C˙p,λ(Rn) for 0≤λ<1/n0≤λ<1/n. The aim of this paper is to introduce some characterizations of C˙p,λ(Rn) for −1/p<λ<0−1/p<λ<0, via the boundedness of commutator operators of Hardy type. Some more explicit decompositions of these operators and the space C˙p,λ(Rn), which are different from that of λ≥0λ≥0, are here proposed to overcome the difficulties caused by λ<0λ<0. Moreover, some further interesting boundedness for the Hardy type operators is also obtained.