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∈Lp,q(X,μ)，并且有0<p<∞和0<q<∞。也证明函数αp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。
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∈Lp,q(X,μ) with 0<p<∞, and 0<q<∞.We also prove that the function αp can not be improved for some sense. When q=∞, the above equality does not hold.
[1] Grafakos L. Classical and modern Fourier analysis[M]. Upper Saddle River, NJ:Pearson/Prentice Hall, 2004.
[2] Lu S Z, Ding Y, Yan D Y. Singular integral and related topics[M]. Singapore: World Scientific, 2007.
[3] Wiener N. The ergodic theorem[J]. Duke Mathematical Journal, 1939, 5(1):1-18.DOI:10.1215/S0012-7094-39-00501-6.
[4] Wheeden R L, Zygmund A. Measure and integral[M]. 2nd ed. Boca Raton, Florida: CRC Press, 2015.
[5] Xu S Z, Yan D Y. Limiting property of the distribution function of Lp function at endpoints[J]. Journal of Mathematical Research with Applications, 2016, 36(2):177-182.
[6] Janakiraman P. Limiting weak-type behavior for singular integral and maximal operators[J]. Transactions of the American Mathematical Society, 2006, 358(5):1937-1952.DOI:10.1090/s0002-9947-05-04097-3.
