Lusin面积积分函数在Lipα(R^n)(0<α<1)上的有界性

BOUNDEDNESS OF LUSIN AREA INTEGRAL ON LIPα(R^n) (O<α<1)

最近,Shilin Wang 在文[1]中得到了 Littlewood—Palegy 函数的一些性质;本文证明了 Lusin面积积分函数 s(f)也具有同样的性质,即当 f∈Lipa(R^n)(0<a<1/2)时,f 的 s 函数要么处处为∞,要么处处有限,如属后者则 s(f)∈Lipa(R^n)(0<a<1/2),且‖s(f)‖≤C‖f‖这里 C 表示仅与维数 n、a 有关的常数。

Recently,Shilin Wang have obtained some Properties of Littlewood—paley g—function.In this paper,We prove that Lusin area integral s(f) also have the same properties,That is,If f∈Lipα(R^n) (0<α<(1/2)),then the s(f) is either equal to infinite almost everywhere or fimite almost in last case,s(f)∈Lipα(R^n)(0<α<(1/2))and ‖s(f)‖_(A_α)≤c‖f‖_(A_α) Where c denot a constant only depend on n、α.