In this paper, some new upper bounds for Kullback-Leibler divergence(KL-divergence) based on $L^1, L^2$ and $L^\infty$ norms of density functions are discussed. Our findings unveil that the convergence in KL-divergence sense sandwiches between the convergence of density functions in terms of $L^1$ and $L^2$ norms. Furthermore, we endeavor to apply our newly derived upper bounds to the analysis of the rate theorem of the entropic conditional central limit theorem.

翻译：本文讨论了基于密度函数$L^1$、$L^2$和$L^\infty$范数的Kullback-Leibler散度（KL散度）的一些新上界。我们的研究揭示了KL散度意义下的收敛性被密度函数在$L^1$和$L^2$范数下的收敛性所夹逼。此外，我们尝试将新推导的上界应用于熵条件中心极限定理的速率定理分析中。