洛伦兹空间上的分布函数的极限性质

用一个新颖的方法证明以下等式：$mathop {lim }limits_{alpha to {0^ + }} {alpha ^P}{d_f}(alpha ) = mathop {lim }limits_{alpha to infty } {alpha ^P}{d_f}(alpha ) = 0$其中 f∈L^p,q(X,μ)，并且有0<p<∞和0<q<∞。也证明函数α^p在某种意义下不能再提升。特别地，当q=∞时，以上等式是不一定成立的。

Limiting property of distribution function in Lorentz space

In this paper, we give a novel proof for the following equality$mathop {lim }limits_{alpha to {0^ + }} {alpha ^P}{d_f}(alpha ) = mathop {lim }limits_{alpha to infty } {alpha ^P}{d_f}(alpha ) = 0$for f∈L^p,q(X,μ) with 0<p<∞, and 0<q<∞.We also prove that the function α^p can not be improved for some sense. When q=∞, the above equality does not hold.