In this paper, we derive estimates for scalar curvature type equations with more singular right hand side. As an application, we prove Donaldson's conjecture on the equivalence between geodesic stability and existence of cscK when $Aut_0(M,J)\neq0$. Moreover, we show that when $Aut_0(M,J)\neq0$, the properness of $K$-energy with respect to a suitably defined distance implies the existence of cscK.

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Given a compact set of real numbers, a random $C^{m + \alpha}$ -diffeomorphism is constructed such that the image of any measure concentrated on the set and satisfying a certain condition involving a real number $s$ , almost surely has Fourier dimension greater than or equal to $s / (m + \alpha)$ . This is used to show that every Borel subset of the real numbers of Hausdorff dimension $s$ is $C^{m + \alpha}$ -equivalent to a set of Fourier dimension greater than or equal to $s / (m + \alpha )$ . In particular every Borel set is diffeomorphic to a Salem set, and the Fourier dimension is not invariant under $C^{m}$ -diffeomorphisms for any $m$ .

In this paper, combining the $p$-capacity for $p\in (1, n)$ with the Orlicz addition of convex domains, we develop the $p$-capacitary Orlicz-Brunn-Minkowski theory. In particular, the Orlicz $L_{\phi}$ mixed $p$-capacity of two convex domains is introduced and its geometric interpretation is obtained by the $p$-capacitary Orlicz-Hadamard variational formula. The $p$-capacitary Orlicz-Brunn-Minkowski and Orlicz-Minkowski inequalities are established, and the equivalence of these two inequalities are discussed as well. The $p$-capacitary Orlicz-Minkowski problem is proposed and solved under some mild conditions on the involving functions and measures. In particular, we provide the solutions for the normalized $p$-capacitary $L_q$ Minkowski problems with $q>1$ for both discrete and general measures.

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