Lip_α(R^n)(0<α<1)上的经典Lusin面积积分函数

Lusin area integral function on Lip α(R n)(0<α<1)

证明了当ｆ∈Ｌｉｐα（Ｒｎ）（０＜α＜１／２）时，ｆ的Ｓ函数或处处有限，或处处为∞；如属前者，则Ｓ（ｆ）∈Ｌｉｐα（Ｒｎ）（０＜α＜１／２），且Ｓ（ｆ）∧α≤Ｃｆ∧α，其中Ｃ是与维数ｎ，α有关的常数．

It is proved that the S(f) is either equal to infinite almost everywhere or fimite almost everywhere when f∈Lip α(R n)(0<α<1/2). In the latter, S(f)∈Lip α(R n)(0<α<1/2) and‖S(f)‖ ∧α ≤C‖f‖ ∧α .Where C denotes a constant only depending on n,α.