一些平方函数的点态有限性

Pointwise finiteness of some square functions.

通过仔细的点态估计 ,证明了 :设 N 为一自然数 ,φ∈ CN(R1 ) ,φ(0 ) =0 ,| φ(x) | + | φ(N ) (x) | =O((1+ | x| ) - N - 1 -δ) (对某一 δ>0 ) ,f(x) (1+ | x| ) - N - 1∈ L1 (R1 ) ,如果 gφ(f)在 N个点有限 ,则 gφ(f)为 a.e.-有限 .

By careful pointwise estimates, it is proved that: Let N be a positive integer, φ∈C N (R 1), (0)=0, |φ(x)|+|φ (N) (x)| =O((1+|x|) -N-1-δ ) (for some δ>0), f(x)(1+|x|) -N-1 ∈L 1 (R 1), if g φ(f) is finite at N points, then g φ (f) is a.e. finite. This result generalizes and improves a series of known results.