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
