Section: New Results

The Hardy-Hodge decomposition

Participant : Laurent Baratchart.

(This is joint work with T. Qian and P. Dang from the university of Macao.) It was proven in previous year that on a smooth compact hypersurface $\Sigma$ embedded in ${ℝ}^n$, a ${ℝ}^n$-valued vector field of $L^p$ class decomposes as the sum of a harmonic gradient from inside $\Sigma$, a harmonic gradient from outside $\Sigma$, and a tangent divergence-free field. This year we extended this result to Lipschitz surfaces for $2-\epsilon <p<2+{\epsilon }^{\text{'}}$, where $\epsilon$ and ${\epsilon }^{\text{'}}$ depend on the Lipschitz constant of the surface. We also proved that the decomposition is valid for $1<p<\infty$ when $\Sigma$ is $VMO$-smooth (i.e. $\Sigma$ is locally the graph of Lipschitz function with derivatives in $VMO$). By projection onto the tangent space, this gives a Helmholtz-Hodge decomposition for vector fields on a Lipschitz hypersurface, which is apparently new since existing results deal with smooth surfaces. In fact, the Helmholtz-Hodge decomposition is valid on surfaces (not just hypersurfaces), and an article is currently being written on this topic. The Hardy-Hodge decomposition generalizes the classical Plemelj formulas from complex analysis.