Is there a reasonable/canonical way to mollify a Borel probability measure without changing its marginals. Let $\pi \in \mathcal{P}(\mathbb{R}^2)$ with marginals $\mu,\nu$. I want to smooth out $\pi$ up to scale $\varepsilon$, which is equivalent to cutting it off at frequency $|k|\approx \varepsilon^{-1}$ in Fourier space, but I do not want to change is marginals. The resulting measure should have a smooth density. It goes without saying that the usual procedure (convolution with a smooth, positive, bump function) does not work.

Is there a reasonable/canonical way to mollify a Borel probability measure without changing its marginals. Let $\pi \in \mathcal{P}(\mathbb{R}^2)$ with marginals $\mu,\nu$. I want to smooth out $\pi$ up to scale $\varepsilon$, which is equivalent to cutting it off at frequency $|k|\approx \varepsilon^{-1}$ in Fourier space, but I do not want to change is marginals. The resulting measure should have a smooth density. It goes without saying that the usual procedure (convolution with a smooth, positive, bump function) does not work. Of course, you can assume $\mu,\nu$ are absolutely continuous with smooth densities.