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I know De Philippis and Figalli have a paper studying the regularity of the Kantorovich potential. In Theorem 3.3, the authors show that the potential is $C^{k+2, \beta}$ if the density functions of both the probability density functions are $C^{k,\beta}$, but only when the cost function $c(x, y) = |x-y|^2/2$. In Theorem 3.4, the authors mention the case for a general cost function, but all the smoothness properties are in local sense. My questions are

  1. If $c\in C^{k+2,\alpha}$ is a general cost function, does the results Theorem 3.3 still holds?

  2. Can the parameter $\beta$ be $1$? I ask this question since this paper claims this result for a quadratic loss in the proof of Corollary 7.

I'm interested in these questions due to the recent dual formulation of Gromov--Wasserstein distance, see Theorem 3.1 in this paper and Theorem 4.2.5 in this paper.

Thank you.

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