In this paper we consider the commutators of fractional integrals ∫R^n b(x)-b(y)/|x-y|^n-α f(y)dy on the Besov spaces Bp^s,q, where b is a locally integrable function and 0 < α < n. We first establish the equivalence between the boundedness of the commutators and the paraproduct of J. M. Bony. The nwe obtain two conditions on the boundedness of the commutators. One of these conditions is necessary and the other is sufficient.
Let μ be a Borel measure on Rd which may be non doubling. The only condition that μ must satisfy is μ(Q) ≤ c0l(Q)n for any cube Q ? Rd with sides parallel to the coordinate axes and for some fixed n with 0 < n ≤ d. The purpose of this paper is to obtain a boundedness property of fractional integrals in Hardy spaces H1(μ).
In this paper,we obtain the(H1,Ln/(n-β)) and ■ type estimates for the commutator of Marcinkiewicz integral with the kernel satisfying the logarithmic type Lipschitz conditions.
In this paper, it was proved that the commutator Hβ,b generated by an n-dimensional fractional Hardy operator and a locally integrable function b is bounded from Lp1(Rn) to Lp2 (Rn) if and only if b is a C(M)O(Rn) function, where 1/p1 - 1/p2 = β/n, 1 ＜ p1 ＜∞, 0 ≤β＜ n. Furthemore,the characterization of Hβ,b on the homogenous Herz space (K)qα,p(Rn) was obtained.
Zun-wei FU~(1,2) Zong-guang LIU~3 Shan-zhen LU~(1+) Hong-bin WANG~3 ~1 School of Mathematical Sciences,Beijing Normal University,Beijing 100875,China
