非双倍测度下的Littlewood-Paley算子的有界性

Boundedness of Littlew ood-Paley operators with non-doubling measures

设μ是一个Rd上的Radon测度,仅满足增长条件:μ(B(x,r))≤C0rn,0<n≤d,x∈Rd,r>0。假设Littlew ood-Paley g函数在L2(μ)上有界,利用非双倍测度下的Calderón-Zygmund分解证明了Littlewood-Paley g函数是L1(μ)到L1,∞(μ)上有界的,并且它是H1(μ)到L1(μ)上有界的。

Let μ be a positive Radon measure on Rd which may be non-doubling. The only condition that μ must satisfy is μ （ B （ x, r） ） ≤ Co r, for x E Rd, r 〉 0 and some fixed constants Co 〉 0 and 0 〈 n ≤ d. Supposing Littlewood-Paley g is bounded on L2 （μ,） , by using the Calder6n-Zygmund decompsition under non-doubling measure, the boundedness of g from L1 （μ） to L1 ,∞ （μ） is obtaind. And then the boundedness of g from Hardy space H1 （μ） to L1 （μ） is established.