Limiting case of the Sobolev inequality in BMO, with application to the Euler equations (Q5940514)

The authors prove (Theorem 1) \[ \|f |L_\infty \|\leq c \left[ 1 + \|f |\text{BMO} \|\left( 1 + \log^+ \|f |W^s_p \|\right) \right], \] where \(1 < p < \infty\) and \(s > \frac{n}{p}\). Here \(W^s_p (R^n)\) are the Sobolev spaces in \(\mathbb{R}^n\). They apply this result to Euler equations for imcompressible fluid motions in \(\mathbb{R}^n\) (Theorem 2).