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Radon measures belonging to the negative Sobolev space $H^{-1}(\mathbb{R}^2)$are important from the point of view of fluid mechanics as they model vorticityof vortex-sheet solutions of incompressible Euler equations. In this note wediscuss regularity conditions sufficient for nonnegative Radon measuressupported on a line to be in $H^{-1}(\mathbb{R}^2)$. Applying the obtainedresults, we derive consequences for measures on $\mathbb{R}^2$ with arbitrarysupport and prove elementarily, among other things, that measures belonging to$H^{-1}(\mathbb{R}^2)$ may be supported on a set of Hausdorff dimension $0$. Wecomment on possible numerical applications.

Nonnegative measures belonging to $H^{-1}(\mathbb{R}^2)$