I see some discussion that the reasoning for the construction of a tax rate function is the utility function. Whould $u=\ln\ln w$ function where $u$ is the utility and $w$ the wealth give a regressive tax rate?
I don't see how the log-log utility function you specify would justify a regressive tax rate. In a simple setup, where government engaged in lump sum spending that had to be financed by a consumption or labor income tax, the log-log utility function would have all the same properties as more typical linear, quadratic, CRRA, and CARA utility functions. quasi-concave, increasing, weakly declining marginal utility, etc.
Usually the motivation for a regressive tax rate isn't the utility function over wealth. It is some other feature of the model, like taxes on wealth discouraging capital formation and therefore harming the productivity of poor households by making them less productive (Judd (1985)). Or some feature of the production function is tied to the income of the rich or high productivity specifically, consider this new paper Jones (2018):
This paper considers the taxation of top incomes when the following conditions apply: (i) new ideas drive economic growth, (ii) the reward for creating a successful innovation is a top income, and (iii) innovation cannot be perfectly targeted by a separate research subsidy — think about the business methods of Walmart, the creation of Uber, or the “idea” of Amazon.com. These conditions lead to a new term in the Saez (2001) formula for the optimal top tax rate: by slowing the creation of the new ideas that drive aggregate GDP, top income taxation reduces everyone’s income, not just the income at the top. When the creation of ideas is the ultimate source of economic growth, this force sharply constrains both revenue-maximizing and welfare-maximizing top tax rates. For example, for extreme parameter values, maximizing the welfare of the middle class requires a negative top tax rate: the higher income that results from the subsidy to innovation more than makes up for the lost redistribution. More generally, the calibrated model suggests that incorporating ideas and economic growth cuts the optimal top marginal tax rate substantially relative to the basic Saez calculation