Let μ be a Borel measure on R^d which may be non doubling. The only condition that μ must satisfy is μ(Q) ≤ col(Q)^n for any cube Q belong to R^d 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 H^1(μ).
In this paper, we obtain the (H^1,L^n/(n-β) and (HKq1^n(1-1/q2),p,Kq2^n(1-1/q1),p) 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