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In their seminal work, Chen and Cheng proved a priori estimates for the constant scalar curvature metrics on compact Kähler manifolds. They also prove C3,α{C}^{3,\alpha }-estimate for the potential of the Kähler metrics under boundedness assumption on the scalar curvature and the entropy. The goal of this article is to replace the uniform boundedness of the scalar curvature to the Lp{L}^{p}-boundedness of the scalar curvature.

Purpose: This paper according to the logic of the "digital access divide--digital capability divide--digital outcome divide" aims to systematically discuss the impact of the digital divide on individual happiness in China, accounting for the variations that exist across different groups, as well as the corresponding mechanisms. Design/methodology/approach: This paper presents cross-sectional analyses of the relationship between the digital divide and individual happiness in China. The analyses are based on data from the Chinese General Social Survey 2017, which academic institutions run on the Chinese Mainland. This database contains information on respondents' Internet access, skills and consequences of use, which can measure the digital divide of Chinese individuals at three levels. Findings: First, individual happiness declined when they experienced the digital access divide in China. For the digital capability divide, the lower the usage skills, the more individual happiness declined. When analyzing the digital outcome divide, the greater the negative consequences, the more individual happiness declined. Second, the impacts of digital access, capability and outcome divide vary according to age, gender, education degrees, hukou, region and sub-dimensions. Third, the digital access and capability divide reduce individuals' happiness by lowering their self-rated social and economic status, whereas the digital outcome divide reduce individual happiness by lowering their fairness perception and social trust. Originality/value: The authors believe that this is the first study to examine the impact and its variations among different groups of the three-level digital divide on individual happiness, as well as its mechanisms. [ABSTRACT FROM AUTHOR]