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

Let 1 <= p < infinity and -1/p < lambda < 1/n. Then a function f is an element of L-loc(p)(R-n) is said to belong to the central Carnpanato space C-p,C-lambda(R-n) if parallel to f parallel to C-p,C-lambda(R-n) = sup(r>0)1/vertical bar B(0,r)vertical bar(lambda) (1/vertical bar B(0,r)vertical bar integral(B(0,r)) vertical bar f - f(B(0,r))vertical bar(p) dx)(1/p) < infinity, where B(0, r) denotes the ball centered at 0 with radius r > 0 and f(B)(0,r) = 1/vertical bar B(0,r)vertical bar integral(B(0,r)) f(y)dy. There are many known characterizations of C-p,C-lambda (R-n) for 0 <= lambda < 1/n. The aim of this paper is to introduce some characterizations of C-p,C-lambda (R-n) for -1/p < lambda < 0, via the boundedness of commutator operators of Hardy type. Some more explicit decompositions of these operators and the space C-P,C-lambda(R-n), which are different from that of lambda >= 0, are here proposed to overcome the difficulties caused by lambda < 0. Moreover, some further interesting boundedness for the Hardy type operators is also obtained. Crown Copyright (C) 2015 Published by Elsevier Inc. All rights reserved.